The more I learn, the more I need to learn

5 downloads 0 Views 4MB Size Report
Jun 6, 2005 - could quickly set up a microcanonical pre-equilibrium with its rotamer I nt2. This near-equilibrium system either can lose a H−atom forming final ...
The m or e I learn, t he m ore I need t o learn.

© 2006 Facult eit Wet enschappen, Geel Huis, Kast eelpark Ar enberg 11, 3001 Hever lee ( Leuv en) Alle recht en voorbehouden. Niet s uit deze uit gave m ag worden v erm enigv uldigd en/ of openbaar gem aakt worden door m iddel v an dr uk, fot ok opie, m icr ofilm , elekt ronisch of op w elk e andere w ij ze ook zonder v oorafgaandelij ke schr ift elij k e t oest em m ing van de uit gev er. All r ight s r eserv ed. No part of t he publicat ion m ay be repr oduced in any form by print , phot opr int , m icr ofilm , elect r onic or any ot her m eans w it hout w r it t en perm ission fr om t he publisher. I SBN 978−90−8649−062−2 Wet t elij k depot D/ 2006/ 10.705/ 59

Katholieke

Faculty of Science

Universiteit

Department of Chemistry

Leuven

Division of Quantum Chemistry and Physical Chemistry

Quantum Chemical and Theoretical Kinetics Study of Reactions of O(3P) Oxygen Atoms with Unsaturated C2-C6 Hydrocarbons

Ph. D. Thesis

Thanh Lam NGUYEN Supervisors:

Prof. Dr. Jozef PEETERS Dr. Luc VEREECKEN

December 2006

Mem bers of exam inat ion com m it t ee: 1) Prof. Dr. Reinhard Zellner , Universit y of Duisburg- Essen, Germ any. 2) Prof. Dr. Luc Vanquickenbor ne ( Chairm an) , Universit y of Leuven, Belgium . 3) Prof. Dr. Jozef Peet ers ( Superv isor) , Univ ersit y of Leuv en, Belgium . 4) Prof. Dr. Minh Tho Nguy en, Univ ersit y of Leuv en, Belgium . 5) Prof. Dr. Shaun A. Car l, Univ ersit y of Leuv en, Belgium . 6) Dr. Luc Ver eecken ( Super visor ) , Univ ersit y of Leuv en, Belgium .

List of Pu blica t ion s ( du r in g m y Ph D w or k )

1) Tropopause chem ist r y rev isit ed: HO2 - init iat ed ox idat ion as an efficient acet one sink . I . Herm ans, Tha nh La m N guye n, P.A. Jacobs, and J. Peet ers J. Am . Chem . Soc. 126 ( 2 0 0 4 ) 9908. 2) Com put at ional st udy of t he st abilit y of α- hydr operoxy l or α- alky lperoxy l subst it ut ed alky l radicals. L. Ver eeck en, Th a n h La m N gu ye n, I . Herm ans, and J. Peet ers Chem . Phys. Let t . 393 ( 2 0 0 4 ) 432. 3) Theor et ical st udy of t he k inet ics of hydr ogen abst ract ion in react ions of sim ple hydrogen com pounds w it h t riplet difluorocarbene. X. J. Hou, Th a n h La m N guy e n , S. A. Car l, J. Peet ers, and M. T. Nguy en. Chem . Phys. Let t . 402 ( 2 0 0 5 ) 460. 4) Kinet ics of alpha- hydrox y - alky lper oxy l radicals in ox idat ion processes. HO2 - init iat ed ox idat ion of ket ones/ aldehy des near t he t r opopause I . Herm ans, J- F. Müller, Th a nh La m N guy e n, J. A. Jacobs, and J. Peet ers J. Phys. Chem . A 109 ( 2 0 0 5 ) 4303. 5) Theor et ical and experim ent al st udy of t he pr oduct branching in t he react ion of acet ic acid w it h OH radicals. F. DeSm edt , V. B. Xuan, Tha n h La m N gu ye n , J. Peet ers, and L. Vereecken J. Phys. Chem . A 109 ( 2 0 0 5 ) 2401. 6) Aut ox idat ion of cyclohexane: Convent ional views challenged by t heory and exper im ent I . Herm ans, Tha nh La m N guye n, P. A. Jacobs, and J. Peet ers Chem . Phys. Chem . 6 ( 2 0 0 5 ) 637. 7) Pot ent ial energy surfaces, product dist r ibut ions and t herm al rat e coefficient s of t he r eact ion of O( 3 P) wit h C2 H 4 : A com prehensive t heor et ical st udy . Th a n h La m N gu ye n, L. Vereecken, X. J. Hou, M. T. Nguy en, and J. Peet ers J. Phys. Chem . A 109 ( 2 0 0 5 ) 7489.

8) Exper im ent al and t heor et ical st udies of t he C2 F4 + O react ion: Nonadiabat ic react ion m echanism Th a n h La m N gu ye n, B. Dils, S. A. Carl, L. Ver eeck en and J. Peet ers J. Phys. Chem . A 109 ( 2 0 0 5 ) 9786. 9) Theor et ical st udy of t he r eact ion of k et eny l and nit rogen diox ide radicals ( HCCO + NO2 ) M. T. Hien, Th a n h Lam N gu ye n , S. A. Car l, and M. T. Nguy en Chem . Phys. Let t . 416 ( 2 0 0 5 ) 199. 10) Quant um chem ical and t heor et ical k inet ics st udy of t he O( 3 P) +

C2 H 2

react ion: A m ult i- st at e process Th a n h La m N gu ye n, L. Vereecken, and J. Peet ers J. Phys. Chem . A 110 ( 2 0 0 6 ) 6696. 11) A Quant um Chem ical and St at ist ical Rat e St udy of t he React ion of O( 3 P) w it h Allene: O- addit ion and H- abst ract ion Channels Th a n h La m N gu ye n, J. Peet ers, and L. Ver eecken J. Phys. Chem . A 110 ( 2 0 0 6 ) 12166. 12) A Theoret ical Re- inv est igat ion of t he O( 3 P) + Chem ical and St at ist ical Rat e Calculat ions Th a n h La m N gu ye n, J. Peet ers, and L. Ver eecken J. Phys. Chem . A ( 2 0 0 6 ) , subm it t ed.

C6 H 6 React ion: Quant um

Ack n ow le dge m e n t s I grat efully acknowledge m y super visors, Pr of. Jozef Peet ers and Dr. Luc Vereeck en, for t he opport unit y t hey gav e m e t o accom plish m y st udy in t his laborat ory . I will nev er forget t heir k ind help, warm est support , careful guidance, det ailed explanat ions and br illiant ideas. Wit hout t hose, none of t his wor k w ould have been possible. I express m y t hanks t o Pr of. Minh Tho Nguy en, who guided m e t o t he im aginat ive, wonderful wor ld of m ot ions and int eract ions of elect rons and nuclei in a m olecule. Also, I am very grat eful t o Pr of. Nguyen for br inging m e t o Prof. Peet ers’s and Pr of. Mebel’s at t ent ion. I wish t o t hank Professors S. H. Lin, Alexander M. Mebel and M. Hay ashi for t heir support , help and guidance during m y st ay at I AMS, Taipei from 10/ 1999 t o 04/ 2002. Many t hanks go t o Dr. I ve Herm ans, Dr. Bart Dils, Dr. Frank DeSm edt and Prof. Shaun A. Car l for efficient , enj oyable collaborat ion and useful discussions in som e art icles. Also I would lik e t o t hank Vung X. Bui, Rehab Moham ed I brahim Elsam ra and St ij n Vranckx for shar ing social life, com ical st or ies and pleasing friendship. Special t hanks go t o Mrs. Rit a Jungblut h, Mr. Chr ist ophe Coeck, Mr. Paul Wij nant s and Dr. Hans Vansweev elt for all t heir kind help, quick response and assist ance in m any aspect s. Financial assist ance of t he BOF ( K. U. Leuv en) is grat efully ack now ledged. I also w ish t o expr ess m y special appreciat ion t o t he com m unit y of Viet nam ese st udent s ( ViNaKul) , scholars and fellows in Leuven w it h w hom I enj oy ed m any exper iences ot her t han quant um chem ist ry and chem ical k inet ics. Finally, I am grat eful t o m y w ife ( Tuy en) and t wo lovely daught ers ( Tr an and Thu) for sharing love, hopefulness, pat ience and difficult ies dur ing t hese y ears. I would lik e t o dedicat e t his work t o t he m em ory of m y parent s, who const ant ly expect ed and hoped t o see it w hen t hey w ere alive.

Ta ble of Con t e nt s Su m m ar y……………………………………………………………………………………………………………. i Sa m e nv a t t ing……………………………………………………………………………………………………. v Ch a pt e r I : I nt r odu ct ion ………………………………………………………………………………….. 1 I .1. Mot ivat ion……………………………………………………………………………………………………… 1 I .2. Sources of sm all unsat urat ed hydr ocarbons……………………………………………….. 5 I .3. Sources of t r iplet gr ound st at e ox ygen at om s……………………………………………. 8 Ch a pt e r I I : Th e or e t ica l M e t h odologie s…………………………………………………………13 I I .1. Quant um Chem ical Calculat ions…………………………………………………………………. 13 I I .1.1. CCSD( T) / CBS calculat ions……………………………………………………………. 15 I I .1.2. Theor et ical Model Chem ist ry ( G2M, G3, and CBS) …………………….. 17 I I .2. St at ist ical Rat e Calculat ions……………………………………………………………………….. 19 I I .2.1. Transit ion st at e t heory ( TST) ………………………………………………………. 19 I I .2.2. Rice- Ram sperger - Kassel- Marcus t heory ( RRKM) ……………………….. 21 I I .2.3. One- dim ensional hindered int er nal rot at ion……………………………….. 22 I I .2.4. Mast er equat ion calculat ions……………………………………………………….. 30 Ch a pt e r I I I :………………………………………………………………………………………………………. 41 Qu a nt u m Ch e m ica l a nd Th e or et ical Kin e t ics St ud y of t h e O( 3 P) + C2 H 2 Re a ct ion : a M ult i- St a t e Proce ss I I I .1. I nt roduct ion…………………………………………………………………………………… 41 I I I .2. Met hodology ………………………………………………………………………………….. 43 I I I .3. Result s and Discussion………………………………………………………………….. 47 I I I .4. Conclusions…………………………………………………………………………………….. 65 Ch a pt e r I V :………………………………………………………………………………………………………… 69 Pot en t ia l

En e rgy

Su r fa ces,

Produ ct

D ist ribu t ions a nd

Th e r m al

Ra t e

Coe fficie n t s of t h e Re a ct ion of O( 3 P) w it h C2 H 4 ( X 1 A g ) : A Com pre h e n sive Th e or e t ica l St ud y I V.1. I nt roduct ion……………………………………………………………………………………. 69 I V.2. Theoret ical appr oaches………………………………………………………………….. 72 I V.3. Result s and Discussion……………………………………………………………………. 77 I V.4. Conclusions……………………………………………………………………………………… 91

Ch a pt e r V :…………………………………………………………………………………………………………. 95 Ex pe rim e n t a l a n d Th e ore t ica l St udie s of t he C2 F 4 + O Re a ct ion: N on Adia ba t ic Re act ion M e ch a nism V.1. I nt roduct ion……………………………………………………………………………………… 95 V.2. Exper im ent al Result s……………………………………………………………………….. 96 V.3. Theoret ical m et hods…………………………………………………………………………. 96 V.4. Theoret ical Result s and Discussions……………………………………………….. 100 V.5. Conclusions………………………………………………………………………………………. 111 Ch a pt e r V I :………………………………………………………………………………………………………… 115 A Qu a n t u m Ch em ical an d St at ist ica l Ra t e St u dy of t h e Re act ion of O( 3 P) w it h Alle ne : O- a ddit ion a nd H - a bst r a ct ion Ch a nn e ls VI .1. I nt roduct ion……………………………………………………………………………………. 115 VI .2. Met hodology …………………………………………………………………………………… 117 VI .3. Result s and Discussion………………………………………………………………….. 120 VI .4. Concluding rem ar ks……………………………………………………………………….. 141 Ch a pt e r V I I :……………………………………………………………………………………………………… 145 A Th e or e t ica l Re - inv e st igat ion of t h e O( 3 P) + C6 H 6 Re act ion : Qua n t u m Ch e m ica l a n d St a t ist ica l ra t e Ca lcu la t ions VI I .1. I nt roduct ion…………………………………………………………………………………… 145 VI I .2. Met hodology………………………………………………………………………………….. 150 VI I .3. Result s and Discussion………………………………………………………………….. 156 VI I .4. Concluding r em ar ks………………………………………………………………………. 177 Ge n e r a l Con clu sions…………………………………………………………………………………………. 181

Sum m a r y

The pot ent ial energy sur faces of t he lowest - lying t riplet and singlet elect r onic st at es for t he r eact ions of t r iplet oxygen at om s wit h sev eral sm all unsat urat ed hydrocarbons, w hich play an im port ant r ole in flam es and in hydrocarbon com bust ion in general, were t heoret ically char act erized using v ar ious high- level, st at e- of- t he- art

quant um

chem ist r y

m et hods

including

DFT- B3LYP,

QCI SD,

CCSD( T) , CBS- QB3, G2M, G3, CASSCF, CASPT2 and MRCI in com binat ion w it h different basis set s.

For each of t hese m ult i- w ell m ult i- channel react ions, t he

prim ary product dist r ibut ion on t he indiv idual surfaces – considered as being adiabat ic – was separ at ely det erm ined by RRKM st at ist ical rat e t heory and weak collision m ast er equat ion analysis using t he ex act st ochast ic sim ulat ion m et hod. I n addit ion, ov erall t her m al rat e coefficient s were det er m ined using conv ent ional t ransit ion st at e t heor y. A num ber of im port ant result s com e out fr om t his st udy and can be sum m ar ized as follows: For t h e O + C2 H 2 r e a ct ion : An efficient r eact ion pat hway on t he elect ronically excit ed

3

A′ surface r esult ing in H( 2 S) + HCCO( A2 A′) was new ly ident ified and is

predict ed t o play an im port ant r ole at higher t em perat ures. Allow ing for nonst at ist ical behav ior of t he int er nal rot at ion m ode of t he init ial com put ed

pr im ary - pr oduct

dist ribut ions

exper im ent al r esult s, i.e. ca. 80% 3

CH 2 ( X B1 ) + CO( X

1

+

2

agree

H( S) +

w ell 2

3

A″ adduct s, our

wit h

HCCO( X A″ +

t he

av ailable

2

A A′) and 20%

) independent of t em per at ur e and pressure ov er t he w ide

300- 2000 K and 0- 10 at m ranges. The t herm al rat e coefficient k ( O + C2 H 2 ) at 200- 2000 K was com put ed using m ult i- st at e t r ansit ion st at e t heor y: k ( T) = 6.14 × 10 −15 × T

1. 28

× ex p( −1244 K/ T) cm 3 m olecule −1 s −1 ; t his expression, obt ained

aft er r educing t he CBS−QCI / APNO ab init io ent rance bar riers by 0.5 k cal/ m ol, quasi- perfect ly m at ches t he exper im ent al k ( T) dat a ov er t he ent ir e 200−2000 K range, spanning t hree orders of m agnit ude. For t h e O + C2 H 4 re a ct ion: The t herm al rat e coefficient s k( O + C2 H 4 ) in t he T = 200- 2000 K range w er e com put ed using m ult i- st at e t ransit ion st at e t heor y and fit t ed by a m odified- Arr henius expr ession as k ( T) = 1.7 × 10 −16 × T

1.66

× exp( −330

K/ T) cm 3 m olecule −1 s −1 , in good agreem ent w it h t he available ex per im ent al dat a. Wit h regard t o t he product dist ribut ion, we could show t hat int ersyst em crossing of t he “ hot ” CH 2 CH 2 O t riplet adduct t o t he singlet surface m ust occur at a fast

rat e of § × 10 11 s −1 in order t o account for about half of t he observ ed product s

i

at room t em perat ur e. Our t hus com put ed product dist r ibut ions as a funct ion of t em perat ur e agree w ell wit h all available exper im ent al r esult s. Pr oduct yields ar e com put ed t o show a m onot onous dependence on t em perat ure. The m aj or product s ( wit h pr edict ed yields at T = 300 K / 2000 K) ar e: CH 3 + CHO ( 48 / 37% ) , H + CH 2 CHO ( 40 / 19% ) , and CH 2 ( X3 B1 ) + H 2 CO ( 5 / 29% ) , w her eas H + CH 3 CO, H 2 + H 2 CCO and CH 4 + CO ar e all m inor ( ≤ 5% ) . For t h e O + C2 F 4 r e a ct ion : Earlier, t he C2 F4 + O react ion was inv est igat ed exper im ent ally using m olecular beam –t hr eshold ionizat ion m ass spect rom et ry ( MB- TI MS) in our laborat ory .

The m aj or pr im ary pr oduct s wer e observ ed t o be

CF2 O ( + CF2 ) and CF3 ( + CFO) , wit h m easured appr ox im at e y ields of +11

versus 16−7 % , respect ively , neglect ing m inor product s.

84 +−711 %

Our pr esent t heoret ical

calculat ions show t hat t he observed product ion of CF3 ( + CFO) can only occur on t he singlet surface, in parallel w it h form at ion of ca. 5 t im es m ore CF2 O( X) + CF2 ( X1 A1 ) .

This r equir es fast int ersyst em crossing ( I SC) from t he t r iplet t o t he

singlet surface at a rat e of ca. 4 × 10 12 s –1 . The t heoret ical calculat ions com bined wit h t he exper im ent al result s t hus indicat e t hat t he y ield of t r iplet CF2 ( ã 3 B1 ) +

CF2 O form ed on t he t riplet surface pr ior t o I SC is ” ZKHUHDV VLQJOHW

CF2 ( X1 A1 ) + CF2 O is produced w it h yield • DIWHU ,6&  ,Q DGGLWLRQ WKH t herm al rat e coefficient s k ( O + C2 F4 ) in t he T = 150- 1500 K range wer e com put ed using

m ult i- st at e

t ransit ion

st at e

t heory

and

can

be

expressed

as

k (T ) = 1.67 × 10−16 × T 1.48 cm 3 m olecule –1 s–1 ; t hey ar e in agreem ent wit h t he available exper im ent al r esult s in t he T = 298- 500 K range. For t h e O + CH 2 = C= CH 2 re a ct ion : The elect rophilic O- addit ion pat hways on t he

cent ral

and

t erm inal

carbon

at om

are

dom inant

up

to

com bust ion

t em perat ur es. Maj or pr edict ed end- product s for t he addit ion rout es include CO + C2 H 4 ,

3

CH 2 + H 2 CCO, and CH 2 = C• −CHO + H • , in agr eem ent wit h exper im ent al

ev idence. CO + C2 H 4 ar e m ainly generat ed fr om t he low est - ly ing singlet surface aft er an int er - syst em crossing pr ocess from t he init ial t riplet surface. Efficient Habst ract ion pat hways are new ly ident ified, and occur on t w o differ ent elect r onic st at e surfaces,

3

A″ and

3

A′, r esult ing in OH + propargy l radicals; t hey ar e

predict ed t o play an im port ant r ole at higher t em perat ur es in hy drocarbon com bust ion chem ist ry and flam es, wit h est im at ed cont ribut ions of ca. 35% at 2000 K. The ov erall t herm al rat e coefficient k( O + C3 H 4 ) at 200- 1000 K was com put ed using m ult i- st at e t ransit ion st at e t heory : k ( T) = 1.6 × 10 −17 × T

ii

2.05

×

exp( −90 K/ T) cm 3 m olecule −1 s −1 , in good agr eem ent w it h t he exper im ent al dat a available for t he 300- 600 K range. For t h e O + C6 H 6 re a ct ion : Our calculat ions show t hat t he elect rophilic Oaddit ion m echanism ont o a carbon at om in benzene is dom inant up t o com bust ion t em perat ur es ( 2000 K) , w it h t he ent rance rout e m ainly on t he lowest - ly ing

3

A′

t riplet surface, in cont radict ion t o earlier w or k. The predict ed m aj or end- pr oduct s of t he addit ion, ov er a wide range of condit ions, are phenoxy radical + H• and phenol and/ or benzene ox ide/ ox epin, in agreem ent •

w it h t he ex per im ent al



ev idence. While c- C6 H 5 O + H are near ly exclusiv ely form ed on t he lowest - ly ing 3

A′ t r iplet surface, phenol and/ or benzene ox ide/ ox epin ar e m ainly generat ed

from t he low est - ly ing singlet sur face aft er an int er - syst em crossing pr ocess from t he init ial t r iplet surface. CO + c- C5 H 6 can be form ed in flam es, w it h m oderat e yields at pr essures around at m ospher ic or low er. The O + C6 H 6

c- C5 H 5 • + • CHO

channel is predict ed t o be unim port ant under all relev ant com bust ion condit ions, in cont rast w it h prev ious t heor et ical conclusions ( Hodgson et al. J. Phy s. Chem . A 2 0 0 1 , 105, 4316) . Efficient H- abst ract ion pat hways are new ly ident ified, and occur on t wo differ ent elect ronic st at e surfaces, 3 B1 and 3 B2 , result ing in hydrox y l plus pheny l radicals; t hey ar e pr edict ed t o play an im port ant role at t he higher t em perat ur es in hydr ocarbon com bust ion, w it h est im at ed cont r ibut ions of ca. 50% at 2000 K. The overall t herm al rat e coefficient k( O + C6 H 6 ) at 300- 800 K was com put ed using m ult i- st at e t ransit ion st at e t heor y : k ( T) = 3.7 × 10 −16 × T

1.66

× ex p( −1830 K/ T) cm 3 m olecule −1 s−1 , in fair agreem ent w it h t he exper im ent al dat a available.

I n general, t he pr esent t heoret ical st udy of t he m echanism s and kinet ics of react ions of O( 3 P) at om s wit h unsat urat ed hydrocarbons ( HC) has led t o t he following new or deepened insight s: ( i)

For alm ost all unsat urat ed HC, t he O- addit ion can occur on t wo differ ent t riplet surfaces; for t he alk yne C2 H 2 t he react ion on t he excit ed t r iplet surface cont r ibut es in an im port ant way t o t he ov er all rat e at flam e t em perat ur es.

( ii)

For alk enes and ar om at ics, specifically C2 H 4 , C2 F4 , and benzene, as well as for allene, t he init ial chem ically act iv at ed t r iplet O- adduct s undergo fast int er syst em crossing ( I SC) , result ing in low - energy singlet isom ers. For t he nascent t r iplet adduct s CH 2 CH 2 O and CF2 CF2 O, t he I SC rat es could be est im at ed at 1.5 × 10 11 and

4 × 10 12 s–1 , r espect iv ely. How ev er, for t he iii

sm all alk yne C2 H 2 such I SC is negligibly slow com pared t o t he v ery fast isom er izat ion/ fragm ent at ion of t he t r iplet HCCHO adduct . ( iii)

For allene and benzene, H- abst ract ion by O( 3 P) is a m aj or r eact ion pat h at higher t em perat ur es; for benzene it ev en account s for > 50% of t he product form at ion at t em perat ur es abov e 2000 K.

( iv )

Theor et ical energy bar r iers com put ed at t he CBS- QB3 and ev en m ore so t he CBS- APNO m odel chem ist ry lev els appear t o closely appr ox im at e t he

real values, wit hin §0. 5 kcal/ m ol. This is wit nessed by t he fact t hat t he first - pr inciples rat e coefficient s com put ed on t his basis using m ult i- st at e TST t heor y, w it h inclusion of all react ant st at es and all cont r ibut ing react ion t ransit ion st at es, are in good t o quasi- perfect agr eem ent w it h t he ex per im ent al dat a, ov er wide t em perat ur e ranges, and t his for a ser ies of react ions. (v)

I n t he line of t he abov e, it can be st at ed t hat t heor et ical k inet ics has

grow n int o a m at ur e science, w hich í when applied cor rect ly and

com prehensiv ely í RIIHUV LPSRUWDQW DGGHG YDOXH LQ FRPSOHPHQW RI ex perim ent al k inet ics st udies.

iv

Sa m e n va t t ing

Deze

doct oraat st hesis

behelst

een

t heor et ische,

kwant um chem ische

en

kwant um st at ist isch- k inet ische st udie van de react ies v an gr ondt oest and t r iplet zuurst ofat om en m et een aant al onv erzadigde C2 - C6 k oolwat erst ofm oleculen, en dit ov er br ede t em per at uurs- en dr ukgebieden. De onderzocht e react ies verv ullen een belangr ij ke rol in v lam m en en in koolwat erst ofv erbrandingsprocessen in het algem een. De pot ent iële- energie opperv lakk en ( PES) van de laagst gelegen t riplet en singlet elect r onische

t oest anden

voor

de

bedoelde

r eact ies

werden

t heoret isch

gekarakt er izeerd m et behulp van verscheidene hoogwaardige, st at e- of- t he- art kwant um chem ische m et hodes, zoals onderm eer DFT- B3LYP, QCI SD, CCSD( T) , CBS- QB3, G2M, G3, CASSCF, CASPT2 en MRCI , in com binat ie m et div erse basisset s. Voor elk van deze m eerk anaals- r eact ies werd dan uit gaande van de bekom en m eer - m inim a PES en de bij behor ende r o- v ibrat ionele param et ers van st at ionaire punt en en overgangst oest anden, de product dist r ibut ie v oor speld op de

afzonderlij k e í als adiabat isch beschouwde í opperv lak ken, dit m et behulp van

de kwant um st at ist ische RRKM t heor ie in com binat ie m et m ast er equat ion analyse in de zwakk e- bot singsbenadering, gebr uik m ak end van de exact e st ochast ische sim ulat iem et hode

( ESM) .

snelheidscoëfficiënt en

Bovendien

bepaald

werden

m et

st eeds

behulp

de

van

t ot ale de

t herm ische

m ult i- t oest and

ov ergangst oest andst heor ie ( MS- TST) . De

belangr ij kst e

r esult at en

volgend

uit

deze

st udie

worden

hier onder

sam engevat . D e C2 H 2 + O r e a ct ie Een

nieuw

en

aangeslagen

3

efficient

addit iereact iepad,

verlopend

op

het

2

elect ronisch

2

A′ opper vlak en result erend in H( S) + HCCO( A A′) , werd hier

geïdent ificeerd. Aanget oond w erd dat dit pad een voor nam e r ol v er v ult bij hoger e t em perat ur en; wij v oor spellen b.v. dat het bij 2000 K v oor 25% zou bij dragen t ot de t ot ale react iesnelheid. Voor

wat

de

react ie

op

het

grondt oest ands- t r iplet

opper vlak

bet reft ,

argum ent eren w ij een niet - st at ist isch gedrag van de int er ne r ot at iem odus v an de init iële

3

A″ adduct en, m et voor nam e im plicat ies voor de com pet it ie t ussen de

t wee uit gangskanalen. Onze op deze basis afgeleide pr im aire- pr oduct dist ribut ies st em m en goed over een m et de beschik bar e exper im ent ele r esult at en, nam elij k ca.

80%

H( 2 S)

+

HCCO( X2 A″ +

A2 A′)

en

20%

CH 2 ( X3 B1 )

+

CO( X1

+

),

v

onafhankelij k van de t em perat uur en de druk over de br ede 300- 2000 K and 0- 10 at m gebieden. De t her m ische snelheidscoëfficiënt k ( O + C2 H 2 ) v oor 200 t ot 2000 K w erd ber ek end op basis van de m eert oest and TST t heor ie: k ( T) = 6.14 × 10 −15 × T

1.28

× exp( −1244 K/ T) cm 3 m olecule −1 s −1 . Deze uit dr uk king, bekom en na

reduct ie v an beide CBS−QCI / APNO ab init io ingangsbarrier es m et 0.5 kcal/ m ol, reproduceert de exper im ent ele k ( T) gegevens vrij w el perfect ov er het hele 200−2000 K gebied, waarin k( T) m et een fact or 1000 t oeneem t . D e C2 H 4 + O r e a ct ie Met bet r ek k ing t ot de product dist ribut ie van deze react ie, w erd bev onden dat int ersyst eem k r uising van het init iële chem isch geact iv eerde CH 2 CH 2 O t r iplet

adduct naar het singlet opperv lak , aan een hoge snelheid van § × 10 11 s−1 , de enige zinvolle v er klar ing biedt v oor circa de helft van de bij k am ert em perat uur waargenom en product en. Onze op die basis t heoret isch afgeleide prim air eproduct dist ribut ies als funct ie van de t em perat uur zij n in goede ov ereenst em m ing m et alle beschikbar e exper im ent ele product m et ingen. Onze aldus voorspelde product opbr engst en ver t onen een uit gespr ok en, m onot one afhank elij k heid v an de t em perat uur. De v oornaam st e product en ( m et voorspelde opbrengst en bij T = 300 K / 2000 K) zij n: CH 3 + CHO ( 48 / 37% ) , H + CH 2 CHO ( 40 / 19% ) , en CH 2 ( X3 B1 ) + H 2 CO ( 5 / 29% ) , t erw ij l H + CH 3 CO, H 2 + H 2 CCO en CH 4 + CO alle onbelangr ij k zij n ( ≤ 5% ) ov er gans dit gebied. De t herm ische snelheidscoëfficiënt k( O + C2 H 4 ) over het t em perat uursgebied van 200 t ot 2000 K w erd berekend m et de m ult i- t oest and overgangst oest andst heor ie en kan weergegev en w orden door een gem odificeerde Arrhenius uit dr ukk ing, als k( T) = 1.7 × 10 −16 × T

1.66

× exp( −330 K/ T) cm 3 m olecule −1 s−1 , in goede

ov er eenk om st m et de v oorhanden zij nde exper im ent ele k inet ische gegevens.

D e C2 F 4 + O r e act ie De C2 F4 + O react ie w erd eerder in dit laborat or ium exper im ent eel onderzocht , gebr uikm ak end

van

m oleculair e- bundelst aalnam e

/

dr em pel- ionizat ie

m assaspect rom et rie ( MB- TI MS) ( Bart Dils, Doct oraat st hesis, okt ober 2003) . Als voor naam st e pr im aire product en

w erden

CF2 O ( +

CF2 )

en

CF3 +7

(+

CFO) +11

waargenom en, m et gem et en approx im at iev e opbr engst en van 84 −11 % en 16−7 % , respect iev elij k, bij v erw aar lozing van onbelangr ij k e nevenpr oduct en. I n de voor liggende t heor et ische st udie w ordt aanget oond dat de v ast gest elde vor m ing van CF3 ( + CFO) enk el op het singlet opperv lak kan v er lopen, en dit dan

vi

sam en m et de v orm ing van ca. 5 m aal m eer CF2 O( X) + CF2 ( X1 A1 ) .

Dit v ereist

opnieuw int ersyst eem kr uising ( I SC) van het init iële t riplet CF2 CF2 O adduct naar het singlet opper vlak, m aar m et een snelheid van ca. 4 × 10 12 s–1 , d.i. m eer dan een gr oot t e- orde sneller nog dan v oor het hogerst aande CH 2 CH 2 O geval.

Onze

t heor et ische evaluat ies gecom bineerd m et de exper im ent ele result at en gev en aldus aan dat de opbrengst van t riplet CF2 ( ã 3 B1 ) + CF2 O gevorm d op het t r iplet

opperv lak vóór I SC ” bedraagt , t erw ij l singlet CF2 ( X1 A1 ) + CF2 O ont st aan

m et opbrengst • ná I SC. Tenslot t e

w erd

de

t herm ische

snelheidscoëfficiënt

k( O

+

C2 F4 )

in

het

t em perat uursgebied van 150- 1500 K berekend m et behulp v an m ult i- t oest and TST

t heor ie;

k (T ) = 1.67 × 10

de

−16

×T

1.48

result at en

k unnen

uit gedrukt

worden

als

cm 3 m olecule –1 s–1 . Ze zij n in r edelij k e ov ereenk om st m et

de schaarse beschikbar e exper im ent ele m et ingen, in het T = 298- 500 K gebied. D e CH 2 = C= CH 2 + O r e a ct ie Bev onden werd dat de elect rofiele O- addit iepaden op de cent rale en t erm inale koolst ofat om en dom inant zij n t ot zelfs bij verbrandingst em perat ur en. Voorspelde hoofdproduct en uit die addit ierout es zij n CO + •

C2 H 4 ,

3

CH 2 +

H 2 CCO, en



CH 2 = C −CHO + H , in ov er eenst em m ing m et de exper im ent ele ev ident ie. CO + C2 H 4

ont st aan

voornam elij k

op

het

laagst gelegen

singlet

opper vlak,

na

int ersyst eem k r uising van de init iële t r iplet adduct en. Er w erden bov endien efficiënt e H- abst ract ie paden nieuw geïdent ificeerd. Deze paden liggen op t wee v erschillende elect ronische t oest andsopperv lak k en, 3 A″ and 3

A′, en leiden t ot OH + propargy l ( • CH 2 CCH) r adicalen. Voorspeld w ordt dat deze

paden

een

v oor nam e

rol

zullen

v er v ullen

bij

hoger e

t em per at ur en

in

koolwat erst ofv erbrandingsprocessen en v lam m en, m et geëst im eerde bij dragen van ca. 35% bij 2000 K. De t ot ale t herm ische snelheidscoëfficiënt k ( O + C3 H 4 ) bij 200- 1000 K werd afgeleid m et behulp v an m ult i- t oest and ov ergangst oest andst heor ie: k ( T) = 1.6 × 10 −17 × T

2.05

× exp( −90 K/ T) cm 3 m olecule −1 s−1 , in goede over eenk om st m et de

exper im ent ele gegev ens, beschikbaar voor de 300- 600 K gebied. D e c- C6 H 6 + O r e act ie Onze

t heor et ische

st udie

t oont

aan

dat

elect r ofiele

O- addit ie

op

een

koolst ofat oom het dom inant e m echanism e van de benzeen + O r eact ie is t ot ongev eer

2000

laagst gelegen

3

K, m et A′ t riplet

het

belangrij kst e ingangskanaal ver lopend op het

opperv lak ,

dit

laat st e in

t egenst elling

t ot

eerder

vii

t heor et isch werk v an een ander e groep ( Hodgson et al. J. Phys. Chem . A 2 0 0 1 , 105, 4316) . Onze nieuwe t oew ij zingen, die st eunen op syst em at ische en doorgedreven I RC react iepad- analyses, leiden t ot belangrij k e m echanist ische verschillen

t . o.v.

het

eerder

w erk.

De door

ons v oorspelde v oornaam st e

eindproduct en van de O- addit ie, over zeer br ede druk - en t em perat uur sbereik en, zij n fenoxy radicaal ( c- C6 H 5 O• ) + H • en fenol en/ of benzeenox ide/ ox epin, in ov er eenk om st m et de exper im ent ele bev indingen. Terw ij l c- C6 H 5 O• + H • bij na uit sluit end ont st aan op het laagst gelegen 3 A′ t r iplet opperv lak, w orden fenol en/ of benzeenox ide/ ox epin hoofdzakelij k gegenereerd op het laagst gelegen singlet opperv lak na int ersyst eem k ruising ( I SC) vanuit het t riplet ingangsopper vlak . De product en CO + c- C5 H 6 kunnen gevorm d worden in v lam m en, m aar slecht s m et m at ige opbrengst , en enkel bij druk ken lager dan ca. 1 at m . Volgens onze c- C5 H 5 • +

voorspellingen is het O + C6 H 6



CHO kanaal v erwaar loosbaar in alle

relevant e verbrandingsom st andigheden, in t egenst elling t ot de conclusies v an het reeds ver m elde werk van Hodgson et al. Tev ens biedt onze gedet ailleer de analyse van

de

O- addit ierout es

een

v er k lar ing

v oor

de

schij nbaar

t egenst r ij dige

waar nem ingen aangaande de observ at ie v an fenol en CO + c- C5 H 6 als product en in t wee gekr uist e m oleculair e bundel experim ent en; w e t onen aan dat de apparent e discrepant ie t erug t e v oer en is t ot de uit zonder lij k lange 100 µs levensduur van het chem isch geact iv eerde singlet fenol ( of benzeenox ide/ ox epin) − dat ont st aat na het hogerv erm eld I SC pr oces − in de om st andigheden van zulk e exper im ent en. Bov endien werden efficiënt e H- abst ract ie paden nieuw geïdent ificeerd, verlopend op t wee verschillende elect ronische- t oest andsopperv lak ken,

3

B1 and

3

B2 , beide

result er end in hydr ox yl + feny l r adicalen. We voorspellen dat deze H- abst ract ie kanalen

v oor nam e

react iepaden

w orden

bij

de

hogere

t em perat ur en

van

koolwat erst ofv erbrandingsprocessen, m et bij dr agen bij 2000 K geëst im eerd op ca. 50% . De

t ot ale

t herm ische

snelheidscoëfficiënt

k( O

+

C6 H 6 )

voor

het

t em perat uursgebied van 300- 800 K w erd ook hier afgeleid op basis v an de m ult it oest and TST t heor ie. De result at en k unnen uit gedruk t w orden als k ( T) = 3.7 × 10 −16 × T

1.66

× exp( −1830 K/ T) cm 3 m olecule −1 s−1 . Onze r esult at en, uit sluit end

verkr egen uit first principles, reproduceren de beschikbar e exper im ent ele k ( T)

waarden binnen een fact or §2 r ond 300 K, en binnen 10% rond 800 K.

viii

Algem een lev erde het onder hav ig t heoret isch onderzoek aan de m echanism en en de k inet ica van de react ies van O( 3 P) at om en m et onv erzadigde k oolwat erst offen ( KWS) v ooral de volgende nieuw e of verdiept e inzicht en op: ( v i)

Voor

v r ij w el

alle

onverzadigde

KWS

v erloopt

O- addit ie

op

t wee

onderscheidene t r iplet opperv lak ken; v oor het alky ne C2 H 2 draagt de react ie op het aangeslagen t r iplet opperv lak in belangr ij k e m at e bij t ot de t ot ale react iesnelheid bij v lam t em perat ur en. ( v ii)

Voor alk enen en arom at en, m et nam e C2 H 4 , C2 F4 , en benzeen, en t ev ens voor alleen, ondergaan de init iële chem isch geact iv eerde t r iplet adduct en

snelle

int er syst eem kr uisingen

( I SC)

r esult er end

in

O-

laag-

energet ische singlet isom er en. Voor de nascent e t r iplet adduct en CH 2 CH 2 O en CF2 CF2 O kon de I SC snelheid geraam d wor den op respect iev elij k 1.5 ×

1011 en 4 × 10 12 s–1 . Voor het k leine alky ne C2 H2 is zulk e I SC echt er verwaar losbaar

t r aag

t en

opzicht e

van

de

zeer

snelle

isom er isat ie/ fragm ent at ie v an het t r iplet HCCHO adduct . ( v iii)

Voor alleen en benzeen, is H- abst ract ie door O( 3 P)

een v oor naam

react iepad bij hoger e t em perat ur en; bij benzeen st aat het zelfs in voor > 50% v an de pr oduct v orm ing bij t em perat uren vanaf 2000 K ( ix )

Theor et ische energiebarr ier es berek end volgens de sam engest elde CBSQB3 en m eer nog de CBS- APNO m odel m et hodes blij k en de r ealit eit zeer dicht t e benader en, binnen 0.5 kcal/ m ol. Get uige hiervan zij n de op deze basis,

lout er

uit

first

principles

afgeleide

m ult i- t oest and

TST

snelheidsconst ant en, rekening houdend m et alle r eagenst oest anden en bij dragende r eact ie- overgangst oest anden, die voor een ganse r eeks van react ies, in goede t ot zelfs quasi- perfect e ov er eenk om st zij n m et de ex perim ent ele gegev ens, ov er zeer br ede t em perat uursgebieden. (x)

Aansluit end bij het bovenst aande, kan gest eld dat t heoret ische kinet ica uit gegroeid is t ot een v olwaardige r ij pe wet enschap, die, m it s cor rect e en volledige t oepassing, een belangr ij k e m eerwaarde biedt als com plem ent van experim ent ele k inet ische st udies.

ix

Cha pt e r I : I n t r odu ct ion

I .1 . M ot iva t ion Fir e, civ ilizat ion’s first great energy invent ion, was discover ed by m ank ind a long t im e before t hey could read and wr it e. Man learned t o ignit e w ood, a ubiquit ous fuel, t o m ak e fir es for heat ing, cook ing and scaring w ild anim als away. Nowadays, we r ely overw helm ingly on energy t o m ak e our lives com fort able, pr oduct iv e and enj oyable.

People

consum e

energy

daily

for

t ransport at ion

( car s,

t rucks,

airplanes, t rains, boat s, and ot hers) ; for r esident ial uses ( cook ing, heat ing and cooling room s, light ing, TV, r efrigerat ors, personal com put ers) ; for indust rial purposes ( fact ories, burners, boilers, furnaces, ot hers) ; and for t he com m ercial sect or ( superm ark et s, t heat ers, hot els, rest aur ant s and ent ert ainm ent ) in order t o m aint ain our qualit y of life. The sun is t he cent ral energy source of our solar syst em . However, t he difficult ies lie in convert ing solar energy int o ot her energy sources and also st ore t hem for fut ur e use. Phot ovolt aic devices and ot her m eans t o ut ilize solar energy hav e int ensiv ely been st udied and developed, but not y et at t he lev el of our energy dem ands. At pr esent and in t he for eseeable fut ur e, our m aj or energy sources ar e st ill fossil fuels ( oil,

nat ural gas,

and

coal)

as well as nuclear

energy .

Not w it hst anding r ecent concer ns on t he safet y of nuclear power plant s, nuclear wast es as well as t er ror ist t hreat s, som e new generat ion nuclear power plant s will newly be built in t he U.S., 1 China, 2 and ot her count r ies in order t o m eet fut ur e energy dem ands. I n addit ion, higher fossil fuel prices and w or r ies of unst eady oil supplies in recent year s also support renew ed int er est in expanding t he use of nuclear pow er t o gener at e elect r icit y. According t o t he I nt er nat ional Ener gy Out look 2006, 3 World m ark et ed energy consum pt ion will incr ease on average by 2.0% per year fr om 2003 t o 2030, i.e. from 421 quadr illion Br it ish t herm al unit s ( Bt u) in 2003 t o 722 quadr illion Bt u in 2030 as a result of r obust econom ic gr ow t h. I n t his per iod fossil fuels should cont inue t o supply m uch of t he energy used worldwide. Oil st ill rem ains t he dom inant energy source, giv en it s im port ance in t he t ransport at ion and indust r ial end- use sect ors ( see Figur e I .1) . 3

Chapt er I : I nt roduct ion

1

Figure I .1: Fuel shares of Wor ld m ark et ed ener gy use, 2003, 2015, and 2030

Nev ert heless, t he m aj or lim it at ion is t he lim it ed nat ure of our fossil fuel resources. Som e realist ic est im at es 4 indicat ed t hat our ov erall w orldwide fossil fuel resources would com plet ely be used up in 200- 300 y ears, of w hich oil and gas would last less t han a cent ury . This is not a long per iod in t he hum an hist or y; it is t her efore essent ial t o find new solut ions, for exam ple ext ensive use of renewable energy sources ( solar, w ind, geot herm al, biom ass, hy dro, and ocean) and/ or nuclear power. Besides, w e m ust efficient ly save fossil r esources t o m aint ain our qualit y of life. I n addit ion, com bust ion of fossil fuels for energy r eleases a large am ount of carbon diox ide ( CO2 ) , one of t he m ost prevalent greenhouse gases in t he at m osphere t hat is linked w it h global warm ing. According t o t he I nt er nat ional Energy Out look 2006, 3 Wor ld carbon diox ide em issions cont inue t o increase st eadily, i. e. from 25.0 billion m et r ic t ons in 2003 t o 33.7 billion m et r ic t ons in 2015 and 43.7 billion m et r ic t ons in 2030. While nat ural processes such as plant phot osynt hesis by t he “ carbon cycle” and dissolut ion in t he oceans can absorb som e of t he ant hr opogenic carbon diox ide em issions, 5 t he r em aining am ount is added t o t he at m ospher e annually . This posit iv e im balance bet w een em issions and absorpt ions result s in t he cont inuing growt h in greenhouse gases in t he at m osphere. I n com put er - based m odels, 5 r ising concent rat ions of greenhouse gases generally pr oduce an increase in t he average t em perat ur e of t he Eart h. Rising t em perat ur es m ay, in t ur n, lead t o changes in w eat her, sea levels, and land use pat t er ns, com m only r efer red t o as “ clim at e change” . Ver y r ecent

2

Chapt er I : I nt roduct ion

ev idences invest igat ed by Shuk m an 6 show t hat clim at e change are already affect ing t he Am azon region t hat has m or e t han half t he planet ’s r em aining rain for est s. Furt herm or e, t he use of fossil fuels for energy also giv es off pollut ant s – carbon m onox ide ( CO) , nit r ogen ox ides ( NO and NO2 ) , sulfur ox ides ( SO2 and SO3 ) , part iculat e m at t er in various form s ( e. g. soot ) , and unbur ned hydrocar bons – t hat cause pr oblem s for our env ironm ent such as air and wat er pollut ion. Pr im ary pollut ion concer ns r elat e t o specific healt h hazards, sm og, acid r ain, global warm ing, and ozone deplet ion. The m ain aim of curr ent com bust ion research is t o dev elop com bust ion st rat egies t hat ar e env ir onm ent - friendly and perm it higher com bust ion efficiencies. The prerequisit e is det ailed and quant it at iv e k nowledge of t he chem ist ry and t he kinet ics of com bust ion processes. The chem ist ry t hat occurs in com bust ion syst em s is t he r esult of num erous com plicat ed and indiv idual chem ical event s, i. e. dom inant

elem ent ary r eact ion st eps occurr ing consecut iv ely

and/ or in

parallel, involv ing t he breaking of old chem ical bonds and/ or t he form ing of new chem ical bonds. Underst anding t hese indiv idual react ions in det ail is necessary in order t o cont rol and opt im ize t he ov erall com bust ion pr ocess as well as t o design novel com bust ion dev ices t hat sim ult aneously get t he m ost out of t he av ailable energy resources and m inim ize pollut ant s r eleased t o our env ironm ent . I n t his cont ext , k inet ic and m echanist ic st udies of sev eral dom inant elem ent ary react ions inv olv ed am ong ot hers in t he oxygen- at om init iat ed ox idat ion of sev er al sm all unsat ur at ed hydr ocarbons occur r ing in hydrocarbon com bust ion chem ist ry processes and flam es ar e being car ried out in our laborat ory , bot h exper im ent ally and t heor et ically . Sev eral ex per im ent al m et hods are used t o m easur e ov erall t herm al rat e coefficient s of elem ent ary radical r eact ions v ery accurat ely as well as to

det ect

t he

var ious product s and

t heir

yields.

On

t he basis of

such

m easur em ent s, an overall react ion m echanism can be der ived. Howev er , for m any react ions, only part ial inform at ion can be gained by exper im ent s. I n such cases,

com put at ional

chem ist ry

m et hods

becom e

highly

useful

t ools

for

com plem ent ing t he r equir ed k inet ic/ m echanist ic inform at ion. I n pr inciple, all inform at ion on a chem ical r eact ion ( e.g. react ion m echanism , rat e coefficient , react ion cr oss sect ion, product dist r ibut ion and ot hers) m ay be obt ained using accurat e quant um dy nam ic calculat ions by direct ly solv ing t he t im e- dependent Schrödinger equat ion. Yet , such calculat ions are very expensiv e and lim it ed t o very sm all sy st em s since a near - com plet e k nowledge of t he pot ent ial energy funct ion

is

r equir ed.

On

t he

ot her

hand,

when

Chapt er I : I nt roduct ion

t he

Bor n- Oppenheim er

3

approx im at ion is applicable, quant um chem ical calculat ions can be used t o gain inform at ion about m olecular propert ies and also ent halpies of r eact ion, r eact ion barr ier height s, and ot hers. Based on t he lat t er infor m at ion, t he m olecular m echanism of a r eact ion can be elucidat ed. Ther efor e, quant um chem ical calculat ions can be used not only t o support , verify and explain ex per im ent al result s, but also t o pr edict in advance m olecular react ion m echanism and m aj or product s. I n addit ion, when t he considered r eact ion is ergodic and adiabat ic, st at ist ical rat e and m ast er equat ion calculat ions can be car r ied out t o gain rat e coefficient s and pr oduct dist ribut ion as well as lifet im es of chem ically act iv at ed int erm ediat es. The r equir ed input inform at ion ( e. g. pot ent ial ener gy surface, harm onic vibrat ional fr equencies and rot at ional const ant s) for st at ist ical rat e calculat ions is obt ained from quant um chem ist r y calculat ions as out lined abov e. The purpose of t he present ly r eport ed r esearch is t o use high- lev el quant um chem ical m et hods ( ab init io/ densit y funct ional t heory ) in com binat ion w it h st at ist ical k inet ic t heories ( TST and RRKM) t o explain and elucidat e chem ical react ion m echanism s as well as t o predict t her m al k ( T) , energy - specific k( E) rat e coefficient s and product dist ribut ions for react ions of t r iplet gr ound st at e O at om s wit h sev eral unsat urat ed hydrocarbons ( C2 H 2 , C2 H 4 ( C2 F4 ) , C3 H 4 , and C6 H 6 ) , w hich play key r oles in hydrocarbon com bust ion chem ist ry processes and flam es. Wher e possible, our pr edict ions ar e also used t o com pare w it h ex perim ent al dat a available. A m aj or t ool in t his work is com put at ional chem ist ry. Generally, t his fair ly new field of chem ist ry is focused on obt aining r esult s relevant t o chem ical problem s, not dir ect ly on developing new t heoret ical m odels. 7,8 There is of course a st rong int erplay bet w een t radit ional t heor et ical chem ist ry and com put at ional chem ist ry . Dev eloping new t heor et ical m odels m ay enable new problem s t o be st udied, and result s from calculat ions m ay r ev eal lim it at ions and suggest im provem ent s in t he underly ing t heory . Com put at ional chem ist ry is t he subfield where m at hem at ical m et hods are com bined wit h fundam ent al laws of physics t o st udy processes of chem ical

relevance

using

com put ers.

Com put at ional

chem ist r y

applied

for

m olecular science has r ecent ly gained great benefit s from st eady and rapid dev elopm ent of com put er sciences for bot h sect ors: soft war e and hardw are. Professors Walt er Kohn and John A. Pople, who w on and equally shar ed t he Nobel Pr ize in Chem ist ry in 1998, 9 ar e t wo pioneer ing scient ist s in st udy ing and dev eloping densit y - funct ional t heory and m olecular orbit al ab init io m et hods in quant um chem ist ry , r espect iv ely, t hat can be used for t heoret ical st udies of t he

4

Chapt er I : I nt roduct ion

propert ies of m olecules and of chem ical processes. Depending on t he accuracy want ed, and t he nat ur e of t he syst em at hand, one can t oday obt ain useful inform at ion for syst em s cont aining fr om a few up t o sev eral t housand at om s. One of t he m ain problem s in com put at ional chem ist ry is select ing a suit able lev el of t heor y for a given pr oblem , and being able t o evaluat e t he qualit y of t he obt ained result s. Today, one can com put e chem ical pr opert ies of a m olecule ( < 10 heavy at om s) such as heat s of form at ion, ionizat ion pot ent ials ( I P) , elect r on affinit ies ( EA) , pr ot on affinit ies ( PA) and ot hers w it h “ chem ical accuracy” , i.e. ca. 1- 2 kcal/ m ol dev iat ion as com par ed t o ex perim ent al dat a av ailable, using t he w ellknow n t heor et ical chem ist ry m odels such as t he Gaussian t heory 10 or t he com plet e basis set ( CBS) appr oach. 11 Ev en “ sub- chem ical accuracy” ( i.e. a few kJ/ m ol deviat ion com pared t o ex perim ent s) for com put at ional t her m ochem ist r y

can be achieved on sm aller sy st em s ( ”  KHDY\ DWRPV  ZKHUH WKH :LJQHU- 3 t heor y 12 or t he coupled- clust er m et hod in com binat ion w it h a com plet e basis set

( i.e. ex t r apolat ed t o t he nearly infinit e basis set lim it ) 13–20 ar e used. I t appears t hat com put at ional chem ist ry appr oaches t oday play a sim ilar ly im por t ant role as exper im ent al chem ist ry m et hods in t eaching, st udy ing, researching, and apply ing in our societ y . I t should be m ent ioned t hat a 1 kcal/ m ol difference in t heoret ical bar rier height s alt ers t he com put ed k( T) v alues by a fact or of 5.5 at r oom t em perat ure. I .2 . Sou r ce s of sm a ll un sa t u r a t e d h yd r oca r bons Acet y lene, et hy lene, benzene, allene and ot her unsat urat ed hydrocarbons ar e t hem selves possible init ial fuels for com bust ion and flam es. Acet y lene flam es are widely used for welding and ot her applicat ions r equir ing high t em perat ures. These hydrocarbons are gener at ed com m ercially by synt het ical processes or by cracking or dist illat ion using fossil m at er ials ( nat ural gas, oil and coal) . 21 I n paraffin com bust ion and flam es, sm all olefins ( m ost ly et hy lene and pr opene but also som e n- but ene

and

i- but ene)

ar e

largely

form ed

as

m aj or

int erm ediat es

unim olecular decom posit ion of unst able high- or der alky l radicals by eq ( I .1) : R1 • + M

olefin + R2 • + M

by

22–24

( I .1)

Wher e R1 • and R2 • are t he higher - and low er - order alky l radicals, respect iv ely . Subsequent ly, t he R2 • alk y l radical can furt her decom pose t o an olefin and anot her sm aller alky l radical or a hydr ogen at om . Decom posit ion of alky l radicals

proceeds via so- FDOOHG - scission, 22–24 w hich m eans t hat t he bond t hat will break

is rem ov ed one bond fr om t he radical sit e. A prim e exam ple is t he β C–C scission of t he CH 3 –CH 2 –CH 2 • n- pr opy l radical t o y ield t he CH 3 • m et hyl r adical and et hylene, CH 2 = CH 2 .

Chapt er I : I nt roduct ion

5

I n radical- poor sit uat ions ( i.e. dur ing ignit ion and induct ion per iods) , alk y l radicals ar e pr im ar ily generat ed by radical- chain init iat ion st eps involv ing t he breaking of a C–C single bond in a paraffin, by eq ( I .2) : R1 CH 2 –CH 2 R2 + M

R1 CH 2 • + R2 CH 2 • + M

( I .2)

This chain init iat ion st ep is undoubt edly dom inant since t he C–C single bond

st rengt h ( ”NFDOPRO LVVXEVWDQWLDOO\ZHDNHUWKDQDQ\RIWKH&+VLQJOHERQGV

( •  NFDOPRO  in t he m olecule. 22 However, at any r easonable com bust ion

t em perat ur es, som e C–H bonds ar e also br ok en and H at om s appear due t o t his init iat ion st ep: RH + M

R• + H • + M

( I .3)

Not e t hat at low er t em perat ures H- abst ract ions by ox ygen m olecules cont r ibut e t o t he chain- init iat ing form at ion of alk y l r adicals: R• + HO2 •

RH + O2

( I .4)

An im port ant point is t hat t he init iat ion st ep by eq ( I .3) pr ov ides H at om s t hat subsequent ly r eact wit h ox y gen m olecules t o begin t he pr edom inant chain branching st ep, H • + O2



OH + • O•

which, by subsequent r eact ions of t he • OH radical and t he • O• diradical, giv es r ise t o a rapid, m ult iplicat iv e built - up of t he pool of chain- propagat ing •



OH, • H and

O• radicals .

Once t he chain- init iat ion and chain- branching lead t o high radical pool lev els, t he disappearance of t he paraffin fuel RH is cont rolled by t he H- abst ract ion r eact ions ( I .5) : 22–24 RH + X ( • OH, • H or O)

R• + HX ( H 2 O, H 2 or • OH)

( I .5)

The r at e of t he H- abst r act ion from t he RH par affin m olecule by t he radical pool species • OH, •



H or O depends generally on t he nat ure of t he at t acking radical



( OH, H or O) as well as on t he C–H single bond t ype and/ or st rengt h in t he m olecule. The order of t he C–H single bond st rengt h is as prim ary > secondar y > t ert iar y CH bonds. 22 The w eaker t he CH bond, t he fast er t he H- abst ract ion. For m at ion of e t hylen e . I t is w ell est ablished t hat et hy lene is m ainly generat ed in hy dr ocarbon com bust ion by t wo com pet ing r eact ions: eit her t he decom posit ion of t he et hy l radical or t he react ion of t his radical wit h m olecular ox ygen. 22–24 • •

C2 H 5 + M C2 H 5 + O2

C2 H 4 + H • + M C2 H 4 + HO2



( I .6) ( I .7)

The et hy l radicals r esult from larger alky l radicals v ia β C–C scission ( see above) and by H- abst ract ion from et hane, w hich it self is form ed in all hy drocarbon flam es by recom binat ion of t he ubiquit ous m et hy l radicals. I n addit ion, et hy lene can be giv en off by decom posit ion of n- and i- C3 H 7 radicals or by a m inor channel

6

Chapt er I : I nt roduct ion

of t he r eact ion of t w o m et hyl radicals or by a react ion of t riplet m et hy lene w it h m et hyl radical. 25 C2 H 4 + • CH 3



C3 H 7



CH 3 + • CH 3

3

( I .8)

C2 H 4 + H 2



CH 2 + CH 3

C2 H 4 + H

( I .9)



( I .10)

For m at ion of a cet yle n e. Acet y lene, a m aj or int erm ediat e of pr edom inant im port ance in all hydr ocarbon flam es, is m ainly form ed by unim olecular Helim inat ion of t he v iny l radical, 22–25 which is a m aj or product in t he H- abst ract ion react ion of et hy lene by t he pool radicals • OH and • H. •

C2 H 2 + • H + M

C2 H 3 + M



C2 H 4 + X ( • OH or • H)

( I .11)

C2 H 3 + HX ( H 2 O or H 2 )

( I .12) •



Addit ionally, H- loss from t he v iny l radical by react ions wit h H, O, OH, O2 and •

CH 3 radicals also cont r ibut es significant ly t o acet y lene for m at ion. 25 •

C2 H 3 + Y ( • H, O, • OH, O2 or • CH 3 ) C2 H 2 + HY ( H 2 , • OH, H 2 O, HO2 • or CH 4 )

( I .13)

For m at ion of a lle n e . According t o a r ecent kinet ic m odeling st udy,

25

at lat er

flam e st ages m aj or allene ( C3 H 4 ) form at ion pat hways are unim olecular decays of propeny l radical ( • C3 H 5 ) , which is produced by eit her t he r eact ion of viny l w it h m et hyl radicals or t he r eact ion of • CH m et hy lidyne radical w it h et hylene. •

C3 H 5 + M



C2 H 3 + • CH 3



C3 H 4 + • H + M • •

CH + C2 H 4

( I .14)

C3 H 5 + • H

( I .15)

C3 H 5

( I .16)





in which t he CH radical is giv en off by react ions of t he pool radicals ( H and • OH) wit h t r iplet m et hy lene, which is a product fr om t he O + C2 H 2 r eact ion. 26,27 3

CH 2 + X ( • H and • OH)

O + C2 H 2 Howev er,

in

3



CH + HX ( H 2 or H 2 O)

( I .17)

CH 2 + CO

ear lier

com bust ion

( I .18) st ages

allene

is

dom inant ly

recom binat ion of t w o radicals, propargy l and hy drogen at om . •

C3 H 3 + • H

C3 H 4

form ed

by

25

( I .19)

For m at ion of be n z en e . Given t hat benzene plays an enorm ously im port ant r ole as a “ building br ick” in form at ion of polycy clic arom at ic hydr ocarbon ( PAH) and soot , react ion m echanism for form at ion of benzene were ext ensiv ely st udied. Most possible react ion pat hway s yielding benzene in com bust ion and flam es wer e excellent ly r ev iew ed by Richt er and Howard, 28 nam ely ( i) t he r ecom binat ion of t wo pr opargyl radicals, which plays a m aj or r ole ( if not t he dom inant one) in t he for m at ion of benzene; 25, 28–31 •

C3 H 3 + • C3 H 3

C6 H 6

( I .20)

Chapt er I : I nt roduct ion

7

( ii) t he r eact ion of n- C4 H 5 • radical produced fr om eq ( I .21) w it h acet ylene, followed by a H- elim inat ion of int erm ediat e c- C6 H 7 • leading t o benzene; 28,31 •

n- C4 H 5 •

C2 H 3 + C2 H 2

( I .21)

n- C6 H 7 •

n- C4 H 5 • + C2 H 2

c- C6 H 7 •

C6 H 6 + • H

( I .22)

( iii) t he addit ion of v inylacet y lene t o v iny l radical, followed by a H- loss t o yield benzene; 25,28, 32 C4 H 4 + • C2 H 3

n- C6 H 7 •

c- C6 H 7 •

C6 H 6 + • H

( I .23)

( iv ) t he r eact ion of 1,3- but adiene w it h v iny l radical, followed by t he elim inat ion of H 2 t o benzene; 33 1,3- C4 H 6 + • C2 H 3 c- C6 H 8

c- C6 H 8 + • H

( I .24)

C6 H 6 + H 2

( I .25) •

( v ) t he react ions of m et hy l radicals w it h eit her i- C5 H 3 ( t he r esonant ly st abilized CH 2 CCCCH radical) 29, 34 or c- C5 H 5 • 29,35 also leads t o benzene; i- C5 H 3 • + • CH 3 •



c- C5 H 5 + CH 3 •

C6 H 6 or c- C5 H 4 CH 2 ( fulv ene)

( I .26)



( I .27)

c- C5 H 4 CH 2 •

c- C5 H 4 CH 2 + H

C6 H 6 + H

( I .28)

wher e i- C5 H 3 • is easily form ed in flam es fr om diacet ylene: 29 3

i- C5 H 3 • + • H

CH 2 + C4 H 2

( I .29)

( v i) t he possible for m at ion of benzene by et hy lene at t ack on cyclopent adiene followed by hydr ogen and m et hy l loss, passing t hr ough norbor nene, a CH 2 bridged cyclohex ene: C6 H 6 + • H + • CH 3

C2 H 4 + c- C5 H 6

( I .30)

This m echanism was fir st ly suggest ed by Dent e et al cont r ibut e significant ly pyroly sis.

t o benzene form at ion

in

36

and recent ly predict ed t o

et hy lene com bust ion

and

37

( v ii) in addit ion, benzene is a k ey int erm ediat e in degradat ion of higher - order arom at ics, for inst ance t he ox idat ion of t oluene leading t o benzene: 38,39 c- C6 H 5 CH 3 •

… •

c- C6 H 5 + H

c- C6 H 5 • + • CH 3

( I .31)

c- C6 H 6 •

c- C6 H 5 CH 3 + H

( I .32) •

c- C6 H 6 + CH 3

( I .33)

I .3 . Sou r ce s of t r iple t gr ou n d st a t e oxyg e n a t om s The m ost im port ant for m at ion of t r iplet O at om in hydr ocarbon com bust ion and flam es is via t he r eact ion of hydrogen at om w it h m olecular oxygen: 31 H • + O2 This react ion,

O• + • OH

¨+r ( 298 K) = + 16.37 k cal/ m ol

t he pr edom inant

rat e- cont rolling

elem ent ary

( I .34) react ion

in

all

com bust ion processes, is t he basic chain- branching process in high- t em perat ur e

8

Chapt er I : I nt roduct ion

com bust ion;

it consum es about 80%

of t he O2 in t ypical hydr ocarbon- air

st oichiom et r ic flam es at at m ospheric pr essure. 23 Hy drogen at om s ar e generat ed eit her by t her m al decom posit ion of hydr ocarbons in t he init iat ion st ep ( see eq ( I .3) ) or, dom inant ly in lat er com bust ion st ages, by decom posit ion of sm all alky l radicals ( see, for exam ple, eqs ( I .6) and ( I .11) ) and of HCO form yl radicals for m ed fr om form aldehyde. The m ole fr act ion of t r iplet O at om in hydrocarbon- fueled flam es is principally cont r olled by eq ( I .34) . I n fuel- lean flam es t he O concent rat ion is fair ly high as a result of high m ole fr act ion of O2 , w her eas it is quit e sm all in v er y fuel- r ich flam es. I n pract ical com bust ion syst em s, t he air / fuel m ixt ur e is approx im at ely st oichiom et r ic ( –equiv alence r at io § t hus t he m ole fract ion of t r iplet O- at om is reasonably high. Ther efor e, t he ox ygen- at om init iat ed ox idat ion of sm all unsat urat ed hydrocarbons plays an im port ant role ( if not t he dom inant one) in hydrocarbon com bust ion processes and flam es. The m aj or case in point is t he addit ion/ fragm ent at ion react ion of acet ylene wit h O at om s, w hich is know n t o be t he dom inant C2 H 2 r em oval process in flam es, 22,23,25–27 due t o t he low propensit y for H- abst ract ion by H or OH, on account of t he except ionally high C–H bond st rengt h of 132 kcal/ m ol in t his m olecule. 40 The C2 H 2 + O react ion plays an im port ant r ole in hydrocarbon com bust ion chem ist ry because it leads t o sev eral highly react ive sm all radicals such as HCCO, t riplet and singlet CH 2 , H, CH and C2 H. Som e of t hose can successfully at t ack closed- shell m olecules such as C2 H 2 and ev en N 2 , or react furt her w it h ot her at om s or r adicals in highly ex ot herm ic react ions, t hus causing m aj or flam e phenom ena such as chem i- ionizat ion and chem i- lum inescence, prom pt - NO form at ion and product ion of PAH- and soot precursors. 27 The highly react ive pr oduct s of t he C2 H 2 + O react ion and t heir subsequent chem ist ry and k inet ics have been t he subj ect of int ensive st udies in t his laborat or y for ov er a decade. 26,27,41–56

Chapt er I : I nt roduct ion

9

Re fe r e nce s ( 1) “ August 8t h, Pr esident Bush signed t he Energy Policy Act of 2005, which included m easures t o encourage t he nuclear indust ry t o build new nuclear pow er plant s” , t ak en fr om t he web- sit e: ht t p: / / www .eia.doe.gov / k ids/ hist or y/ t im elines/ nuclear.ht m l ( 2) “ China t o build 10 new nuclear pow er plant s” in “ Japan Ener gy Scan” published on June 6 t h , 2005 by China Nat ional Nuclear Corp. ( 3) The I nt er nat ional Ener gy Out look 2006 ( I EO2006) , t aken fr om t he web- sit e ht t p: / / www .eia.doe.gov / oiaf/ ieo/ index.ht m l ( 4) For discussion of energy sources and consum pt ion, see Science and Sust ainabilit y , I nt . I nst . For Appl. Anal. Syst . ( NASA 20 t h Anniv ersary ) , Vienna, 1992 and refer ences t her ein. ( 5) Taken fr om t he w eb- sit e ht t p: / / www .eia.doe.gov / neic/ brochur e/ gr eenhouse/ Chapt er1.ht m ( 6) “ Diary : The Am azon rainfor est ” r eport ed by David Shuk m an, BBC New s on Tuesday, 18 July 2006. ( 7) Jensen, F. “ I nt roduct ion t o Com put at ional Chem ist ry ” , John Wiley & Sons: West Sussex, England, 1999. ( 8) Jalbout , A. F; Nazar i, F. ; Turk er, L. J. Mol. St ruct . ( THEOCHEM) 2 0 0 4 , 671, 1. ( 9) Taken fr om w eb- sit e ht t p: / / nobelprizes.com / nobel/ chem ist ry / chem ist ry.ht m l ( 10) Curt iss, L. A.; Raghavachar i, K.; Redfern, P. C. ; Rassolov, V. ; Pople, J. A. J. Chem . Phys. 1 9 9 8 , 109, 7764 and r eferences t her ein. ( 11) Mont gom ery , Jr., J. A.; Ocht erski, J. W.; Pet ersson, G. A. J. Chem . Phys. 1 9 9 4 , 101, 5900 and r efer ences t her ein. ( 12) Boese, A. D.; Or en, M.; At asoy lu, O.; Mart in, J. M. L. , Kallay, M.; Gauss, J. J. Chem . Phys. 2 0 0 4 , 120, 4129 and r eferences t her ein ( 13) Pet erson, K. A.; Woon, D. E. ; Dunning, T. H. J. Chem . Phys. 1 9 9 4 , 100, 7410. ( 14) Mart in, J. M. L. Chem . Phys. Let t . 1 9 9 6 , 259, 669. ( 15) Halk ier, A.; Helgak er , T.; Jorgensen, P.; Klopper, W.; Kock , H.; Olsen, J.; Wilson, A. K. Chem . Phy s. Let t . 1 9 9 8 , 286, 243. ( 16) Klopper , W.; Bak , K. L.; Jorgensen, P.; Olsen, J.; Helgak er T. J. Phys. B: At . Mol. Opt . Phys. 1 9 9 9 , 32, R103- R130. ( 17) Klopper , W. Mol. Phys. 2 0 0 1 , 99, 481. ( 18) Feller , D.; Dix on, D. A. J. Chem . Phy s. 2 0 0 1 , 115, 3484. ( 19) Valeev , E. F.; Allen, W. D.; Hernandez, R.; Sherill, C. D.; Schaefer I I I , H. F. J. Chem . Phy s. 2 0 0 3 , 118, 8594. ( 20) Klopper , W.; Noga, J. Chem . Phys. Chem . 2 0 0 3 , 4, 32. ( 21) Olah, G. A.; Molnar, A. Hydr ocarbon Chem ist ry, Second Edit ion, John Wiley: New Jersey, 2003. ( 22) Glassm an, I . Com bust ion, Second Edit ion, Academ ic Press: London, 1987. ( 23) Edit ed by Gardiner , Jr ., W., C. Com bust ion Chem ist ry, Springer - Verlag: New Yor k, 1984. ( 24) Tur ns, S. R. An I nt r oduct ion t o Com bust ion: Concept s and Applicat ions, Second Edit ion, McGraw - Hill Book : Singapore, 2000. ( 25) Richt er , H.; Howard, J. B. Phys. Chem . Chem . Phys. 2 0 0 2 , 4, 2038 and references t herein. ( 26) Peet ers, J.; Dev r iendt , K. Tw ent y - Sixt h Sy m p. ( I nt .) on Com bust ion 1 9 9 6 , 1001. ( 27) Peet ers, J. Bull. Soc. Chim . Belg. 1 9 9 7 , 106, 337 and r eferences t her ein. ( 28) Richt er, H.; Howar d, J. B. Prog. Ener gy Com bust . Sci. 2 0 0 0 , 26, 565 and references t herein. ( 29) Miller, J. A. Faraday Discuss. 2 0 0 1 , 119, 461 and refer ences t herein. ( 30) Miller, J. A. ; Klippenst ein, S. J. J. Phy s. Chem . A 2 0 0 3 , 107, 7783 and references t herein.

10

Chapt er I : I nt roduct ion

( 31) Miller , J. A.; Pilling, M. J.; Tr oe, J. Proc. Com bust . I nt . 2 0 0 5 , 30, 43 and references t herein. ( 32) Colk et , M. B. Tw ent y- first Sy m p. ( I nt .) Com bust . 1 9 8 6 , 851. ( 33) Lindst edt , R. P.; Skev is, G. Tw ent y - sixt h Sym p. ( I nt . ) Com bust . 1 9 9 6 , 703. ( 34) Pope, C. J.; Miller , J. A. Proc. Com bust . I nst . 2 0 0 0 , 28, 1519. ( 35) Melius, C. F.; Colv in, M. E.; Mar inov, N. M.; Pit z, W. J.; Senkan, S. M. Proc. Com bust . I nst . 1 9 9 6 , 26, 685. ( 36) Dent e, M. ; Ranzi, E.; Goossens A. G. Com put er Chem . Eng. 1 9 7 9 , 3, 61. ( 37) Farav elli, T. ; Goldaniga, A.; Ranzi, E. Tw ent y- sev ent h Sym p. ( I nt .) Com bust . 1 9 9 8 , 1489. ( 38) Brezinsky, K. Prog. Ener gy Com bust . Sci. 1 9 8 6 , 12, 1. ( 39) Dav is, S. G.; Wang, H.; Br ezinsk y , K.; Law, C. K. Tw ent y - six t h Sym p. ( I nt .) Com bust . 1 9 9 6 , 1025. ( 40) Fr om NI ST w eb page: ht t p: / / srdat a.nist .gov / cccbdb/ , wit h:

∆H f (C2 H ) = 135.0 ± 1 kcal/ m ol; ∆H f (C2 H 2 ) = 54.5 ± 0.1 kcal/ m ol; ∆H f ( H ) = 51.6 kcal/ m ol, 0

0

0

leading t o BDE( HC2 –H) = 132.1 ± 1.1 k cal/ m ol. ( 41) Peet ers J. ; Schaek ers M.; Vinck ier C. J. Phys. Chem . , 1 9 8 6 , 90, 6552. ( 42) Peet ers J. ; Vanhaelem eersch S.; Van Hoeym issen J.; Borm s R.; Verm ey len D. J. Phys. Chem ., 1 9 8 9 , 93, 3892. ( 43) Boullart W.; Peet er s J. J. Phys. Chem ., 1 9 9 2 , 96, 9810. ( 44) Peet ers J. ; Langhans I .; Boullart W. I nt . J. Chem . Kinet ., 1 9 9 4 , 26, 869. ( 45) Boullart W.; Dev riendt K.; Borm s R. ; Peet ers J. J. Phys. Chem . 1 9 9 6 , 100, 998 ( 46) Nguy en, M. T.; Boullart W.; Peet ers J. J. Phys. Chem . 1 9 9 4 , 98, 8030. ( 47) Boullart W.; Nguy en, M. T.; Peet ers J. J. Phys. Chem . 1 9 9 4 , 98, 8036. ( 48) Peet ers J.; Langhans, I .; Boullart W. ; Nguyen, M. T.; Dev r iendt , K. J. Phys. Chem . 1 9 9 4 , 98, 11988. ( 49) Peet ers J. ; Boullart W.; Dev r iendt , K. J. Phys. Chem . 1 9 9 5 , 99, 3583. ( 50) Devr iendt , K.; Van Look , H.; Ceurst ers, B.; Peet ers J. Chem . Phys. Let t . 1 9 9 6 , 261, 450. ( 51) Car l, S. A.; Sun, Q.; Peet ers, J. J. Chem Phys. 2 0 0 1 , 114, 10332 ( 52) Dev r iendt , K.; Peet ers J. J. Phys. Chem . 1 9 9 7 , 101, 2546. ( 53) Ver eeck en, L.; Sum at hy, R. ; Peet ers J. Chem . Phys. Let t . 2 0 0 1 , 344, 400. ( 54) Car l, S. A.; Sun, Q.; Teugels, L.; Peet ers, J. Phys. Chem . Chem . Phys. 2 0 0 3 , 5, 5424. ( 55) Car l, S. A.; Van Poppel, M.; Peet ers, J. J. Phys. Chem A 2 0 0 3 , 107, 11001. ( 56) Nguy en, T. L.; Ver eeck en, L.; Peet ers, J. J. Phys. Chem A 2 0 0 6 , 110, 6696.

Chapt er I : I nt roduct ion

11

Cha pt e r I I : Th e or e t ica l M e t h odologie s

I I .1 . Qu a n t u m Che m ica l Ca lcu la t ions First - pr inciples quant um m echanic t reat m ent s of m olecular pr oper t ies ar e a widely used t ool in t he st udy of chem ical r eact ions. Fully describing t he quant um chem ical, t im e- dependent int eract ions bet w een all part icles in a m olecule or all com ponent s

in

a

r eact ive

m ix t ur e

is

int ract ably

com plex .

For

curr ent

com put at ional st udies aim ing t o describe chem ical kinet ics, t he quant um chem ical m et hodologies

are

usually

based

on

t he

Bor n- Oppenheim er

( BO)

approx im at ion. 1-3 Accor ding t o t his approx im at ion, 1–3 t he m ot ions of elect r ons described by eq ( I I .2) can be separat ed from t hose of at om ic nuclei described by eq ( I I .3) in t he t im e- independent Schrödinger equat ion ( I I .1) because nuclei ar e m uch heav ier t han elect rons.

wit h

Hˆ tot Ψ tot ( R, r ) = Etot Ψ tot ( R, r )

( I I .1)

Hˆ e Ψ e ( R, r ) = Ee ( R ) Ψ e ( R, r )

( I I .2)

(Tˆn + Ee ( R )) Ψ n ( R ) = Etot Ψ n ( R )

( I I .3)

Hˆ tot = Hˆ e + Tˆn ; Hˆe = Tˆe + Vˆne + Vˆee + Vˆnn ; and Ψ tot ( R, r ) = Ψ e ( R, r )Ψ n ( R )

wher e nuclear coordinat es ar e denot ed as R and subscript n, while elect ron coordinat es ar e present ed as r and e;

Hˆ, Tˆ and Vˆ are t he Ham ilt onian, k inet ic

energy, and pot ent ial energy operat ors, respect iv ely ; E is t he t ot al energy anG  is t he wav e- funct ion w hich in com put at ional st udies is usually described as a

linear com binat ion of basis wavefunct ions in a so- called basis set . Thus, approx im at e solut ions t o t he elect r onic Schrödinger equat ion ( I I .2) can be obt ained for a ser ies of calculat ions at fix ed nuclear geom et r ies. These point s ar e t hen fit t ed or ot herw ise int erpolat ed t o giv e an elect r onic pot ent ial ener gy surface ( PES) , w hich is independent of t he m asses of t he nuclei. 4 I deally, elect ronic st ruct ur e calculat ions perfor m ed w it h an effect ively com plet e basis set at t he full configurat ion int eract ion ( Full- CI ) lev el of t heor y involv ing ev ery possible elect r on configurat ion ( see eq ( I I .4) ) w ill t heoret ically generat e an exact solut ion t o t he Schrödinger equat ion wit hin t he t im e- independent , nonrelat iv ist ic BO appr ox im at ion. Such calculat ions ar e im pract ical but for t he sm allest of chem ical pr oblem s since full- CI calculat ions are only feasible for sm all basis set s due t o t he lim it s of com put at ional r esources av ailable.

Chapt er I I : Theor et ical Met hodologies

13

occ vir

occ vir

Ψ full −CI = c0 Ψ HF + ∑∑ airψ ir + ∑∑ aijrsψ ijrs + ... i

r

( I I .4)

i< j r 0.2 for a singlet elect ronic st at e. I n such cases, m ult ireference m et hods are em ploy ed. I n t his work, we first use t he CASSCF m et hod t o r e- opt im ize t he geom et ry and t o com put e it s wave- funct ion. I f t he wav efunct ion possesses m ult i- r efer ence charact er ( i.e. t he non- dy nam ic elect ronic corr elat ions are im port ant ) , t he CASPT2 43 or MRCI 44,45 m et hods t hat appropriat ely include dy nam ic elect ron correlat ions w ill t hen be em ploy ed t o refine t he pot ent ial energy ; if m ult i- reference charact er is found t o be negligible, t he earlier value obt ained from t he t heor et ical m odel chem ist ry will be accept ed. I I .2 . St a t ist ica l Ra t e Ca lcu la t ion s I I .2 .1 . Tr a nsit ion st a t e t he ory ( TST) According t o TST, 46,47 t he t herm al rat e coefficient for a bim olecular react ion of R1 wit h R2 v ia a t ransit ion st at e TS leading t o P1 + P2 ( see Fig. I I .1) at a specific t em perat ur e T is giv en by t he follow ing expression:

kTST (T ) =

kbT Q≠ × TS exp( − E0 / RT ) h QR1QR2

Chapt er I I : Theor et ical Met hodologies

( I I .14)

19

Figure I I .1: Pot ent ial ener gy sur face for t he R1 + R2

P1 + P2 react ion.

wher e k b is Bolt zm ann’s const ant ; h is Planck’s const ant ; R is t he univ ersal gas const ant ; E0 is t he bar rier height of t he react ion ( i.e. t he int er nal energy of a t ransit ion st ruct ur e r elat iv e t o t he init ial r eact ant s) ; and Q is a com plet e part it ion funct ion per unit v olum e, w hich is appr ox im at ely com put ed as a product of t he indiv idual elect r onic ( Qe ) , t ranslat ional ( Qt ) , rot at ional ( Qr ) , and v ibrat ional ( Qv ) part it ion funct ions, assum ing t hat t hese m ot ions are decoupled. 47

Q = Qe × Qt × Qr × Qv

( I I .15) 3

 2π mkbT  2 wit h t he t ranslat ional part it ion funct ion per unit of volum e Qt =   ; 2  h  3

 8π 2kbT  2 8π 2 IkbT Q = π Qr = for linear m olecules or r  h 2  I A I B I C for non- linear h2   s

m olecules;

Qv = ∏ i =1

1 . 1 − exp( − hcωi / kbT )

Not e t hat t he conv ent ional TST eq ( I I .14) does not t ake int o account quant um corr ect ions ( i.e. t unneling cor rect ions/ non- classical r eflect ions) and non- react iv e reflect ion t raj ect or ies back t o t he r eact ant s. The lat t er corr ect ions lead t o var iat ional TST, w hich det erm ines a k inet ic bot t le- neck at a m inim um of chem ical react ion flux ( i.e. m in[ k ( T) ] ) . I ncluding t hese effect s int o eq ( I I .14) y ields equat ion ( I I .16) :

kTST (T ) = κ (T ) ×

20

≠ kbT Min[QTS ( s ) × exp( − E0 ( s ) / RT )] × h QR1QR2

Chapt er I I : Theor et ical Met hodologies

( I I .16)

wher e

7 is t he one- dim ensional t unneling corr ect ion, w hich is com put ed as

below by assum ing an asym m et r ic Eckart pot ent ial: 48

κ (T ) =



1  E − E0  P( E ) exp  − dE ∫ RT 0 RT  

(II.17)

wher e P( E) is t he t unneling pr obabilit y, w hich is giv en as follows: 49

P( E ) = 1 − wit h

cosh 2π (a − b) + cosh 2π d cosh 2π ( a + b) + cosh 2π d

2π a = 2[ χξ ]0.5 × ( χ −0.5 + β −0.5 ) −1 ;

( I I .18)

2π b = 2[ χξ + β − χ ]0.5 × ( χ −0.5 + β −0.5 ) −1 ;

2π d = 2[ χβ − π 2 / 4]0.5 ; ξ = E / EF ; χ = 2π EF / hω ∗ ; β = 2π ER / hω ∗ ; EF and ER ar e t he for ward and r ev erse bar rier height s r elat ive t o t he react ant s and product s, respect ively , as present ed in Fig. I I .1;

is t he im aginar y fr equency, w hich

corr esponds t o t he r eact ion coordinat e t ranslat ional m ode. I t should be not ed t hat when d is im aginary, t he funct ion cosh2πd in eq ( I I .18) becom es cos2π| d| .

I I .2 .2 . Rice - Ra m spe r ge r - Ka ssel- M a r cu s t he or y ( RRKM ) The energy - dependent specific unim olecular rat e coefficient k( E) for a react ant wit h an int er nal ener gy E ( see Fig. I I .2) is given by t he RRKM st at ist ical t heor y : 46, 47,50–53

k(E) =

α G≠ (E − E ≠ ) × h ρ (E)

(II.19)

Figure I I .2: Pot ent ial ener gy sur face for t he R1 + R2 react ion.

Chapt er I I : Theor et ical Met hodologies

21

wher e α is t he r eact ion pat hway degeneracy, h is Planck’s const ant , E≠ is t he barr ier height for t he r eact ion Adduct → TS2 → Product , G≠( E−E≠) is t he sum of vibrat ional st at es of t he t ransit ion st at e ( i.e. TS2 in t his case) for energies from 0 up t o E−E≠, and ρ( E) is t he densit y of st at es for a r eact ant m olecule w it h int ernal energy E. The Bey er - Sw inehart - St ein- Rabinovit ch algor it hm 54,55 was used t o calculat e t he sum and densit y of st at es in eq ( I I .19) em ploy ing a gr ain size of ≈0.003 kcal/ m ol ( 1 cm −1 ) . Analogously, t he conv ent ional RRKM t heory does not account for quant um corr ect ions and non- react ive r eflect ion fluxes. The lat t er effect s can be recovered by var iat ional t reat m ent s using t he var iat ional RRKM t heor y, equat ion ( I I .20) , while t he form er effect is approxim at ely account ed for by t he one- dim ensional t unneling

corr ect ion,

which

is

com put ed

assum ing

an

asym m et ric

Eckart

pot ent ial, eq ( I I .21) . 53

k(E) =

α Min[G ≠ ( E − E ≠ ( s ))] × ρ (E) h E −E≠

k(E) =

α × h



( I I .20)

κ (ε t ) ρ ≠ ( E − E ≠ − ε t )d ε t

−E≠

ρ (E)

( I I .21)

wher e εt is t he t ranslat ional energy in t he r eact ion coordinat e and κ( εt ) is t he t unneling probabilit y, w hich is given as t he following expr ession: 53,56

κ (ε t ) = wit h

a=

sinh( a ) × sinh(b) sinh [(a + b) / 2] + cosh 2 ( c ) 2

( I I .22)

−1 −1 4π 4π × ε t + EF × (E F−0.5 + ER0.5 ) ; b = ∗ × ε t + ER × (EF−0.5 + E R0.5 ) ; ∗ hω hω

0.5

 E E 1 c = 2π  F ∗ R2 −  ; and EF= E .  (hω ) 16  I I .2 .3 . On e - dim e n siona l hin der e d in t e r n a l rot a t ion Quit e oft en t he v ibrat ional m ot ions of som e m olecules have v er y low force const ant s, corr esponding t o relat iv ely low fr equency m ot ions about bonds t hat are usually of single- or der or less. I n such cases, t he m olecule does not ex ecut e sim ple harm onic m ot ion, but inst ead, t he absence of a significant resist ing pot ent ial allows one part of t he m olecule t o t wist and t ur n r elat iv e t o t he ot her part . Such m ot ions are referr ed t o as t or sional r ot at ions, w hich could be consider ed as t he loss of v ibrat ional m ot ions.

22

Chapt er I I : Theor et ical Met hodologies

I f t he int eract ion of t he int er nal r ot at ion m ot ion w it h t he ot her r ot at ional and vibrat ional m odes is assum ed t o be negligible, t he separable Schrödinger equat ion descr ibing t his m ot ion about a single axis is 57



h2 ∂ 2 Ψ (ϕ ) × + V (ϕ ) Ψ (ϕ ) = E Ψ (ϕ ) 8π 2 I hr ∂ 2ϕ

( I I .23)

wher e ϕ is t he int er nal rot at ion angle ( 0”ϕ< 2π) ; I hr is t he r educed m om ent of inert ia for t his r ot at ion, giv en by 1/ I hr = 1/ I 1 + 1/ I 2 w her e I 1 and I 2 are t he separat e gr oup m om ent s of inert ia of t he t wo count er - rot at ing m oiet ies. I n general, I hr var ies as a funct ion of t he r ot at ional angle ( ϕ) . I n m any cases in pract ical applicat ions, I hr changes only slight ly wit h ϕ and can be t r eat ed as a const ant ( i.e. r igid rot or ) ; E is t he quant ized hinder ed r ot at ional energy lev el; V( ϕ) is t he pot ent ial energy t hat is appr ox im at ely pr esent ed as a sinusoidal funct ion, a m odified Fourier series ( also see Fig. I I .3) :

V (ϕ ) = Vo (1 − cos σϕ ) / 2

( I I .24)

wher e Vo is t he classical barr ier height for t he hindered int er nal r ot at ion, and σ is t he r ot at ional sym m et ry num ber.

Figure I I .3: Pot ent ial ener gy cur ve, V/ Vo= ( 1- cos3 ϕ) / 2.

Equat ion ( I I .23) can be re- form ulat ed in anot her form , equat ion ( I I .25) :

Chapt er I I : Theor et ical Met hodologies

23

− Bhr ×

∂ 2 Ψ (ϕ ) + V (ϕ )Ψ (ϕ ) = E Ψ (ϕ ) ∂ 2ϕ

( I I .25)

wher e Bhr is t he r ot at ional const ant , giv en by corr elat ed r elat ion:

Bhr = h 2 /(8π 2 I hr ) or in a useful

Bhr (cm −1 ) ≈ 16.85763/ I hr (amu × Å 2 ) .

Now we w ill consider t wo ext rem es of a v er y high bar r ier ( Vo >>E or at very low t em perat ur es) and a v er y low bar rier ( Vo E) , t he eigenvalues of eq ( I I .25) approx im at ely obt ained are t he quant ized harm onic oscillat or energy lev els: 57,58

E (v ) = σ Vo Bhr (v + 1/ 2) wit h ν- t he v ibrat ional quant um num ber = 0, 1, 2,… So

t he

fundam ent al

frequency

of

har m onic

oscillat ion

is

giv en

as:

hωhr = σ Vo Bhr , and t he quant um part it ion funct ion for t his harm onic oscillat or is giv en by : ∞

Qvq (T ) = ∑ exp[− E (ν ) / kbT ] = v =0

wit h

exp( −u / 2) 1 = 1 − exp( −u ) 2sinh(u / 2)

( I I .26)

u = hωhr / kbT = σ Vo Bhr / kbT .

For t he lat t er case ( Vo 0) . Based on t he part it ion funct ion of t he classical one- dim ensional hinder ed int er nal rot or ( eq ( I I .30) ) suggest ed by Pit zer and Gwinn, Knyazev 74, 75 recent ly der iv ed analyt ical form ula for com put ing t he densit y of st at es as t he follow ing expression:

Chapt er I I : Theor et ical Met hodologies

29

 2Κ (   πσ ρ hr ( E ) =   2Κ (  πσ 

E / Vo ) BhrVo Vo / E ) B hrVo

,0 < E < Vo ( I I .50a)

, E > Vo

wher e K( x ) is t he com plet e ellipt ic int egral of t he first kind, 76 giv en by 2 2 2  π  1  1× 3  2  1× 3× 5  3 Κ ( x ) = 1 +   x +   x +  x + ... 2   2   2×4  2×4×6 

,

wit h

| x| < 1 ( I I .50b)

Forst

77

also obt ained a for m ula for densit y of st at es using t he Tr uhlar’s part it ion

funct ion ( see eqs ( I I .31) and ( I I .32) ) . How ev er , it erat ive num erical solut ions for a non- linear equat ion are requir ed in order t o com put e

, a posit iv e variable w hich

depends on t he int er nal rot at ion energy. I I .2 .4 . M a st er e q ua t ion ca lcu la t ion s Many im port ant react ions occurr ing in com bust ion, part icular ly t hose inv olved in t he form at ion of arom at ics, polycyclic ar om at ic hydrocarbons ( PAH) , and soot , ar e com plicat ed pr ocesses t hat t ak e place ov er m ult iple, int erconnect ed int erm ediat es ( pot ent ial energy wells) . These int erm ediat es can eit her be t her m alized by collisions w it h bat h gas m olecules, isom er ize t o ot her int erm ediat es, or dissociat e t o form pr oduct s or r egenerat e t he r eact ant s. I n t he gas phase, bot h t he rat e coefficient s and t he dist ribut ion of pr oduct s ( i.e. of st abilized int erm ediat es and dissociat ion pr oduct s) can depend not only on t em perat ur e, but also on pressure. The dependences can be quit e com plex , y et t hese rat e- and product - y ield dat a are essent ial in m odeling com bust ion syst em s and sim ulat ions of engines. The react ion condit ions inv olved in com bust ion applicat ions are oft en challenging t o reproduce exper im ent ally , such t hat direct m easurem ent s at t he requir ed high pressur es and t em perat ur es can be ext r em ely difficult . Ther efor e, it is essent ial t o be able t o t heor et ically m odel t he r eact ions r ealist ically. These m odels prov ide not only t he basis for t he dev elopm ent of a fundam ent al underst anding of t hese int erest ing react ions, but ar e also an essent ial t ool for ext rapolat ion of av ailable exper im ent al dat a and for r epr esent ing t hose dat a in com bust ion m odels. I n t his sect ion, t he m ast er equat ion m odel calculat ions will be described and used t o address chem ically act iv at ed r eact ions in com bust ion in next chapt ers.

30

Chapt er I I : Theor et ical Met hodologies

Figure I I .4: Pot ent ial ener gy surface for t he chem ically act iv at ed react ion, R1 + R2 . Suppose t hat w e are consider ing a react ion of R1 w it h R2 species as show n in Fig. I I .4. First , t he chem ically act ivat ed adduct int erm ediat e I nt 1 is form ed by a recom binat ion of R1 and R2 and has an int er nal energy of Ei . Second, t he adduct I nt 1 when form ed can t hen carr y out several possible r eact ion st eps, nam ely ( i) Colliding w it h t he bat h gas, gaining or losing energy. I f it s r em aining energy is insufficient t o surm ount t he lowest - lying t ransit ion st at e ( i.e. TS2 in t his case) , it will be collect ed int o a sink as a t herm ally st abilized pr oduct ; ( ii) Regenerat ing t he init ial react ant s v ia TS1; and ( iii) I som er izing t o I nt 2 v ia TS2. Sim ilar ly, I nt 2 can eit her be st abilized by collisions w it h t he bat h gas or decom pose t o t he final product via TS3 or convert back t o I nt 1. I n general, t he k inet ic behavior of I nt 1 and I nt 2 is very com plicat ed, inv olv ing t hr ee com pet ing sim ult aneous processes: decom posit ion react ions, isom erizat ions and energy t ransfers. The equat ion describing t hese changes can be present ed as t he follow ing coupled int egr odifferent ial equat ion ( I I .51) , t he so- called one- dim ensional ( energy - dependent ) m ast er equat ion, which describes a t im e- ev olut ion of populat ions of I nt 1 and I nt 2 species. 46,47,50–52

Chapt er I I : Theor et ical Met hodologies

31

∞ ∞ ∂C1 ( Ei , t ) = + ∫ Pi ← j Z LJ [ M ]C1 ( E j , t )dE j − ∫ Pi → j Z LJ [ M ]C1 ( Ei , t )dE j 0 0 ∂t

− [k Re ( Ei ) + k12 ( Ei )]C1 ( Ei , t )

+ k21 ( Ei )C2 ( Ei , t ) + k R1R2 (T )[ R1 (t )][ R2 (t )]F ( Ei )dEi

( I I .51a)

∞ ∞ ∂C2 ( Ei , t ) = + ∫ Pi ← j Z LJ [ M ]C2 ( E j , t )dE j − ∫ Pi → j Z LJ [ M ]C2 ( Ei , t )dE j 0 0 ∂t

− [k Pr o ( Ei ) + k 21 ( Ei )]C2 ( Ei , t ) + k12 ( Ei )C1 ( Ei , t )

( I I .51b)

wher e t he first t wo t er m s in t he right - hand side of eq ( I I .51a) present energy t ransfer processes: t he first t erm pr esent s populat ion gain at Ei by collisional energy t ransfer pr ocesses from an energy Ej w her eas t he second t erm describes populat ion loss at Ei by collisional energy t ransfer processes t owards Ej ; t he next t erm pr esent s populat ion loss by chem ical react ions of I nt 1 at Ei ; t he fourt h t erm describes populat ion gain by isom er izat ion react ion at Ei fr om I nt 2 t o I nt 1; and t he final one is t he source t erm at Ei arising fr om t he chem ically act iv at ed react ion of R1 w it h R2 . C1 ( Ei ) and C2 ( Ei ) ar e t he fract ional populat ions of I nt 1 and I nt 2 at Ei lev el, r espect iv ely ; Pi →j is t he pr obabilit y densit y funct ion for energy t ransfer fr om Ei t o Ej lev el; Z LJ is t he Lennard- Jones collision frequency ; [ M] is t he m olar concent rat ion of t he bat h gas; k ( E) is t he energy- specific rat e coefficient ; and F( E) is t he pr obabilit y densit y funct ion of energy dist ribut ion. Equat ion ( I I .51) is appr opr iat e for analyt ic solut ion, but such an approach is only generally feasible wit h sim ple analyt ic form s for Pi →j , k ( E) and in t he low- pressur e lim it . 52 For applicat ion t o realist ic syst em s, a num er ical solut ion is generally requir ed. Thus, it is essent ial t o re- form ulat e t he abov e m ast er equat ion. The usual appr oach is t o part it ion t he energy lev els int o a cont iguous set of “ bands” ,

wit h t he w idt h of t he bands being pr eset ( generally , ∆E ”FP–1 is sufficient ly

sm all 51,52 t o m ake sur e t hat t he propert ies of t he m olecules w it hin a band are m or e or less ident ical, such t hat t he choice of t he w idt h of band does not affect t he result ; ∆E = 10 cm –1 is used t hrough t his t hesis) , leading t o equat ion ( I I .52) , t he so- called energy - gr ained m ast er equat ion ( EGME) . 46,52

32

Chapt er I I : Theor et ical Met hodologies

Emax Emax ∂C1 ( Ei , t ) = + ∑ Pi ← j Z LJ [ M ]C1 ( E j , t )∆E − ∑ Pi → j Z LJ [ M ]C1 ( Ei , t )∆E ∂t 0 0

− [k Re ( Ei ) + k12 ( Ei )]C1 ( Ei , t ) + k21 ( Ei )C2 ( Ei , t )

+ k R1R2 (T )[ R1 (t )][ R2 (t )]F ( Ei )∆E

( I I .52a)

Emax Emax ∂C2 ( Ei , t ) = + ∑ Pi ← j Z LJ [ M ]C2 ( E j , t )∆E − ∑ Pi → j Z LJ [ M ]C2 ( Ei , t )∆E ∂t 0 0

− [k Pr o ( Ei ) + k21 ( Ei )]C2 ( Ei , t ) + k12 ( Ei )C1 ( Ei , t )

( I I .52b)

wher e Em ax is t he ceiling energy, w hich in t his wor k is chosen t o be high enough abov e t he lowest - ly ing adduct , depending on t em perat ur e range of int er est in order t o ensur e t hat t here is no significant populat ion t r uncat ed at t he t ail of t he probabilit y densit y funct ion for energy dist r ibut ion. Equat ion ( I I .52) can t hen be int egrat ed t o obt ain t he t im e- dependent populat ions of var ious species by using det erm inist ic approaches, for exam ple using a st iff ODE int egrat or such as VODE78, 79 or diagonalizing t he m at rix of m ast er equat ion t o find all eigenv alues and eigenfunct ions, 80–84 of w hich expansions can t hen be carr ied out t o obt ain t he t im e- ev olut ion of t he populat ion. Som e soft ware packet s available em ploy ing t hese t echniques are Chem Rat e, 85 UNI MOL, 86 and VariFlex . 87 On t he ot her hand, t he t im e- dependent populat ions in eq ( I I .52) can also be achiev ed by num er ically int egrat ing, using t he Mont e- Car lo t echnique. I n t his t hesis, t he ex act st ochast ic m et hod ( ESM) dev eloped early by Gillespie 88–90 w ill be em ploy ed t o solv e num er ically t he m ast er equat ion ( I I .52) . Not e t hat t he ESM was im plem ent ed very ear ly in t he first version of t he URESAM program by Huyberecht s 91 and was significant ly im prov ed recent ly by Ver eeck en and Peet ers. 92,93 I n addit ion, Vereeck en 92, 93 new ly dev eloped t he advanced CSSPI m et hod t o com put e t he st eady - st at e populat ion of int erm ediat es at gr eat ly reduced com put at ional cost , reform ing t he solut ion of t he m ast er equat ion as a single eigenvect or pr oblem generat ed by t he descript ion as a cont inuous t im e Mark ov chain. 92, 93 Anot her ESM im plem ent at ion is also available in lit erat ur e, t he so- called Mult iWell program suit e dev eloped by Bark er . 94–96 Let ’s consider a chem ical react ion schem e ( at a part icular t im e t ) , w hich has n unim olecular r eact ions, consist ing of chem ical react ions and ener gy t ransfer processes by collision wit h bat h gas. According t o Gillespie, 88–90 a Mont e Car lo sim ulat ion of t he kinet ics of such a react ion schem e involv es t w o m ain quest ions, nam ely : ( i) w hen ( τ) w ill t he next react ion occur? and ( ii) w hat k ind of react ion ( µ) w ill it be? τ, t he cont inuous random variable ( 0”τ< ∞) describing t he t im e lapse t ill t he next react ion, is com put ed from equat ion ( I I .53) ; µ, t he discret e

Chapt er I I : Theor et ical Met hodologies

33

random var iable ( µ= 1, 2, 3, … , n- 1, n) select ing t he part icular react ion occur r ing, can be derived fr om equat ion ( I I .54) .

τ=

1 n

∑h c

ln(1/ r1 )

( I I .53)

i i

i =1

µ −1

n

µ

i =1

i =1

i =1

∑ hi ci < r2 × ∑ hi ci ≤ ∑ hi ci

( I I .54)

wher e h i is t he num ber of r eact ant m olecules in t he i t h react ion channel; ci is t he unim olecular rat e coefficient ( e.g. in t he EGME it is eit her k( E) or collision fr equency ) ; r 1 and r 2 ar e t wo random num bers from a uniform dist r ibut ion w it hin ( 0,1) . The uniform pseudorandom num ber generat or ( RNG) Mersenne Tw ist er ( MT19937) 97 w ill be used t o generat e random num bers, t his uniform RNG has t he advant ages of m oder n RNG such as fast execut ion t im e, an ext r em ely long per iod of 2 19937 –1, and good st at ist ical pr opert ies of t he generat ed random num bers ( “ high qualit y” ) . I n t he follow ing sect ion, w e w ill br iefly sum m ar ize funct ions and param et ers which are used in t he EGME calculat ions ( please see r efer ences 91, 92 and 93 for det ails) . ( i) The probabilit y densit y funct ion of t he ener gy dist ribut ion of form at ion of t he init ial adduct s F( E) , which is der iv ed by t em porar ily consider ing t her m ody nam ic equilibr ium bet w een R1 + R2 and I nt 1 ( e.g. see Fig. ( I I .4) ) , can be expr essed as: 50

F ( E )dE =

G ≠ ( E − E ≠ ) exp( − E / RT ) ∞

∫G



dE , for E•E , ot herw ise F( E) = 0 

( E ’ − E ≠ ) exp( − E ’ / RT )dE ’

E≠

I n t he EGME, t his form ula becom es:

F ( E ) ∆E =

G ≠ ( E − E ≠ ) exp( − E / RT )∆E Emax

∑G





( I I .55)

( E − E ) exp( − E / RT ) ∆E ’



E≠

wher e G ( E- E ) is t he sum of v ibrat ional st at es of t he t ransit ion st at e ( i.e. TS1 in 



t his case) for energies from 0 up t o E−E≠. ( ii) The collision fr equency wit h t he bat h gas is m odeled as a pseudo- first order process at a giv en const ant pr essur e, and can be com put ed as follows: 98

kColl = Z LJ [ M ] ( in s–1 )

34

Chapt er I I : Theor et ical Met hodologies

( I I .56)

wher e Z LJ is t he Lennard- Jones collision fr equency ( in cm 3 m olecule –1 s –1 ) ,

Z LJ = σ 2 N A (8π RT / µ ) × Ω 2,2 ; 0.5

[ M]

is

t he

concent rat ion

of

t he

bat h

gas

( m olecules/ cm 3 ) , [ M] = P/ RT; Ω2, 2 is t he r educed collision int egral, approx im at ely giv en

Ω 2,2 = [0.636 + 0.246ln(kbT / ε )] ; 99 −1

by

µ

is

t he

r educed

m ass,

µ= MA×MM/ ( MA+ MM) ; σ and ε ar e t he Lennard- Jones param et ers, ex pressed as σ= ( σA+ σM) / 2 and ε= ( εAεM) 0.5 . ( iii) Troe’s bi- exponent ial m odel is used for energy t ransfer process. 100 I n t his m odel t he pr obabilit y densit y funct ions for up- and down- energy t r ansfer ar e giv en as follows:

PUp−transfer = E> E’.

1 1 exp[ −( E ’ − E ) / β ] for E< E’ and PDown −transfer = exp[−( E − E ’) / α ] for N N

α and

β can

be

det er m ined

by

t wo

equat ions:

β= αFEk b T/ ( FEk b T+ α) , in which FE is approx im at ely ev aluat ed as:

α–β= –< ∆E>

and

100



∫ ρ ( E ) exp( − E / RT )dE ’

FE ( E ) ≈ FE ( E0 ) =



E0

ρ ( E0 ) exp( − E0 / RT )kbT



; < ∆E> is t he ov erall av erage

t ransfer red energy ; N is a norm alizat ion const ant at t he E level, giv en by N( E) = NDow n + NUp wit h: E

N Down = ∫ exp[−( E − E ’) / α ]dE ’ = α [1 − exp( − E / α )] , and 0

NUp =

Emax



exp[−( E ’ − E ) / β ]dE ’ = β {1 − exp[ −( Emax − E ) / β ]} ;

E

I n t he ESM sim ulat ion, t he energy gained or lost is ex plicit ly incorporat ed, m odeling specific up- energy

or

dow n- ener gy t ransfer

ev ent s by random ly

select ing a part icular ev ent according t o t he appr opriat e probabilit y densit y funct ions. Thus, in our im plem ent at ion w e use t he t hird r andom num ber ( r 3 ) t o select t his energy t ransfer st ep. Wit h PUp = N Up / N = t he up- t ransit ion pr obabilit y , if

0 < r 3 < PUp , an up- t ransfer energy st ep is select ed; ot herw ise w it h PUp ” r 3 < 1,

a dow n- t ransfer energy st ep is chosen. The fourt h random num ber ( r 4 ) is used t o obt ain a st ep- size of t he energy t ransfer pr ocess: 95

Chapt er I I : Theor et ical Met hodologies

35

E’

1 r4 = exp[−( E ’ − E ) / β ]dE ’ for up- energy t ransfer wit h E < E’; NUp ∫E

r4 =

1 N Down

E

∫ exp[−( E − E ) / α ]dE ’



for dow n- energy t ransfer w it h E > E’.

E’

To obt ain high st at ist ical precision in t he Mont e Car lo sim ulat ions, a large num ber of t r ials need t o be used. I n t his t hesis, w e em ploy ed N t rials ≈10 7 , w hich can yield a relat iv e st at ist ical er ror of ≈0.3% ev aluat ed for a fract ional populat ion of 0.01 using t he follow ing form alism : 95

σi =

Ci (1 − Ci ) –t he st andard dev iat ion N trials

( I I .57)

RSE =

σi Ci

( I I .58)

–t he r elat iv e st at ist ical er ror

wher e Ci is t he fract ional populat ion of t he i t h species. I n t his t ype of Mont e Carlo sim ulat ions, one can t her efor e balance com put at ional expense against accuracy of t he sim ulat ion. For t he syst em s consider ed here, t he st at ist ical uncert aint y incurr ed by t he select ed num ber of t r ials ( Nt rails) is negligible relat ive t o ot her sources of err ors, such as t he uncert aint ies on r elat iv e energies, rov ibrat ional charact er ist ics, and ener gy t ransfer param et ers.

36

Chapt er I I : Theor et ical Met hodologies

Re fe r e nce s ( 1) Szabo, A.; Ost lund, N. S. Moder n Quant um Chem ist ry : I nt roduct ion t o Adv anced Elect r onic St r uct ur e Theory , Dov er Publicat ions: New Yor k, USA, 1996. ( 2) Lev ine, I . N. Quant um Chem ist ry, 4 t h Edit ion, Prent ice- Hall: New Jersey, USA, 1991. ( 3) Jensen, F. I nt roduct ion t o Com put at ional Chem ist ry , John Wiley : West Sussex , England, 1999. ( 4) Lect ure Not es in Chem ist ry , Edit ed by Heidrich, D.; Kliesch, W.; Quapp, W. Propert ies of Chem ically I nt erest ing Pot ent ial Energy Surfaces, Spr inger - Ver lag: Berlin, Germ any , 1991. ( 5) Klopper , W.; Noga, J. Chem . Phy s. Chem . 2 0 0 3 , 4, 32 and references t herein. ( 6) Tay lor , P. R. Coupled- Clust er Met hods in Quant um Chem ist ry, p. 361 in Book I I , European Sum m er - school in Quant um Chem ist ry 2003, Edit ed by Roos, B. O; Widm ark , P. O. ( 7) Lee, T. J.; Scuser ia, G. E. Achiev ing Chem ical Accuracy wit h Coupled- Clust er Theory, p. 47 in Quant um Mechanical Elect ronic St r uct ure Calculat ions wit h Chem ical Accuracy, Edit ed by Langhoff, S. R. Kluw er Academ ic Publishers: Dordrecht , Net herlands, 1995. ( 8) Mercer o, J. M.; Mat xain, J. M.; Lopez, X.; Yor k, D. M.; Largo, A. ; Er iksson, L. A. ; Ugalde, J. M. I nt . J. Mass Spec. 2 0 0 5 , 240, 37. ( 9) Raghavachari, K.; Trucks, G. W.; Pople, J. A.; Head- Gordon, M. Chem . Phys. Let t . 1 9 8 9 , 157, 479. ( 10) Bart let t , R. J.; Wat t s, J. D.; Kucharsk i, S. A.; Noga, J. Chem . Phys. Let t . 1 9 9 0 , 165, 513. ( 11) Feller, D. ; Pet erson, K. A.; Crawford, T. D. J. Chem . Phys. 2 0 0 6 , 124, 054107- 1 and references t herein. ( 12) Klopper , W. J. Chem . Phys. 1 9 9 5 , 102, 6168. ( 13) Klopper, W. ; Bak , K. L. ; Jorgensen, P. ; Olsen, J.; Helgak er , T. J. Phys. B: At . Mol. Opt . Phys. 1 9 9 9 , 32, R103. ( 14) Klopper , W. Mol. Phys. 2 0 0 1 , 99, 481. ( 15) Mart in, J. M. L.; Lee, T. J. Chem . Phys. Let t . 1 9 9 6 , 258, 136. ( 16) Mart in, J. M. L. Chem . Phys. Let t . 1 9 9 6 , 259, 669. ( 17) Helgak er, T.; Klopper, W. ; Koch, H.; Noga, J. J. Chem . Phys. 1 9 9 7 , 106, 9639. ( 18) Halk ier , A.; Helgaker, T.; Jorgensen, P.; Klopper, W. ; Koch, H.; Olsen, J.; Wilson, A. K. Chem . Phy s. Let t . 1 9 9 8 , 286, 243. ( 19) Feller , D.; Dix on, D. A. J. Chem . Phys. 2 0 0 1 , 115, 3484. ( 20) Dunning, Jr.; T. H. J. Chem . Phys. 1 9 8 9 , 90, 1007. ( 21) Kendall, R. A.; Dunning, Jr ., T. H. J. Chem . Phys. 1 9 9 2 , 96, 6796. ( 22) Woon, D. E.; Dunning, Jr., T. H. J. Chem . Phys. 1 9 9 3 , 98, 1358. ( 23) Woon, D. E.; Dunning, Jr., T. H. J. Chem . Phys. 1 9 9 4 , 100, 2975. ( 24) Woon, D. E.; Dunning, Jr., T. H. J. Chem . Phys. 1 9 9 5 , 103, 4572. ( 25) Wilson, A. K. ; Woon, D. E.; Pet erson, K. A.; Dunning, Jr ., T. H. J. Chem . Phys. 1 9 9 9 , 110, 7667. ( 26) Dunning, Jr ., T. H.; Pet erson, K. A.; Wilson, A. K. J. Chem . Phys. 2 0 0 1 , 114, 9244. ( 27) Pet erson, K. A.; Dunning, Jr., T. H. J. Chem . Phys. 2 0 0 2 , 117, 10548. ( 28) Schwart z, C. Phys. Rev. 1 9 6 2 , 126, 1015. ( 29) Kut zelnigg, W.; Morgan, J. D. J. Chem . Phy s. 1 9 9 2 , 96, 4484. ( 30) Sensiain, J. P.; Klippenst ein, S. J.; Miller, J. A. J. Phys. Chem . A 2 0 0 5 , 109, 6045. ( 31) Sensiain, J. P.; Klippenst ein, S. J.; Miller, J. A. J. Phys. Chem . A 2 0 0 6 , 110, 6960. ( 32) Nguy en, T. L.; Ver eeck en, L.; Peet ers, J. J. Phys. Chem . A 2 0 0 6 , 110, 6696.

Chapt er I I : Theor et ical Met hodologies

37

( 33) Pet erson, K. A.; Wilson, A. K.; Dunning, Jr., T. H. J. Chem . Phys. 1 9 9 4 , 110, 7410. ( 34) Wilson, A. K.; Dunning, Jr ., T. H. J. Chem . Phys. 1 9 9 7 , 106, 8718. ( 35) Curt iss, L. A.; Raghavachar i, K.; Redfer n, P. C.; Rassolov , V.; Pople, J. A. J. Chem . Phys. 1 9 9 8 , 109, 7764 and r eferences t her ein. ( 36) Mont gom er y, Jr. , J. A.; Ocht ersk i, J. W. ; Pet ersson, G. A. J. Chem . Phys. 1 9 9 4 , 101, 5900 and r efer ences t her ein. ( 37) Mont gom er y, Jr., J. A.; Fr isch, M. J.; Ocht erski, J. W.; Pet ersson, G. A. J. Chem . Phys. 1 9 9 9 , 110, 2822. ( 38) Mebel, A. M.; Morokum a, K.; Lin, M. C. J. Chem . Phys. 1 9 9 5 , 103, 7414. ( 39) Becke, A. D. J. Chem . Phys. 1 9 9 3 , 98, 5648. ( 40) Lee, C.; Yang, W.; Parr, R. G. Phys. Rev. B 1 9 8 8 , 37, 785. ( 41) Schlegel, H. B. J. Chem . Phy s. 1 9 8 6 , 84, 4530. ( 42) Chen, W.; Schlegel, H. B. J. Chem . Phys. 1 9 9 4 , 101, 5957. ( 43) Celani, P.; Wer ner , H. J., J. Chem . Phys. 2 0 0 0 , 112, 5546. ( 44) Werner, H. J.; Knowles, P. J. J. Chem . Phy s., 1 9 8 8 , 89, 5803. ( 45) Know les, P. J.; Werner, H. J. Chem . Phys. Let t ., 1 9 8 8 , 145, 514. ( 46) Gilbert , R. G.; Sm it h, C. S. Theor y of Unim olecular and Recom binat ion React ions ( Blackwell Scient ific, Ox ford, 1 9 9 0 ) . ( 47) St einfeld, J. I .; Francisco, J. S.; Hase, W. L. Chem ical Kinet ics and Dynam ics ( Prent ice- Hall, Englewood Cliffs, NJ, 1 9 9 9 ) . ( 48) Eckart , C., Phys. Rev. 1 9 3 0 , 35, 1303. ( 49) Johnst on, H. S.; Heicklen, J., J. Phys. Chem . 1 9 6 6 , 66, 532. ( 50) Forst , W. Theory of Unim olecular React ions ( Academ ic Press, New York , 1973). ( 51) Robinson, P. ; Holbrook , K. Unim olecular React ions ( Wiley - I nt erscience, London, 1 9 7 2 ) . ( 52) Holbr ook, K.; Pilling, M. ; Robert son, S. Unim olecular React ions, 2nd edit ion ( Wiley, New Yor k, 1 9 9 6 ) . ( 53) Baer, T. ; Hase W. L. Unim olecular React ion Dy nam ics: Theory and Exper im ent ( Ox ford Universit y Pr ess, Oxford, 1 9 9 6 ) . ( 54) Bey er , T.; Sw inehart D. F. Com m . Assoc. Com put . Machines 1 9 7 3 , 16, 379. ( 55) St ein, S. E.; Rabinov it ch B. S. J. Chem . Phys. 1 9 7 3 , 58, 2438. ( 56) Miller, W. H. J. Am . Chem . Soc. 1 9 7 9 , 101, 6810. ( 57) Guillory , W. A. I nt roduct ion t o Molecular St r uct ure and Spect roscopy, Allyn and Bacon: Bost on, USA, 1 9 7 7 . ( 58) McClurg, R. B.; Flagan, R. C.; Goddard, W. A. J. Chem . Phys. 1 9 9 7 , 106, 6675. ( 59) Pit zer, K. S.; Gw inn, W. D. J. Chem . Phys. 1 9 4 2 , 10, 428. ( 60) Tr uhlar, D. G. J. Com p. Chem . 1 9 9 1 , 12, 266. ( 61) Forst , W. Unim olecular React ions: A Concise I nt roduct ion, Cam br idge Univ ersit y Press: Cam br idge, UK, 2 0 0 3 . ( 62) Ayala, P. Y.; Schlegel, H. B. J. Chem . Phy s. 1 9 9 8 , 108, 2314. ( 63) Wit schel, W.; Hart wigsen, C. Chem . Phys. Let t . 1 9 9 7 , 273, 304. ( 64) Leeb, W. R. Algorit hm 537: Charact er ist ic Values of Mat hieu’s Differ ent Equat ion, ACM Transact ions on Mat hem at ical Soft war e 1 9 7 9 , 5, 112. ( 65) Marst on, C. C.; Balint - Kurt i, G. G. J. Chem . Phy s. 1 9 8 9 , 91, 3571. ( 66) Balint - Kurt i, G. G.; Dixon, R. N. ; Marst on, C. C. I nt . Rev. Phys. Chem . 1 9 9 2 , 11, 317. ( 67) Mey er , R. J. Chem . Phys. 1 9 6 9 , 52, 2053. ( 68) Light J. C.; Ham ilt on I . P.; Lill J. V. J. Chem . Phys., 1 9 8 4 , 82, 1400. ( 69) Colbert D. T.; Miller W. H. J. Chem . Phys., 1 9 9 2 , 96, 1982. ( 70) Shok hir ev, N. V. ; Krasnoperov, L. N. ROTATOR: A Com put er Code t o Calculat e Ener gy Levels of a One- Dim ensional Rot or w it h Arbit rary Pot ent ial, 2003-2004. ( 71) Sum at hi, R.; Carst ensen, H. H.; Green, W. H. Jr. J. Phys. Chem . A 2 0 0 1 , 105, 6910.

38

Chapt er I I : Theor et ical Met hodologies

( 72) Halpern, A. M.; Glendening, E. D. J. Chem . Phy s. 2 0 0 4 , 121, 273. ( 73) Csaszar, A. G.; Szalay , V.; Senent , M. L. J. Chem . Phys. 2 0 0 4 , 120, 1203. ( 74) Knyazev, V. D.; Dubinsky , I . A. ; Slagle, I . R.; Gut m an, D. J. Phys. Chem . 1 9 9 4 , 98, 5279. ( 75) Knyazev, V. D. J. Phys. Chem . 1 9 9 8 , 102, 3916. ( 76) Abram ow it z, M.; St egun, I . A. Handbook of Mat hem at ical Funct ions; Nat ional Bur eau of St andards: Washingt on, DC, 1 9 6 8 . ( 77) Forst , W. J. Com p. Chem . 1 9 9 6 , 17, 954. ( 78) Miller, J. A. Faraday Discuss. 2 0 0 1 , 119, p. 258. ( 79) Br ow n, P. N.; Byr ne, G. D.; Hindm arsh, A. C. SI AM J. Sci. St at . Com put . 1 9 8 9 , 10, 1038. ( 80) Miller, J. A. Faraday Discuss. 2 0 0 1 , 119, 461 and refer ences t herein. ( 81) Frankcom be, T. J.; Sm it h, S. C. J. Theo. Com p. Chem . 2 0 0 3 , 2, 179 and references t herein. ( 82) Knyazev, V. D.; Tsang, W. J. Phys. Chem . A 2 0 0 0 , 104, 10747. ( 83) Venk at esh, P. K. ; Dean, A. M.; Cohen, M. H.; Car r, R. W. J. Chem . Phys. 1 9 9 7 , 107, 8904. ( 84) Venk at esh, P. K. ; Dean, A. M.; Cohen, M. H.; Car r, R. W. J. Chem . Phys. 1 9 9 9 , 111, 8313. ( 85) Mor ushin, V.; Tsang, W. Chem Rat e. A Calculat ional Dat a Base for Unim olecular React ions; Nat ional I nst it ut e of St andards and Technology : Gait hersburg, MD, 2 0 0 0 . ( 86) Gilbert , R. G.; Jordan, M. J. T.; Sm it h, S. C. UNI MOL Pr ogram Suit e, Sydney , Aust ralia, 1 9 9 0 . ( 87) Klippenst ein, S. J.; Wagner, A. F. ; Rober t son, S. H. ; Dunbar , R.; Wardlaw , D. M. Var iFlex Soft ware, v ersion 1.0, Argonne Nat ional Laborat ory, 1 9 9 9 . ( 88) Gillespie D. T. J. Com put . Phy s. 1 9 7 6 , 22, 403. ( 89) Gillespie D. T. J. Phys. Chem . 1 9 7 7 , 81, 2340. ( 90) Gillespie D. T. J. Com put . Phy s. 1 9 7 8 , 28, 395. ( 91) Huyber echt s G. PhD Dissert at ion, Depart m ent of Chem ist ry , Facult y of Science, KULeuv en, 1988. ( 92) Ver eeck en L.; Huy ber echt s G.; Peet ers J. J. Chem . Phy s. 1 9 9 7 , 106, 6564. ( 93) Ver eecken L. PhD Dissert at ion, Depart m ent of Chem ist ry, Facult y of Science, KULeuven, 1999. ( 94) Bark er , J. R. 1 9 8 3 , 77, 301. ( 95) Bar ker, J. R. Mult iWell Com put er Pr ogram Suit e, I nt . J. Chem . Kinet . 2 0 0 1 , 33, 232. ( 96) Bark er , J. R. Mult iWell Program Suit e, v er sion 1.3.2; Ann Arbor, MI , 2 0 0 3 . ( 97) Mat sum ot o M.; Nishim ura T. ACM Trans. Model. Com put . Sim ul. 1 9 9 8 , 8, 3. ( 98) Tardy , D. C. ; Rabinov it ch, B. S. Chem . Rev . 1 9 7 7 , 77, 369. ( 99) Troe J. J. Chem . Phys. 1 9 7 7 , 66, 4758. ( 100) Troe J. J. Chem . Phys. 1 9 7 7 , 66, 4745.

Chapt er I I : Theor et ical Met hodologies

39

Cha pt e r I I I : Qu a n t u m

Ch e m ica l a n d The or e t ica l Kin e t ics

St u dy of t h e O( 3 P) + C2 H 2 Re a ct ion: a M u lt i- St a t e Pr oce ss †

I I I .1 . I n t r odu ct ion Chem ical pr ocesses occurr ing in com bust ion and flam es go t hr ough com plex react ion net w orks norm ally consist ing of sev eral hundr eds and ev en t housands of coupled

elem ent ary

react ions

occur r ing

consecut iv ely

and/ or

in

parallel.

Charact er izing t he dom inant elem ent ary react ions is v ery im port ant in order t o underst and

t he overall r eact ion

m echanism s as well as t o

opt im ize t he

com bust ion process. Acet y lene is k now n t o be a m aj or int erm ediat e in alm ost all hydrocarbon- fueled

flam es. 1

Also,

it

is

well

est ablished

t hat

t he

m aj or

consum pt ion pat hway of acet y lene is r eact ion w it h an oxygen at om in it s gr ound st at e. 2 This react ion plays an

elect r onic t r iplet

im port ant

r ole in

hydrocarbon com bust ion chem ist ry because it leads t o sev er al highly r eact iv e sm all radicals such as HCCO, t r iplet and singlet CH 2 , H, CH and C2 H. Som e of t hose can successfully at t ack closed- shell m olecules such as C2 H 2 and even N2 , or react furt her w it h ot her at om s or radicals in highly exot herm ic react ions, t hus causing

m aj or

flam e

phenom ena

such

as

chem i- ionizat ion

and

chem i-

lum inescence, pr om pt - NO form at ion and product ion of PAH- and soot precursors. 2 Exper im ent al 3- 12 and t heoret ical 13, 14 st udies agree t hat t he pr im ar y product s of t he O( 3 P) + C2 H 2 r eact ion are produced t hr ough t wo channels as present ed below . Exper im ent al react ion ent halpies15 ( ∆r H( 0 K) , in kcal/ m ol) ar e giv en, w hile t he values in par ent heses are obt ained by us using quant um chem ical calculat ions at t he CCSD( T) / 6−311+ + G( 3df,2p) +

ZPE[ CCSD( T) / 6- 311+ + G( d,p) ] lev el of

t heor y ( v ide infra) . O( 3 P) + C2 H 2 → H( 2 S) + HCCO( X2 A″) 3

3

O( P) + C2 H 2 → CH 2 ( X B1 ) + CO

−19.7 ( −19.6)

( I I I .1a)

−47.5 ( −49.1)

( I I I .1b)

I t should be m ent ioned t hat t he hydrogen abst ract ion channel ( O( 3 P) + C2 H 2 → OH( X2 ∏) + HCC( X2 ∑) ) is so highly endot herm ic ( ∆r H( 0 K) ≈ 31 kcal/ m ol, see Table I I I .1) t hat it cannot com pet e w it h t he addit ion/ elim inat ion r out es, even at T = 3500 K. As a r esult , t he abst ract ion is unim port ant under all com bust ion condit ions.



Thanh Lam Nguyen et al, J. Phys. Chem. A 2006, 110, 6696–6706. Chapt er I I I : The O + C2 H 2 React ion

41

The product branching rat io for t he O( 3 P) + C2 H 2 react ion was a subj ect of cont r ov ersy for a long t im e, as det ailed in som e m ore recent work . 2,7,11 Suffice it t o m ent ion her e t hat in som e ear ly w or k t he m inor channel ( I I I .1b) was t hought t o be t he dom inant one. How ev er, all r ecent exper im ent al det erm inat ions, by Michael et al. 7 and by us 8,9 agr ee t hat t he pr oduct s H( 2 S) +

HCCO ar e

3

predom inant over CH 2 ( X B1 ) + CO, wit h t he y ield of t he form er being 80 ± 10% and nearly independent of t em perat ur e for T = 290−1200 K. These r esult s wer e reconfirm ed v er y recent ly in m olecular beam ex per im ent s by Casav ecchia et al. 11 for collision energies of 9.5 kcal/ m ol. The pr oduct dist ribut ion for t his react ion was t heor et ically com put ed ear lier by Harding and Wagner, 14 w it h t he y ield for +10

product s H( 2 S) + HCCO( X2 A″) pr edict ed t o be 70 −35% , closely for ecast ing t he m or e recent ex per im ent al result s m ent ioned. These t heor et ical calculat ions w er e based on an appr ox im at e pot ent ial energy sur face const r uct ed m ainly at t he CI SD+ Q level of t heory but m odified t o m at ch som e exper im ent al ent halpies. 14 Considering t he accuracies obt ainable w it h curr ent quant um chem ical m et hods such as t he coupled- clust er t heory 16 and t he com binat ion m et hods ( Gaussian- 3 t heor y ( G3) , 17 Com plet e Basic Set

( CBS) 18 m odel chem ist ry ) , a t horough

reinv est igat ion of t he pot ent ial energy surface for t he O( 3 P) + C2 H 2 r eact ion from first pr inciples appears t o be in order. Therm al rat e coefficient s for t he react ion of ground st at e at om ic oxygen w it h acet y lene w ere m easur ed ov er a w ide range of t em perat ure: 200−284 K by Bohn and St uhl,

19

21

e.g. at low T =

at m oderat e T by Sheaffer and Zit t el ( 295−873

K) , 20 am ong m any ot hers, and at higher T = 290- 1510 K by Mahm ud and Font ij n 5 as well as at T = 850−1950 K by Michael and Wagner. 7 From t hese r eport s, t he exper im ent al Arr henius act ivat ion energy is derived t o be about 3−3.5 k cal/ m ol. At r oom t em perat ur e, t he t herm al rat e coefficient is well k now n t o be ~ 1.4 × 10 −13 cm 3 m olecule −1 s−1 . 21 Ov erall t herm al r at e coefficient s w er e t heoret ically com put ed

using

conv ent ional

Transit ion

St at e Theor y

exper im ent al act ivat ion energy of 3.3 kcal/ m ol.

7,14

( TST)

adopt ing

t he

I n t hese st udies, t heory and

exper im ent were in good agr eem ent at t em perat ures below 1000 K, t hough t he predict ed rat es w er e about half t hat observ ed in high t em perat ur e shock t ube exper im ent s. The r eason for t his discrepancy w as not clear. 7,14 Theor et ical ab init io quant um chem ical inv est igat ions of t he pot ent ial energy surface of t he O( 3 P) + C2 H 2 r eact ion ar e few. As m ent ioned abov e, Harding et al. 13,14 invest igat ed t he t wo low est - ly ing t r iplet surfaces as part of his t heor et ical kinet ic work, qualit at ively elucidat ing t he r eact ion m echanism as well as showing t he predom inance of channel ( I I I .1a) over channel ( I I I .1b) . Som e st at ionary point s on t he t r iplet PES were charact er ized using t he BAC- MP4 m et hod 22 and t he

42

Chapt er I I I : The O + C2 H 2 React ion

CCSD( T) / cc−pVTZ/ / B3LYP/ 6−31G( d,p) lev el of t heor y. 23 However, t he t r iplet surfaces clear ly need t o be refined furt her t o gain energies wit h sufficient accuracy for accurat e kinet ic calculat ions, in part icular t o invest igat e higher energy pat hways w hich m ight clear up t he under est im at ion of t he pr edict ed rat e coefficient at higher t em perat ur es. Consider ing t he param ount im por t ance of t he react ion of acet y lene w it h at om ic ox ygen in hy drocarbon com bust ion and flam es, we set out t o re- inv est igat e t his r eact ion using coupled- clust er t heor y and t he CBS−QCI / APNO com binat ion m et hod t o const ruct t he t wo low est - ly ing t r iplet surfaces, and t o use t hese in high- lev el t heor et ical kinet ic analyses. The com put ed r esult s will t hen be com par ed w it h t he av ailable exper im ent al dat a. I I I .2 . M et h odology I I I .2 .1 . Qu a n t u m Che m ica l Ca lcu la t ion s Geom et ries and Hessians of st at ionary DFT−B3LYP/ 6−311G( d,p) lev el of t heory

point s wer e obt ained fir st

24,25

at

t he

and t hen used as init ial guesses for

opt im izing at t he coupled- clust er lev el of t heor y [ CCSD( T) ] 26 in com binat ion w it h t he 6−311+ + G( d,p) basis set . 27 Num er ical Hessian calculat ions wer e carr ied out at t he sam e level t o v er ify t he st at ionary point s locat ed ( one im aginary fr equency for a t ransit ion st ruct ure and all posit iv e fr equencies for a m inim um ) and t o obt ain zer o- point vibrat ional energies ( ZPE) and harm onic v ibrat ional fr equencies. To obt ain m or e accurat e r elat ive energies, t he CCSD( T) m et hod in com binat ion wit h t he m uch larger ext ended 6−311+ + G( 3df,2p) basis set 27 was em ployed t o com put e single- point energies ( see Table I I I .1) . Not e t hat t he ZPE[ CCSD( T) ] ar e used unscaled t o cor r ect t he relat ive energies. For som e st at ionar y point s which influence t he react ion k inet ics st rongly, t he effect of basis set size on t he opt im ized geom et r ies and energies was also inv est igat ed by r e- opt im izing at larger basis set s such as 6−311+ + G( 2df,2pd) or cc−pVTZ, 27 follow ed by single- point 6−311+ + G( 3df,3pd)

basis

set

or

CCSD( T) energy calculat ions using t he an

ext r apolat ion

to

a

respect ively . For t he ex t rapolat ions, cc−pCVTZ and cc−pCVQZ

basis 27

set

lim it ,

basis set s were

em ploy ed: 28 CCSD ( T ) HF corr (T ) Elim = EccCCSD it − pCV ( TQ ) Z = Ecc − pCVQZ + Ecc − pCV ( TQ ) Z

wher e,

Ecccorr − pCV ( TQ ) Z =

3 corr 3 Ecccorr − pCVQZ × 4 − Ecc − pCVTZ × 3

43 − 33

( I I I .2)

, w it h

CCSD ( T ) HF corr CCSD ( T ) HF Ecccorr − pCVQZ = Ecc − pCVQZ − Ecc − pCVQZ , and Ecc − pCVTZ = Ecc − pCVTZ − Ecc − pCVTZ .

Because of t he lim it at ions of our curr ent com put at ional r esources, w e could not do single−point energy calculat ions at t he CCSD( T) / cc−pCVQZ lev el. We t her efor e

Chapt er I I I : The O + C2 H 2 React ion

43

approx im at ed energies at t his lev el by a linear ext r apolat ion using t he MP2/ ccpCVQZ lev el 29,30 w it h an addit iv e schem e present ed as follows: CCSD( T) / cc−pCVQZ = CCSD( T) / cc−pCVTZ + [ MP2/ cc−pCVQZ − MP2/ cc−pCVTZ] The r elat ive energies, t abulat ed in Table I I I .2, com put ed at t he CCSD( T) lev el using t hr ee differ ent basis set s for t he im port ant st at ionary point s, ar e in excellent agreem ent w it h each ot her , i.e. a discrepancy of only ~ 0.5 k cal/ m ol and lit t le sensit iv it y t o t he basis set used, indicat ing t hat t he com put ed r elat ive energies in t his w ork ar e nearly conv erged. Table I I I .1: Com put ed Relat ive Energy ( kcal/ m ol) at T = 0 K for Species in t he O( 3 P) + C2 H 2 React ion using t he CBS−QCI / APNO and CCSD( T) / 6−311+ + G( 3df,2p) Lev els of Theory . The Available Exper im ent al Dat a and Values in Lit erat ure Ar e Given for t he Purpose of Com par ison. Species O( 3 P) + C2 H2 H( 2 S) + HCCO( X2 A″) H( 2 S) + HCCO( A2 A′) CH2 ( X3 B1 ) + CO CH2 ( a 1 A1 ) + CO OH( X2 ∏) + HCC( X2 ∑g ) HC( X2 ∏) + HCO( X2 A′) H2 + CCO( X3 ∑g ) Ket ene, H2 CCO( C2v ,X1 A1 ) I n t 1 ( Cs, 3 A″) , OC2 H 2 - t rans I n t 2 ( Cs, 3 A″) , OC2 H 2 - cis I n t 3 ( Cs, 3 A″) , H2 CCO TS1 ( Cs, 3 A″) TS2 ( Cs, 3 A″) TS3 ( C1 , 3 A) TS4 ( Cs, 3 A″) TS5 ( C1 , 3 A) TS6 ( Cs, 3 A″) TS7 ( C1 , 3 A) TS8 ( Cs, 3 A″) I n t 1 - e x( Cs, 3 A′) I n t 2 - e x( Cs, 3 A′) TS1 - e x ( Cs, 3 A′) TS3 - e x ( Cs, 3 A′) TS6 - e x ( Cs, 3 A′)

CBS- QCI / APNO

0. 0 −20. 7 −47. 7 −39. 0 31.9 36.9 −24. 6 −125.6 −51. 7 −50. 4 −71. 4 3. 5 31.7 f) −46. 0 −9. 5 −10. 4 −16. 6 −2. 5 −44. 2 −22. 5 −23. 2 6.1 e) −21. 2 −5. 0

CCSD( T) - 1

a)

0.0 −19.6 −16.5 −49.1 −39.4 31.9 35.9 −24.7 −122.4 −50.6 −49.1 −69.2 5.3

Expt l.

b)

Est im at ion

c)

0.0 −19.7

0.0 19.1 ± 2.5

−47.5 −38.5 30.4 38.4 −22.4 −124.2

−46. 9 ± 0.5

−44.7 −7.1 −7.8 −14.8 0.3 −43.9 −21.0 −20.3 −4.2

a

−59.5 ± 5 −58.7 ± 5 −75.9 ± 4 3.3

CCSD( T)

d)

0. 0

−121.7 −52. 0 −50. 8 −72. 4 −45. 1

−15.6 ± 5 −16.1 ± 5

−7. 0

−42.0 ± 2 −32.1 ± 5

6. 3 ± 1.3

CCSD( T) / 6 −311+ + G( 3df, 2p) / / CCSD( T) / 6 −311+ + G( d, p) + b ZPE[ CCSD( T) / 6 −311+ + G( d, p) ] . Mainly t aken from t he w eb- page: ht t p: / / srdat a.nist . gov / cccbdb/ ; all v alues w ere obt ained at 0 K: ∆H0 f ( 0) = 58.98 k cal/ m ol; ∆H0 f ( C2 H2 ) = 54.48 kcal/ m ol; ∆H0 f ( H) = 51.63 kcal/ m ol; ∆H0 f ( HCCO) = 42.0 k cal/ m ol; 59 ∆H0 f ( CH2 ( X3 B1 ) ) = 93.2 kcal/ m ol; ∆H0 f ( CH2 ( a 1 A1 ) ) = 102.2 kcal/ m ol; ∆H0 f ( CO) = −27.2 kcal/ m ol; ∆H0 f ( OH) = 8.84 kcal/ m ol; ∆H0 f ( HCC) = 135.0 kcal/ m ol; ∆H0 f ( HC) = 141.98 kcal/ m ol; ∆H0 f ( HCO) = 9. 95 kcal/ m ol; ∆H0 f ( CCO( X3 ∑g) ) = 91.1 kcal/ m ol; 60 ∆H0 f ( H2 CCO) = 10.66 k cal/ m ol. c Ref 14. d Ref 23. e Derived from t he G2M( CC,MP2) approach, CBSQCI [ TS1- ex ] = CBS- QCI [ TS1] + { G2M[ TS1- ex] – G2M[ TS1] } = 3.5 + { 7.2 – 4. 6} = 6. 1. f This value = ( G2M + CBS- QB3 + G3 + G3B3) / 4.

44

Chapt er I I I : The O + C2 H 2 React ion

Table I I I .2: Com put ed Relat iv e Energy ( kcal/ m ol) for Som e St at ionary Point s Which Kinet ically Cont r ol t he O( 3 P) + C2 H 2 React ion Using Var ious Lev els of Theory . Species

CCSD( T) - 2

d)

CCSD( T) - 3

e)

CBS- QCI / APNO b)

0. 0 −51. 3 4. 6 7. 2 −8. 4 −9. 3 −16. 2 −1. 2

0.0 −51.7 3.5

0.0 −50.6 5.3

0.0 −51. 1 5.2

0. 0 −50. 2 5. 9

−9.5 −10.4 −16.6 −2.5

−7.1 −7.8 −14.8 0.3

−7.1 −8.4 −14.7 −0.4

−6. 9 −8. 0 −14. 2 0. 3

O( 3 P) + C2 H2 I n t 1 ( Cs, 3 A″) TS1 ( Cs, 3 A″) TS1 - e x ( Cs, 3 A TS4 ( Cs, 3 A″) TS5 ( C1 , 3 A) TS6 ( Cs, 3 A″) TS7 ( C1 , 3 A)

CCSD( T) - 1

c)

G2M ( CC,MP2) a)

a) G2M( CC,MP2) = CCSD( T) / 6- 311+ + G( d, p) + [ MP2/ 6- 311+ + G( 3df, 2p) – MP2/ 6311+ + G( d,p) ] + ZPE[ B3LYP/ 6- 311+ + G( 3df,2p) ] , based on t he B3LYP/ 6- 311+ + G( 3df,2p) opt im ized geom et ry. b) Replacing t he correct ed ZPE[ HF] in t he original CBS- APNO approach by t he ZPE obt ained at t he QCI SD/ 6- 311G( d,p) level and scaled down by 0. 9776. 61 c) CCSD( T) - 1 = CCSD( T) / 6- 311+ + G( 3df, 2p) / / CCSD( T) / 6- 311+ + G( d,p) + ZPE[ CCSD( T) / 6- 311+ + G( d,p) ] . d) CCSD( T) - 2 = CCSD( T) / 6- 311+ + G( 3df,3pd) / / CCSD( T) / 6- 311+ + G( 2df,2pd) + ZPE[ CCSD( T) / 6- 311+ + G( d,p) ] . e) Ext rapolat ing t he CCSD( T) approach t o an infinit e basis set using energies at t he CCSD( T) / cc- pCVQZ and CCSD( T) / cc- pCVTZ levels based on t he CCSD( T) / cc- pVTZ opt im ized geom et ry. CCSD( T) - 3 = HF/ cc- pCVQZ + Ecorr + ZPE[ CCSD( T) / 6- 311+ + G( d,p) ] , where Ecorr = { 4 3 x Ecorr ( cc- pCVQZ) – 3 3 x Ecorr ( cc- pCVTZ) } / { 4 3 – 3 3 } , and CCSD( T) / ccpCVQZ ≈ CCSD( T) / cc- pCVTZ + [ MP2/ cc- pCVQZ – MP2/ cc- pCVTZ] .

Addit ionally, v ar ious com binat ion m et hods such as CBS−QCI / APNO, 18 CBS−QB3, 31 G3B3, 32 G3, 17 and G2M( CC,MP2) 33 w er e applied for t he purpose of com parison wit h t he direct

coupled- clust er calculat ions.

We chose t he CBS−QCI / APNO

approach t o com par e w it h t he CCSD( T) r esult s and for kinet ic calculat ions, as t his level is t hought t o be t he best in t his ser ies of CBS fam ily. Not e t hat we m odified t he

or iginal

CBS−QCI / APNO

appr oach

by

r eplacing

t he

HF−ZPE wit h

t he

QCI SD−ZPE cor r ect ion, as suggest ed ear lier by Radom et al. for radical syst em s. 34 Table I I I .1 shows t hat t he CBS−QCI / APNO r esult s are in good agreem ent w it h t he CCSD( T) values and also w it h t he av ailable exper im ent al dat a. However, a discrepancy of ~ 1 k cal/ m ol as com pared t o ex per im ent st ill r em ains for fragm ent radicals, for w hich heat s of form at ion hav e an uncert aint y of ± 1 k cal/ m ol, or ev en higher. 15 To

check

t he

effect s

of

m ult i−configurat ion

or

near−degeneracies

of

t he

wavefunct ions for st at ionary point s and part icular ly for t he t ransit ion st ruct ur es, we

r e−opt im ized

all

st at ionary

CASSCF( 8,8) / 6−311+ + G( d,p)

level

point s of

t heor y

in 35,36

Fig. and

I I I .1

using

perform ing

t he

analyt ical

Hessian calculat ions as well. The CASSCF calculat ions confirm ed t hat for each of t he species considered in t his paper t he HF−wav efunct ion is dom inant ( i.e. t he CI -

Chapt er I I I : The O + C2 H 2 React ion

45

coefficient of t he m ost im port ant configurat ion is > 0.9) , indicat ing t hat a single−refer ence m et hod should giv e fair result s in all cases. Finally, int r insic react ion coordinat e ( I RC) 37,38 calculat ions wer e done at t he B3LYP/ 6−311G( d,p) level t o est ablish t he corr ect connect ions bet w een t he react ion int erm ediat es.

Figure I I I .1: The t wo lowest - ly ing t r iplet surfaces for t he O( 3 P) + C2 H 2 react ion const ruct ed using t he CBS−QCI / APNO ( and CCSD( T) / 6 −311+ + G( 3df,2p) ) levels of t heor y. The 3 A″ sur face is show n by solid lines, while t he 3 A′ surface is present ed by dashed lines.

The CCSD( T) , QCI SD, CBS−QCI / APNO, DFT−B3LYP, G2M( CC,MP2) , CBS−QB3, G3B3, and G3 calculat ions were perform ed using t he Gaussian 03 package, 39 while t he CASSCF geom et r ies and v ibrat ional fr equencies wer e com put ed using t he Dalt on 40 and Molpr o 41 packages. I I I .2 .2 . RRKM / M a st e r Equ a t ion ca lcu la t ion s Product dist r ibut ions as a funct ion of t em perat ur e and pressur e ( P ”DWP7  298−2000 K) for t he O( 3 P) + C2 H 2 r eact ion proceeding on t he adiabat ic t r iplet surface w er e obt ained by solut ion of t he w eak−collision m ast er equat ion using t he

46

Chapt er I I I : The O + C2 H 2 React ion

exact st ochast ic sim ulat ion m et hod ( ESM) , ex plained in det ail in t he Chapt er I I . The Lennard−Jones collision param et ers for t he bat h gas He are σ = 2.55 Å and ε/ k B = 10 K. 42 Since no collision param et ers for [ C2 H 2 O] ar e available in t he lit erat ur e, t he values σ = 4.08 Å and ε/ k B = 421 K ar e est im at ed based on t hose of et hy lene ox ide C2 H 4 O. 42 Thus, t he collision fr equency Z LJ [ M] was est im at ed at

§×10 10 s −1 at 1 at m ospher e and r oom t em perat ure. An average energy t ransfer red per collision < ∆E> all of −130 cm −1 was adopt ed. As TS6 in t he I nt 2 ( :CHCHO) → TS6 → H( 2 S) + HCCO( X2 A″) st ep is a loose t ransit ion st ruct ur e ( see nex t sect ion) , w e used var iat ional t ransit ion st at e t heor y 43- 48 t o locat e t he k inet ic bot t leneck. The CCSD( T) / 6−311+ + G( d,p) and QCI SD/ 6−311G( d,p) lev els of t heory w ere em ployed t o opt im ize geom et r ies and num er ically

calculat e

harm onic

v ibrat ional

fr equencies

along

t he

react ion

coordinat e ( RC) using const rained opt im izat ions for var ious fixed C−H bond lengt hs; energies along t he RC w er e r efined at t he CCSD( T) / 6−311+ + G( 3df,2p) and CBS−QCI / APNO levels of t heory . Using t his PES, k( E) rat e coefficient s at ev ery posit ion along t he RC w er e com put ed for int er nal chem ical act ivat ion energies E of 54.4 ( = 49.1 + 5.3) or 53.9 ( = 50.4 + 3.5) kcal/ m ol ( see Fig. I I I .1) , r espect iv ely . The m inim al k ( E) was found for a C−H bond dist ance of 1.8 Å, and t he charact er ist ics at t his point along t he RC w ill be used in t he subsequent kinet ic calculat ions. I I I .3 . Re su lt s a n d D iscu ssion I I I .3 .1 . Pot e nt ia l En e rgy Sur faces Unless st at ed ot herw ise, t he CCSD( T) result s will be used for discussion in t his sect ion. The t it le r eact ion is init iat ed eit her by H−at om abst ract ion from acet ylene by t he oxy gen at om , or by elect r ophilic O- addit ion ont o a C−at om in acet ylene, following t he spin conser vat ion r ule ( see Fig. I I I .1) . The H−abst ract ion channel proceeds t hr ough TS2 , which is a v ery lat e, pr oduct −lik e t r ansit ion st r uct ur e w it h a long C−H bond dist ance of 1.567 Å and a short O−H bond of 1.031Å ( see Fig. I I I .3) . Consequent ly, TS2 lies v er y close t o t he pr oduct s OH + HCC, 31.7 kcal/ m ol abov e t he init ial react ant s as com put ed at t he CBS−QB3 lev el. We w er e successful t o locat e TS2 at t he B3LYP and MP2 lev els, but unsuccessful at t he higher lev els CCSD( T) and QCI SD. Thus, t he abst ract ion st ep appears t o be barr ier less. I n any case, because of it s high endot herm icit y of + 31.9 kcal/ m ol, t he H−abst ract ion channel cannot com pet e w it h t he addit ion/ elim inat ion, below, under any com bust ion condit ions. Addit ion of t he oxy gen at om ont o a C−at om in acet ylene can t ake place on t wo different elect r onic st at e surfaces,

3

A″ and

3

A′, v ia TS1 and TS1 - e x leading t o

Chapt er I I I : The O + C2 H 2 React ion

47

I n t 1 ( 3 A″) and I n t 1 - e x ( 3 A″) , r espect iv ely . I RCMax ( G2M: B3LYP) 49 calculat ions confir m ed t hese connect ions, in w hich TS1 ( 3 A″ st at e) lies 2.6 kcal/ m ol low er in energy t han TS1 - e x ( 3 A′ st at e) . The C−O bond dist ances in t hese t w o t ransit ion st ruct ur es ar e about 1.95 Å. We w ill now discuss t he

3

A" and

3

A' surfaces

indiv idually .

Figure I I I .2: Opt im ized geom et r ies obt ained at t he CCSD( T) / 6 −311+ + G( d,p) level of t heory , unless indicat ed ot herw ise, for som e im port ant m inim a in t he O( 3 P) + C2 H 2 r eact ion.

48

Chapt er I I I : The O + C2 H 2 React ion

Figur e I I I .3: Opt im ized geom et r ies obt ained at t he CCSD( T) / 6 −311+ + G( d,p) level of t heory , unless indicat ed ot herw ise, for t ransit ion st ruct ur es in t he O( 3 P) + C2 H 2 r eact ion.

Chapt er I I I : The O + C2 H 2 React ion

49

Th e 3 As elect r onic st a t e sur fa ce. TS1 is a key k inet ic react ion bot t leneck and it s barr ier

height

and

harm onic vibrat ional

fr equencies will

be

used

for

subsequent k ( T) TST calculat ions. To refine it s com put ed charact er ist ics, we carried out I RCMax ( CCSD( T) / 6−311+ + G( 3df,2p) : CCSD( T) / 6−311+ + G( d,p) ) 49 and I RCMax ( CBS−QCI / APNO: QCI SD/ 6−311G( d,p) ) 49 calculat ions on t his TS, y ielding com put ed

barr ier

height s of

5.3

and

3.5

kcal/ m ol at

t he CCSD( T)

and

CBS−QCI / APNO lev els, respect ively . The lat t er value is in good agr eem ent wit h t he experim ent al Arr henius act ivat ion energy of 3−3.5 kcal/ m ol. 14,19,20

Figure I I I .4: HOMO or bit als obt ained at t he QCI SD/ 6- 311G( d,p) lev el for I n t 1 and I n t 1 - e x .

I n t 1 is sit uat ed at 50.6 kcal/ m ol below t he init ial react ant s. Bot h unpaired elect r ons ar e locat ed on t he C−at om w it h one orbit al ly ing in−plane of t he m olecule and anot her out −of−plane ( see t wo HOMOs of I nt 1 in Fig. I I I .4) . As a result , t he C=O dist ance of 1.233 Å in I n t 1 is a double bond and t he C−C of 1.442 Å is closer t o a single bond. How ev er, t he t w o unpaired elect r ons on t he radical carbon slight ly delocalize along t he C−C−O skelet on, r esult ing in addit ional st abilizat ion of I n t 1 . Ther e ar e four possible react ion pat hways from I nt 1 , nam ely : ( i) int ernal r ot at ion ov er 180° about t he C- C ax is t o form I n t 2 v ia TS3 wit h a low bar r ier height of 5.9 kcal/ m ol; ( ii) 1,2−H m igrat ion t o I nt 3 t hr ough t he a t ight t ransit ion st ruct ur e TS5 w it h a bar r ier of 42.8 kcal/ m ol; ( iii) H−elim inat ion t o form final pr oduct s H( 2 S) + HCCO( X2 A″) v ia TS4 ov ercom ing a barr ier of 43.5 kcal/ m ol. TS4 is som ewhat looser t han TS5 , and lies only 0.7 k cal/ m ol higher,

50

Chapt er I I I : The O + C2 H 2 React ion

such t hat t he pat hway v ia TS4 appears m or e fav orable t han TS5 ; ( iv ) redissociat ion back t o t he init ial r eact ant s t hrough TS1 w it h a high barr ier energy of 55.9 kcal/ m ol which m ak es t his st ep unim port ant at any r elevant t em perat ur e. I t should be m ent ioned t hat I nt 1 could do a 1,2−H shift fr om t he cent ral C−at om t o t he O−at om v ia TS9 ( not show n in Fig. I I I .1) leading t o t riplet HCCOH. Howev er, t his st ep faces a huge barr ier of 64.6 kcal/ m ol com put ed at t he CBS−QB3 level, so it is not r elevant and will not be discussed furt her .

Figure I I I .5: Pot ent ial ener gy curv e as a funct ion of t he dihedral angle HCCO in TS1 com put ed at t he CCSD( T) / cc−pVTZ level of t heory .

I n t 2 has an int ernal energy of 49.1 kcal/ m ol r elat ive t o t he init ial react ant s. I t s elect r onic st r uct ur e is sim ilar t o t hat of I nt 1 . While I n t 1 can be consider ed as a t rans- configurat ion w it h t he t wo H−at om s ly ing on opposit e sides of t he C−C bond, I n t 2 has a cis- configurat ion. The t rans

cis isom erizat ion is expect ed t o

occur rapidly in t he chem ically act ivat ed adduct s since t he int er nal r ot at ion barr ier is sm all, about 5−6 kcal/ m ol. A m icro- canonical I nt 1

I nt 2 pre-

equilibr ium m ay t her efor e be est ablished. Not e t hat I nt 2 can only be pr oduced from I n t 1 , but not form ed direct ly by addit ion of O t o acet y lene. At t em pt s t o Chapt er I I I : The O + C2 H 2 React ion

51

search for a dir ect addit ion TS sim ilar t o TS1 were unsuccessful; opt im izat ion always eit her conv erged back t o TS1 or failed t o com plet e. As t his issue is of im port ance ( see below ) , in order t o check t he pot ent ial ex ist ence of t his TS, w e inv est igat ed t he pot ent ial energy curv e as a funct ion of dihedral angle HCCO. St art ing at t he opt im ized geom et ry of TS1 obt ained at t he CCSD( T) / cc−pVTZ level w it h HCCO = 0°, w e increased t he HCCO angle up t o 180° in st eps of 10°. At ev ery new posit ion, a single- point calculat ion w as done at t he sam e CCSD( T) level. The com put ed r esult s, plot t ed in Fig. I I I .5, show t hat t he cis- configurat ion TS is a second- order saddle point , sm oot hly connect ed t o TS1 on eit her side. Ther efor e, a first - order m inim um energy pat hw ay connect ing t he init ial r eact ant s dir ect ly t o I nt 2 is not expect ed t o ex ist . I n t 2 can elim inat e t he H−at om at t he cent er carbon at om form ing t he pr oduct s H( 2 S) + HCCO( X2 A″) . This channel pr oceeds v ia TS6 and faces a bar r ier height of 34.3 kcal/ m ol. TS6 is a very loose saddle point st ruct ure w it h C−H bond dist ance r( C−H) = 1.95 Å. We used var iat ional t ransit ion st at e t heory t o locat e t he r at elim it ing bot t leneck. This kinet ic bot t leneck st ruct ure is som ew hat t ight er w it h r( C−H) = 1.8 Å, i.e. 0.15 Å short er t han in t he saddle point TS6 . I nt 2 isom er izes by a 1,2−H m igrat ion v ia TS7 , a t ight TS, leading t o t r iplet k et ene ( I nt 3 ) aft er clearing a bar r ier of 49.4 kcal/ m ol. Tr iplet ket ene ( I nt 3 ) as form ed from I n t 1 and I n t 2 possesses a high int ernal energy of 69.2 kcal/ m ol. Hence, it is pr edict ed t o decom pose rapidly int o fragm ent s CH 2 ( X3 B1 ) + CO v ia t he low - ly ing TS8 . This channel faces a barr ier of 25.3 kcal/ m ol, about 36 kcal/ m ol low er t han for isom er izat ion back t o I n t 1 or I n t 2 . Therefor e, r e- isom er izat ion is v er y unlik ely. I n sum m ary , calculat ions for t he 3 A″ st at e present ed in Fig. I I I .1 show t hat init ial O- addit ion t o acet y lene leads ent ir ely t o v ibrat ionally excit ed adduct I nt 1 , w hich could quick ly set up a m icrocanonical pr e- equilibr ium wit h it s r ot am er I nt 2 . This near - equilibrium syst em eit her can lose a H−at om form ing final pr oduct s H( 2 S) + HCCO( X2 A″) or can pr oceed furt her by a 1,2−H shift leading t o t riplet k et ene, followed by fast dissociat ion int o final product s CH 2 ( X3 B1 ) + CO. Since TS5 and TS7 for H−m igr at ion ar e t ight er and lie higher in energy t han TS4 and TS6 for H−elim inat ion, t he st eps via t he lat t er t ransit ion st ruct ur es are m or e fav orable t han t he form er. Consequent ly, t he H( 2 S) +

HCCO( X2 A″) pr oduct s yield is

t heor et ically expect ed t o dom inat e t he CH 2 ( X3 B1 ) + CO y ield, in accord wit h exper im ent . 7- 9 I t should be m ent ioned t hat in t heir t heor et ical st udies, Gigard and Chaquin 23 report ed t he ring st ruct ur es t hat could result fr om C2 H 2 + O( 3 P) t o lie very high in

52

Chapt er I I I : The O + C2 H 2 React ion

energy on t he t r iplet surface, such t hat t hey ar e not expect ed t o play any role in t he r eact ion. Th e

3

Ac e le ct r on ic st a t e su rf a ce . Elect r ophilic addit ion of oxygen at om ont o a

C−at om in acet y lene can also pr oceed on t he 3 A′ st at e surface v ia TS1 - e x leading t o I n t 1 - e x . This st ep needs t o ov ercom e a barr ier of 6.1 kcal/ m ol. TS1 - e x was found t o be a first −order saddle point at t he B3LYP, MP2, and CASSCF lev els of t heor y, but a second−order saddle point w it h t wo im aginary w avenum bers at t he CCSD( T) and QCI SD levels. Hence, t he t r ue TS1 - e x saddle point m ay in realit y belong t o t he C1 sym m et ry point gr oup rat her t han Cs, t hus r esult ing in a slight reduct ion in it s energy which should incr ease t he pr edict ed ov erall t herm al rat e coefficient . This is im port ant at high t em perat ures, w her e t heoret ical calculat ions rem ain underest im at es com pared t o exper im ent 7,14 ( see also below ) . I n t 1 - e x has an int er nal energy of 21.0 kcal/ m ol r elat ive t o t he init ial react ant s. I n t 1 - e x is a biradical feat uring t wo unpair ed elect rons, bot h of which m ov e in t he m olecular plane. One is locat ed on t he O−at om , t he ot her on t he C−at om ( see it s t wo HOMOs in Fig. I I I .4) . Consequent ly , t he C=C dist ance of 1.325 Å is close t o a double bond, while how ev er t he r ( C−O) of 1.345 Å is bet ween a single and double bond ( see Fig. I I I .2) . The appar ent slight delocalizat ion of t he unpair ed elect r on of t he O−at om along t he C−C−O skelet on should prov ide som e addit ional st abilizat ion. Ther e is anot her conform er of I n t 1 - e x , t hat is I n t 2 - e x, which has a cis- form and lies 0.7 k cal/ m ol low er . Fast pre- equilibrat ion of t he I n t 1 - e x and I n t 2 - e x isom ers is ex pect ed as t his st ep, v ia TS3 - e x , faces a barrier of only ~ 1−2 kcal/ m ol. Not e t hat t he configurat ion change from t rans- t o cis- form in t his case proceeds by bending t he HCC angle in t he m olecular plane, unlike t he 3 A″ I nt1

I n t 2 isom er isat ion t hat occurs by int er nal r ot at ion. The HCC angle in

TS3 - e x is exact ly equal t o 180° ( see Fig. I I I .3) . Not e t hat no TS could be found t hat dir ect ly connect s t he r eact ant s t o I n t 2 - e x ; m or eover, t he ener gy of TS1 e x - lik e 3 A′ st ruct ur es as a funct ion of t he HCC angle w as found t o show only a single m inim um , at §ƒLHWKHTS1 - e x geom et ry ( see Fig. I I I .2) . This is not inconsist ent w it h t he “ excit ed” ent rance t ransit ion st at e act ually being of C1 sym m et ry . I n t 2 - e x can lose a H−at om at t he cent er C−at om leading t o elect ronically ex cit ed product s H( 2 S) +

HCCO( A2 A′) , about ~ 3 k cal/ m ol abov e t he gr ound st at e

product s H( 2 S) + HCCO( X2 A″) . This st ep proceeds t hr ough TS6 - e x and faces a barr ier height of 18.2 kcal/ m ol. TS6 - e x is a first −order saddle point at t he CASSCF and B3LYP lev els, but a second−order saddle at t he QCI SD and CCSD( T) levels. Again, t his indicat es t hat t he t r ue TS6 - e x saddle point m ay hav e a C1

Chapt er I I I : The O + C2 H 2 React ion

53

sym m et ry , such t hat t he bar r ier of t his st ep is ev en low er and m or e fav or able for decom posit ion of I nt 2 - e x . I n t 1 - e x could also re- dissociat e back t o t he init ial react ant s via TS1 - e x w it h a barr ier height of 28.6 k cal/ m ol. How ev er , TS1 - e x is t ight er and is ~ 11 kcal/ m ol higher t han TS6 - e x , indicat ing t hat re- dissociat ion is disfav or ed. Finally, int ernal conv ersion of I nt 1 - e x t o t he low est - ly ing t r iplet , I nt 1 is v ery unlik ely since t he lifet im e of I n t 1 - ex is est im at ed t o be only ~ 1 ps. Hence, I nt - ex is expect ed t o alm ost

ent ir ely

fragm ent

int o t he elect r onically

excit ed product s H( 2 S)

+

2

HCCO( A A′) , t hus increasing t he H and HCCO pr oduct s yield. This product - form ing pat hway t hrough t he 3 A′VWDWH

QHZO\LGHQWLILHGDVIDUDVZHDUHDZDUH

VKRXOG

be especially im port ant at high t em perat ur es. Not e t hat in t heir inv est igat ion, Harding and Wagner 14 found no cor relat ion of t he excit ed init ial 3 A′ adduct wit h an accessible product s st at e and so concluded t hat it should pr efer ent ially r edissociat e back int o t he init ial r eact ant s, at least at t em perat ur es > 1000 K.

Figure I I I .6: Percent age populat ions of t he TS1 - e x and TS1 t ransit ion st at es for t he init ial addit ion st eps as a funct ion of t em per at ur e.

54

Chapt er I I I : The O + C2 H 2 React ion

I t is of int erest t o ev aluat e t he r elat ive im port ance of t he r out es on t he 3 A′ and 3

A″ surfaces. The r at io of t he t wo rat e coefficient s is given by t he r elat iv e t herm al

populat ion of t he t wo ent rance- channel t ransit ion st at es TS1 and TS1 - e x , and t herefor e by t heir part it ion funct ion rat io. Thus, t he fract ional cont r ibut ion of t he TS1 - e x r out e can be ex pressed as:

Pex =

QTS1−ex × exp( − ETS1−ex / RT ) QTS1−ex × exp( − ETS 1−ex / RT ) + QTS 1 × exp( − ETS1 / RT )

( I I I .3)

The result s plot t ed in Fig. I I I .6 show t hat Pex depends st rongly on t em perat ur es; Pex r ises from a low ~ 1% at 300 K t o ~ 30% at T = 2000 K, indicat ing t hat t he react ion on t he 3 A′ surface y ielding H( 2 S) + HCCO( A2 A′) cont ribut es subst ant ially in flam es. I I I .3 .2 . Ove r a ll Pr im a ry Pr od uct D ist ribu t ion Te m per a t u r e a n d pr e ssu re de p en de n ce . Based on t he t r iplet

3

A″ elect r onic

st at e surface in Fig. I I I .1, a r educed r eact ion schem e for k inet ic calculat ions is present ed in Schem e 1. First , w e carr ied out product dist r ibut ion calculat ions for t he

3

A″ pat hways by

solv ing t he m ast er equat ion under r eact ion condit ions T = 298−1000 K and P ”

at m , where t he available exper im ent al dat a shows a HCCO + H y ield §DQG

CH 2 ( X3 B1 ) + CO y ield § LQGHSHQGHQWRIWHPSHUDWXUH2XUUHVXOWVEDVHGRQ t he CCSD( T) and CBS−QCI / APNO dat a agr ee well w it h each ot her ( see Table I I I .3) . The differ ence bet ween t he t w o appr oaches is negligible ( ~ 1% ) , indicat ing t hat t he com put ed pr oduct dist ribut ion is not sensit iv e t o t he quant um chem ical m et hodology used in t his work. I n t he discussion below we w ill refer t o t he CCSD( T) r esult s. SCHEME 1.

Chapt er I I I : The O + C2 H 2 React ion

55

Table I I I .3 shows t hat t he com put ed pr oduct dist ribut ion is slight ly dependent on t em perat ur e. The H( 2 S) + HCCO( X2 A″) y ield is 93% at 300 K and drops t o 90% at 1000 K, w hile t he CH 2 ( X3 B1 ) + CO y ield r ises from 7% at 300 K t o 10% at 1000 K. I t should be not ed t hat about 82% of t he product s at 300 K t hus com put ed result direct ly from I n t 2 and only 18% fr om I nt 1 ; t his is m ainly due t o t he ~ 7.0 kcal/ m ol low er TS fr om I n t 2 for form ing t he m aj or product s, H + HCCO ( see TS6 versus TS4 in Fig. I I I .1) and t he v er y high I nt 1

I n t 2 int erconv ersion rat es as

giv en by conv ent ional RRKM. Table I I I .3: Com put ed Product s Dist ribut ion ( % ) at 1at m as a Funct ion of Tem perat ur e for t he O( 3 P) + C2 H 2 React ion occurr ing on t he 3 A″ St at e Sur face using Kinet ic Schem e 1. T( K) 298 500 800 1000

H( 2 S) + HCCO( X2 A″) a)

93.1 ( 92.1) 92.4 ( 91.4) 91.1 ( 90.3) 90.2 ( 89.5)

b)

CH2 ( X3 B1 ) + CO 6.9 ( 7.9) 7.5 ( 8.5) 8.6 ( 9.5) 9.3 ( 10.1)

O( 3 P) + C2 H2 0. 0 ( 0. 0) 0. 1 ( 0. 1) 0. 3 ( 0. 2) 0. 5 ( 0. 4)

OC2 H2 0. 0 ( 0. 0) 0. 0 ( 0. 0) 0. 0 ( 0. 0) 0. 0 ( 0. 0)

a) Using t he CCSD( T) / 6- 311+ + G( 3df,2p) PES wit h ZPE from unscaled CCSD( T) / 6311+ + G( d,p) harm onic v ibrat ional frequencies. b) Using t he CBS- QCI / APNO PES wit h ZPE from QCI SD/ 6- 311G( d, p) harm onic vibr at ional frequencies scaled dow n by 0.9538. 61

We exam ined t he possible im pact of H- at om t unneling, using t he asy m m et r ical Eckart pot ent ial, 50 but t he effect s were found t o be unim port ant . For exam ple, at T= 298 K t unneling t r eat m ent s incr ease t he absolut e y ield of t riplet CH 2 by only 0.6% , from 6.9% t o 7.5% , w hile at T= 1000 K t he incr ease of t he t r iplet CH 2 y ield is only 0.4% . This sm all im pact of t unneling is ex pect ed, as t he adduct s have chem ical- act ivat ion int ernal energies t hat ar e m uch higher t han t he barr iers of t he decom posit ion channels. Thus, our com put ed CH 2 ( X3 B1 ) + CO y ield, of 7 – 10% , is about t wice sm aller t han t he experim ent al dat a ( 15% 8,9 − ~ 20% 7 ) ; it w ill be r educed even furt her when t he 3 A′ r eact ion pat h is t aken int o account for T > 1000 K. To find ot her possible source( s) of t he discrepancy , we invest igat ed t he sensit iv it y of t he quant um chem ical result s t o t he basis set size, w hich could influence t he PES and hence t he pr oduct branching rat ios. As can be seen in Fig. I I I .1, t he product s CH 2 ( X3 B1 ) + CO form ed from act iv at ed t r iplet k et ene ar e kinet ically cont r olled by TS5 and TS7 , while t he product s H( 2 S) + HCCO( X2 A″) are form ed t hr ough TS4 and TS6 . Furt her CCSD( T) calculat ions wit h larger basis set s for

56

Chapt er I I I : The O + C2 H 2 React ion

opt im izat ion and energy, t abulat ed in Table I I I .2, show t hat r elat iv e energies for t hese t ransit ion st at es are not sensit iv e t o t he basis set s used, wit hin ~ 0.5 kcal/ m ol, such t hat t he basis set s are unlik ely t o influence t he pr edict ed pr oduct dist ribut ion bey ond ∼2% . Lik ew ise, t he calculat ions at ot her high lev els of t heor y ( v ide infra) agr ee v ery well w it h our CCSD( T) dat a. Not e t hat t he est im at ed worst - case CCSD( T) relat ive- energy err or is about 2 kcal/ m ol. Shift ing up t he posit ion of TS6 by 2.0 kcal/ m ol r esult s in a 3% increase of t he absolut e CH 2 ( X3 B1 ) + CO y ield at T= 298 K, t o 10% , i.e. st ill ~ t w ice sm aller t han t he exper im ent al yield of 20% . To r epr oduce t he lat t er, TS6 w ould hav e t o be shift ed up by 6 kcal/ m ol, i.e. t o t he sam e posit ion as TS4 . Such large CCSD( T) relat iv e- energy er rors on v ery sim ilar st r uct ur es are ext rem ely unlik ely . Hence, ev en when ot her accurat e quant um chem ical levels of t heor y wer e t o be used, t he change requir ed in relat iv e energies or t ransit ion st at e t ight ness t o m at ch

t he exper im ent al dat a

uncert aint y

on

our

is significant ly

quant um

chem ical

larger

dat a.

t han

Rat her,

t he m argins of t he

sy st em at ic

underest im at ion of t he com put ed pr oduct dist ribut ion seem s t o im ply t hat t he branching rat ios of t he O( 3 P) +

C2 H 2 react ion on t he

3

A″ surfaces behav e

non−st at ist ically , i.e. t hat t he st at ist ical RRKM t heory fails for cert ain int erm ediat e st eps. We now at t em pt t o explain t his suspect ed non−st at ist ical behav ior of t he O( 3 P) + C2 H 2 r eact ion. The product dist ribut ion com put ed abov e act ually im plies fast I nt 1 I n t 2 int erconv ersion rat es >10 13 s −1 as giv en by t he st andard RRKM for m alism , and t her efor e a m icr ocanonical near - equilibr ium I n t 1 / I n t 2 rat io close t o unit y at

t he high t ot al int ernal energies of § NFDOPRO LQYROYHG $V FRPSDUHG WR I nt 1 , t he r ot am er I nt 2 faces a m uch lower barr ier for H−at om elim inat ion but a higher one for isom er izat ion t o t riplet ket ene ( see Fig. I I I .1) , such t hat t he react ion flux t hr ough I nt 2 is high and leads alm ost solely t o H( 2 S) + HCCO( X2 A″) and alm ost no CH 2 ( X3 B1 ) + CO. Thus, t he fast I n t 1

I n t 2 conversion im plied by RRKM

increases t he dev iat ion of t he com put ed pr oduct dist ribut ion from t he ex per im ent . Howev er, t he convent ional RRKM form alism assum es t hat t he t ot al int er nal energy of I n t 1 is st at ist ically dist r ibut ed ov er all int ernal m odes, including t he t orsional v ibrat ion corr esponding t o hinder ed int er nal r ot at ion about t he C−C ax is, which gov er ns t he I n t 1

I n t 2 isom erizat ion. I f t his m ode w ere t o r em ain

under- act ivat ed, t he exper im ent al product

branching rat io could easily be

explained. Such a sub- st at ist ical energy part it ioning would gr eat ly slow down I n t 1 → I nt 2 isom erizat ion, allow ing for a lar ger fract ion of t he pr oduct s being for m ed from I n t 1 and hence incr easing t he yield of CH 2 ( X3 B1 ) + CO. Dy nam ic ( t raj ect ory ) calculat ions are r equir ed t o validat e t his assum pt ion, but ar e far

Chapt er I I I : The O + C2 H 2 React ion

57

beyond t he scope of t his st udy. Nonet heless, for t he case at hand such a nonergodic behav ior cannot be dism issed off- hand. For any approach of t he ox y gen, any out - of- plane excit at ion w ould inv olv e t orsional effect s on t he hydrogens. These at om s, however, are light and furt herm or e locat ed v ery close t o t he ax is of t he acet y lene m olecule ev en at dist ances wher e t he presence of t he oxygen at om already dist ort s t he ax ial acet y lene sy m m et ry . Bot h t hese fact ors result in a very low r elat ive m om ent of inert ia, such t hat any out - of- plane im pulse im part ed by t he O- at om is channeled t owards ov erall m olecular rat her t han int er nal rot at ion. Hence, w e pr opose t hat t he rat e of int er nal- r ot at ion isom er isat ion is ham pered by non- st at ist ical energy part it ioning in t he init ial adduct . I n order t o r eproduce t he exper im ent al product branching rat ios in t he fram e of t his assum pt ion, t he specific I nt 1

I n t 2 int erconversion rat es, as obt ained fr om RRKM t heor y, ar e

reduced by scaling dow n t he effect ive t ot al int er nal energy ( E) t hat is av ailable for st at ist ical part it ioning int o t he t orsional m ode of int erest . An effect iv e- energy scaling fact or of 0.4 m at ches t he exper im ent ally observ ed pr oduct dist ribut ion at 300 K. An efficiency fact or of t his m agnit ude appears reasonable for t he case at hand, and w e w ill adopt it ov er t he ent ire energy range. The com put ed result s are present ed in Table I I I .4. Not e t hat t he energy scaling fact or of 0.4 for t he t orsion m ode ent ails t hat t he I nt 1

I n t 2 isom er izat ion rat es, r elat iv e t o t he product

for m at ion rat es, decr ease by a fact or 200 ov er t he short ( §SV OLIHWLPHRIWKH init ial I n t 1 adduct . Yet , t he st eady - st at e populat ion r at io I nt 2 / I nt 1 decreases m uch less, by a fact or of 6.3; t he fr act ion of product s result ing dir ect ly from I nt 1 increases t o 58%

( inst ead of 18% ) , wher eas t he cont r ibut ion from

I nt 2

decreases t o 42% ( inst ead of 82% ) . Table I I I .4: Com put ed Product s Dist ribut ion ( % ) a) as a Funct ion of Tem perat ur e at P = 1 at m for t he O( 3 P) + C2 H 2 React ion Occurr ing on t he 3 A″ Elect ronic St at e Sur face using React ion Kinet ic Schem e 1 and allow ing for Non- St at ist ical Behavior of I nt er nal Rot at ion in I n t 1 b) . T( K) 298 500 800 1000 1200 1500 1800 2000

H + HCCO( X2 A” ) 79.0 77.3 75.2 74.1 73.3 72.2 71.5 71.0

CH2 ( X3 B1 ) + CO 20.9 22.4 24.0 24.6 24.9 25.0 24.7 24.5

O( 3 P) + C2 H2 0. 1 0. 3 0. 8 1. 3 1. 8 2. 8 3. 8 4. 5

OC2 H2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

a) Using t he CCSD( T) / 6- 311+ + G( 3df,2p) PES wit h ZPE from unscaled CCSD( T) / 6311+ + G( d,p) harm onic v ibrat ional frequencies. b) Effect iv e t ot al int ernal energy of I n t 1 available for st at ist ical part it ioning int o t he int er nal rot at ion m ode scaled dow n by a fact or 0. 4 ( see t ext ) .

58

Chapt er I I I : The O + C2 H 2 React ion

I t should be not ed t hat t reat ing t he int ernal r ot at ion m ode in I n t 1 ( or I n t 2 ) as a hinder ed rot or inst ead of a harm onic oscillat ion should lower t he st at ist ical RRKMcalculat ed rat es of t he I n t 1 ↔ I n t 2 isom er isat ion. How ev er t his decrease w ould be at m ost a fact or of § ZKHUHDV WKH H[SHULPHQWDO &+2 / HCCO branching r at io

requir es §WLPHVORZHUUDWHVWKDQWKHFRQYHQWLRQDO 55.0KDUPRQLFRVFLOODWRU approx im at ion. Such a lim it ed r educt ion of t hese large rat es w ould leav e t he

st eady - st at e populat ion rat io I n t 2 / I n t 1 DOPRVWXQFKDQJHG m icrocanonical pr e- equiliEULXP

LHVWLOOFORVHWRWKH

 DQG KHQFH FDQ VKLIW WKH SUHGLFWHG DEVROXWH

product y ields by at m ost 1 % . I n a sim ilar v ein, one should also consider possible anhar m onicit y effect s influencing t he rat es of crit ical r eact ion st eps, t o rat ionalize t he discrepancy abov e. The pot ent ially m ost sensit iv e r eact ion in t his respect is I nt 2 → H( 2 S) + HCCO( X2 A″) v ia TS6 , which carr ies 80% of t he react ion flux in t he conv ent ional harm onic oscillat or RRKM approx im at ion  t he m or e so as TS6 is t he m ost loose of all t ransit ion st ruct ur es involv ed her e and t her efor e appears m ost suscept ible to

v ibrat ion

anharm onicit ies.

How ev er,

t he

m agnit udes

of

t he

vibrat ion

fr equencies of bot h TS6 and I n t 2 are not suggest iv e of im port ant anharm onic effect s. Mor eov er, a det ailed analysis shows t hat t he rat e of t he I n t 2 → H( 2 S) + HCCO( X2 A″) st ep should be a fact or of 6 low er t han t he st at ist ical RRKM value in order t o m at ch t he m easured CH 2 and HCCO yields at room t em perat ure; such a large anharm onicit y effect appears highly unlikely . Anot her possible alt ernat iv e rat ionalizat ion for t he t oo low pr edict ed CH 2 yield t hat needs t o be exam ined, is t hat t r iplet HCCHO ( I nt 1 or I nt 2 ) m ight also undergo com pet it iv e int ersyst em cr ossing ( I SC) t o singlet HCCHO, w hich should rapidly isom er ize t o t he low - ly ing singlet k et ene H 2 C= CO and so y ield singlet CH 2 ( 1 A1 ) + CO. A ca. 10% prim ary CH 2 ( 1 A1 ) y ield by t his hypot het ical r out e w ould explain t he discrepancy bet ween t heor y and ex per im ent , abov e. I t m ust be not ed t hat sm all am ount s of singlet CH 2 ( 1 A1 ) have been obser ved in C2 H 2 / O/ H “ at om ic flam e” syst em s, in t his laborat ory , by laser - induced fluor escence and as well as m olecular - beam m ass spect rom et ry t echniques; how ev er, t he large body of ev idence gat hered on it s form at ion r out e consist ent ly dem onst rat es it t o be a secondar y pr oduct fr om t he fast HCCO + H react ion, while r uling out any significant pr im ar y product ion by t he C2 H 2 + O r eact ion. 6,8,9,51 The absence of significant t r iplet → singlet I SC in t his r eact ion  t hough im port ant I SC was recent ly confirm ed by us in bot h t he C2 H 4 + O( 3 P) and C2 F4 + O( 3 P) react ions52,53  can be at t ribut ed t o ( i) t he fast er chem ical decom posit ion of t he chem ically act ivat ed t r iplet HCCHO adduct I nt 1 , and ( ii) a slow er I SC on account of t he HCCHO t r iplet and singlet surface cr ossing seam s being rest r ict ed t o a narr ow er

Chapt er I I I : The O + C2 H 2 React ion

59

geom et r y range. I ndeed, as show n in Fig. I I I .7, at t he CBS- QB3 lev el of t heory , t he low est singlet HCCHO ( 1 A st at e, point group C1 ) was found t o lie 4 kcal/ m ol abov e I n t 1 ( 3 A″) , feat ur ing a sharp m inim um for a 90° HCCO dihedral angle, j ust t ouching t he t r iplet sur face at t he TS3 saddle point for t he int er nal rot at ion of I n t 1 ( 3 A″) t o I n t 2 ( 3 A″) . Thus, t he negligible I SC t hat follows fr om t he cit ed exper im ent al ev idence is fully consist ent w it h t he sub- st at ist ical act ivat ion of t he int ernal r ot at ion m ode in I nt 1 as put for ward abov e.

Figure I I I .7: Cr ossing seam on t he t r iplet and singlet surfaces for t he HCCO t orsional coordinat e com put ed at t he CBS- QB3 lev el.

60

Chapt er I I I : The O + C2 H 2 React ion

Table I I I .5: Com put ed Ov erall Pr oduct s Dist ribut ion ( % ) React ion at 1 at m as a Funct ion of Tem per at ur e. T( K) 298 500 800 1000 1200 1500 1800 2000

H( 2 S) + HCCO( X2 A″ + A2 A′) c) 79.2 78.7 78.9 79.1 79.3 79.6 79.9 80.0

( 78.1 ( 72.6 ( 64.3 ( 60.0 ( 56.7 ( 53.0 ( 50.4 ( 49.0

+ + + + + + + +

1.1) 6.1) 14. 6) 19. 1) 22. 6) 26. 6) 29. 5) 31. 0)

CH2 ( X3 B1 ) + CO 20.7 21.1 20.5 19.9 19.3 18.3 17.4 16.9

a,b)

for t he O( 3 P) + C2 H 2

O( 3 P) + C2 H2 0.1 0.2 0.6 1.0 1.4 2.1 2.7 3.1

OC2 H2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

a) Using t he CCSD( T) / 6- 311+ + G( 3df,2p) PES wit h ZPE from unscaled CCSD( T) / 6311+ + G( d,p) harm onic v ibrat ional frequencies. b) Effect iv e t ot al int ernal energy of I n t 1 available for st at ist ical part it ioning int o t he int er nal rot at ion m ode scaled dow n by a fact or 0. 4 ( see t ext ) . c) The first and second values in parent hesis give t he H + HCCO cont ribut ions from t he react ions on t he 3 A″ and 3 A′ surfaces, respect ively.

Ther efor e, it appears t hat t he non- ergodic behavior argued abov e offers t he only viable explanat ion for t he observ ed pr oduct dist ribut ion. The pr oduct dist ribut ion so obt ained above for t he 3 A" surface, is t hen com bined w it h t he branching rat io for t he init ial addit ion of t he O−at om wit h acet ylene v ia t he 3 A″ and 3 A′ st at es, t o der iv e t he ov erall pr im ary pr oduct dist r ibut ion as a funct ion of t em perat ure, wher eby it should be not ed again t hat t he r eact ion on t he ex cit ed

3

A′ t r iplet

surface becom es im por t ant only at higher t em perat ur es. The result ing pr oduct dist ribut ion dat a, pr esent ed in Table I I I .5, shows t hat t he t ot al H( 2 S) + HCCO yield is ~ 80% , near ly independent of t em perat ur e ov er t he w ide range 298−2000 K, while t he y ield of CH 2 ( X3 B1 ) + CO w eak ly decreases fr om ~ 21% at 298 K t o ~ 17% at 2000 K, in good agreem ent w it h t he exper im ent al yields for T = 290−1200 K. Howev er , it should be em phasized t hat t he const ancy of t he com put ed y ields wit h increasing t em perat ur e is largely owed t o t he st rongly increasing cont r ibut ion of t he r eact ion on t he 2

3

A′ sur face, w hich pr oduces only

2

H( S) + HCCO( A A′) ( see Table I I I .5) . The fr act ion of re- dissociat ion of init ial adduct s back t o t he react ant s is m inor ( < 5% ) , w hile t here is no st abilizat ion of t riplet int erm ediat es at at m ospher ic pressur es and below . This last result is easily underst ood giv en t hat t he t ransit ion st at es for decom posit ion of t he t r iplet adduct lie m uch lower in energy t han t he addit ion t ransit ion st r uct ures of t he first react ion st ep. The lifet im e of t riplet adduct HCCHO I n t 1 is est im at ed t o be ~ 15 ps at T = 1000 K, and reduces t o ~ 4 ps at T = 2000 K. Mor eover, it r equir es dozens of collisions t o st abilize t his v ibrat ionally excit ed adduct below t he energy level of t he lowest - lying decom posit ion TS. I t can t herefore be pr edict ed t hat t he

Chapt er I I I : The O + C2 H 2 React ion

61

product dist ribut ion does not depend significant ly on pr essur e for pract ical com bust ion sy st em s. Table I I I .6: Calculat ed Microcanonical Rat e Const ant s ( s−1 ) for t he Various St eps in t he O( 3 P) + C2 H 2 React ion Occurr ing on t he 3 A″ Elect ronic St at e Surface under Collision- Free Condit ions in t he Molecular Beam Exper im ent w it h an I nit ial Collision Energy of 9.5 k cal/ m ol. a) k( E) / s−1 k 1 = 2. 13 × 10 10 a) k −1 = 5.10 × 10 1 0 a) k 2 = 1. 68 × 10 10 k −2 = 4.84 × 10 9 k 3 = 2. 68 × 10 9 k −3 = 3.24 × 10 8 k 4 = 3. 52 × 10 10 k 5 = 4. 21 × 10 11 k 6 = 7. 04 × 10 13

React ion st ep I nt 1 → TS3 → I nt 2 I nt 2 → TS3 → I nt 1 I nt 1 → TS5 → I nt 3 I nt 3 → TS5 → I nt 1 I nt 2 → TS7 → I nt 3 I nt 3 → TS7 → I nt 2 I nt 1 → TS4 → H + HCCO( X2 A″) I nt 2 → TS6 → H + HCCO( X2 A″) I nt 3 → TS8 → CH2 ( X3 B1 ) + CO

a) Effect ive t ot al int ernal energy of I n t 1 available for st at ist ical part it ioning int o t he int er nal rot at ion m ode scaled dow n by a fact or 0. 4 ( see t ex t ) .

Und e r collision - fre e condit ion s. Finally , it is of int er est t o com pare our com put ed pr im ary pr oduct

dist r ibut ion wit h t hat

r ecent ly obser ved in t he

collision- fr ee, energy - specific m olecular beam st udy by Casav ecchia and cowor kers. 11 This exper im ent was carr ied out at a collision energy of 9.5 k cal/ m ol. We assum e here t hat t his is convert ed ent irely t o addit ional int er nal v ibrat ion energy of t he init ially form ed t r iplet adduct :CHCHO. Not e t hat a sim ilar av erage t herm al energy of t he react ant s is acquired at a t em perat ure of about 750 K. Micr ocanonical rat e const ant s for various channels in t he O( 3 P) + C2 H 2 react ion com put ed at an int ernal energy of 9.5 kcal/ m ol abov e t he init ial r eact ant s are display ed in Table I I I .6. As can be seen, t he v alue of k 6 is ~ 7×10 13 s −1 , w hich is m uch fast er t han t he rat e of int ernal energy r edist ribut ion ( 10 12 −10 13 s−1 ) . Hence, non- RRKM behav ior is expect ed as t he st at ist ical energy part it ioning assum pt ion breaks down. However, in t his case t he very lar ge value for k 6 sim ply im plies t hat all t he act iv at ed t r iplet ket ene, once form ed, im m ediat ely decom poses t o product s CH 2 ( X3 B1 ) + CO. Sim ilar t o calculat ions for t he t herm al r eact ion above, we also scaled dow n t he effect ive t ot al int ernal energy t hat is av ailable for st at ist ical part it ioning int o t he t orsional m ode by a fact or of 0.4. Solv ing t he m ast er equat ion for t he r eact ion on t he

3

A″ surface y ields 76%

H( 2 S) +

HCCO( X2 A″) and 24% CH 2 ( X3 B1 ) + CO. To com put e t he ov erall prim ary product dist r ibut ion, t he r elat ive cont r ibut ions of t he init ial addit ion st eps need t o be k now n at a collision energy of 9. 5 k cal/ m ol. This rat io can be evaluat ed if t he react ion cross sect ion for t he react ion of t he

62

Chapt er I I I : The O + C2 H 2 React ion

O−at om w it h acet y lene is available. According t o Marcus, 54,55 Mor ok um a, 56 and Lin, 57 an average r eact ion cr oss sect ion at an int er nal energy E for a bim olecular react ion is expr essed as follows:

< σ ( E ) >=

κ h2 G ≠ ( E − E ≠ ) × 8πµ ε (E)

( I I I .4)

wher e κ is t he t r ansm ission coefficient , µ is t he r educed m ass, G≠ is t he sum of st at es for t he addit ion t ransit ion st r uct ur e, and ε is t he energy densit y funct ion for t he init ial r eact ant s, w hich is giv en as:

ε (E) =

γ +i ∞

1 Qint ( β ) β E e dβ ∫ 2π i γ −i∞ β 2

( I I I .5)

wit h β = 1/ k b T, and Qint t he int er nal part it ion funct ion for t he r eact ant s. Using eq I I I .4, a branching rat io ( BR) at an int er nal energy E for t he 3 A′ and 3 A″ react ion channels can be com put ed as t he rat io of t he sum of st at es for TS1 - e x and t he sum of st at es for TS1 .

BR( E ) =

≠ G ≠ ( E − ETS 1− ex ) ≠ ≠ G ( E − ETS 1 )

( I I I .6)

At a collision energy ( Ec = E) of 9.5 kcal/ m ol, BR is com put ed t o be ~ 0.19, t hus result ing in a populat ion of 16% for t he 3 A′ st at e and of 84% for t he 3 A″ st at e. Finally, com bining t hese v alues w it h t he pr oduct branching rat ios on t he 3 A″ and 3

A’ st at e, t he overall prim ary product dist r ibut ion pr edict ed for Ec = 9.5 kcal/ m ol

is der iv ed as 80% ( 79% ± 5% ) for H + HCCO and 20% ( 21% ± 5% ) for CH 2 ( X3 B1 ) + CO, w it h t he exper im ent al dat a giv en in parent heses for t he purpose of com parison.

Again,

our

com put ed

y ields agree w ell w it h

t hose observ ed

exper im ent ally . I I I .3 .3 . Ove r a ll t he rm a l ra t e coe fficie n t The ov erall t em perat ur e- dependent rat e coefficient k ( T) overall for t he O( 3 P) + C2 H 2 react ion can be com put ed according t o t he following expr ession:

k (T )overall = (1 − γ re ) × kTST ( T)

( I I I .7)

wher e γre is t he fract ion of re- dissociat ion of t he init ial adduct s back t o t he init ial react ant s, O( 3 P) +

C2 H 2 . The v alue of γre is a funct ion of pr essure and

t em perat ur e ( see Table I I I .5) . k TST( T) is t he r at e coefficient der iv ed from m ult ist at e t ransit ion st at e t heory :

k (T )TST =

≠ ≠ ≠ kbT QTS≠ 1 exp( − ETS 1 / RT ) + QTS 1− ex exp( − ETS 1− ex / RT ) × h QOQC2 H 2

Chapt er I I I : The O + C2 H 2 React ion

( I I I .8)

63

wher e Q( T) is a com plet e part it ion funct ion, k b is Bolt zm ann’s const ant , h is Planck’s const ant , R is t he univ ersal gas const ant , and ETS1 ≠ and ETS1- ex ≠ are t he barr ier height s, of 3.0 and 5.6 kcal/ m ol used ( see explanat ion below ) for t he init ial addit ion st eps on t he

3

A″ and

3

A' sur faces, respect iv ely . The rot at ional

sym m et r ies for C2 H 2 and t he t ransit ion st at es are 2 and 1, respect ively , such t hat t he react ion pat h degeneracy is 2. The elect ronic part it ion funct ion of t he O at om explicit ly includes t he t hr ee low est - ly ing elect ronic st at es ( 3 P2 wit h elect ronic degeneracy g= 5,

3

P1 wit h g= 3, and

3

P0 w it h g= 1) , w it h relat iv e energies of

0.0000, 0.4525, and 0.6490 kcal/ m ol, respect ively. 58 I n addit ion, t he elect r onic degeneracy of 3 for TS1 and TS1 - e x, bot h having a t r iplet elect ronic st at e, is also t ak en int o account . Alt hough our com put ed CBS−QCI / APNO bar r ier height of 3.5 kcal/ m ol for TS1 is in

agr eem ent

wit h

t he

ex perim ent al

Arr henius

act ivat ion

ener gy

( 3−3.5

kcal/ m ol) , 7,19,20 t he value of k( T) com put ed at room t em perat ur e using t his barr ier height is 6 × 10 −14 cm 3 m olecule −1 s −1 , ~ 2.3 t im es sm aller t han t hat observ ed in exper im ent s:

1.4 × 10 −13 cm 3 m olecule −1 s −1 . 21 Therefore, we

est im at ed t he r elat iv e energy of TS1 in an alt er nat iv e way by forcing k ( T) com put ed using eq 8 t o m at ch t he exper im ent al k at 300 K, but w hile k eeping t he energy differ ence bet w een TS1 and TS1 - e x at 2.6 kcal/ m ol. I n t his w ay, a relat iv e energy of 3.0 k cal/ m ol was obt ained for TS1 , and hence 5.6 k cal/ m ol for TS1 - e x .

Using

t hese

barrier

height s,

bot h

0.5

kcal/ m ol

below

t he

CBS−QCI / APNO values, we com put ed ov erall t herm al rat e coefficient s for t he wide t em perat ur e

range

200−2000K;

t hey

can

be

m odified- Arr henius expr ession k( T) = 6.14 × 10 −15 × T

sum m ar ized 1.28

by

t he

× exp( −1244 K/ T) cm 3

m olecule −1 s−1 . The rat e predict ions are plot t ed in Fig. I I I .8, t oget her w it h t he m or e r ecent exper im ent al dat a for com par ison. Our k ( T) r esult s are in near - per fect agreem ent wit h t he experim ent al values obt ained since 1990 over t he ent ir e range 200 t o 2000K, spanning ov er t hr ee orders of m agnit ude. This excellent agr eem ent is owed t o a large ext ent t o t he new ly charact erized react ion pat hway on t he excit ed 3 A'surface, w hich car r ies ca. 30% of t he r eact ion flux at 2000 K.

64

Chapt er I I I : The O + C2 H 2 React ion

Figure I I I .8: Ov erall t herm al rat e coefficient s com put ed at t em perat ures in t he range of 200 −2000 K. Exper im ent al dat a are given for t he purpose of com par ison.

I I I .4 . Con clu sion s I n t he pr esent t heor et ical st udy, t he t w o low est - ly ing t r iplet pot ent ial energy surfaces for t he O( 3 P) + C2 H 2 react ion are const ruct ed, uniform ly using high levels of t heory such as CCSD( T) and CBS−QCI / APNO. RRKM−Mast er Equat ion calculat ions t o evaluat e prim ary product dist ribut ion wer e carried out using t he exact st ochast ic sim ulat ion m et hod. I n addit ion, ov er all t herm al rat e coefficient s wer e det erm ined using conv ent ional t ransit ion st at e t heor y. A num ber of im port ant r esult s em erge from t his st udy and can be sum m ar ized as follows: ( i) The O( 3 P) + C2 H 2 r eact ion is confirm ed t o occur near- exclusively v ia an elect r ophilic addit ion m echanism as t he first r eact ion st ep; ( ii) The lev els of t heory used in our quant um chem ical calculat ions yield result s in bet t er

agreem ent

w it h

available exper im ent al dat a com pared

to

prev ious

t heor et ical r esult s;

Chapt er I I I : The O + C2 H 2 React ion

65

( iii)

The new ly charact erized r eact ion pat h on t he 2

elect r onically excit ed product s H( S) +

3

A′ surface result s in

2

HCCO( A A′) , and is predict ed t o be

im port ant at high t em perat ures. I t s cont ribut ion t o overall product form at ion is est im at ed t o be ca. 30% at 2000 K; ( iv ) Conv ent ional- RRKM product y ields depart from t he experim ent al branching fract ions by som e 10 percent age point s, suggest ing a non−st at ist ical energy dist ribut ion in t he chem ically act ivat ed init ial adduct t rans- I nt 1 t hat reduces it s rat e of int ernal- rot at ion t o form cis- I nt 2 . When scaling dow n t he I n t 1

I nt 2

rat es so as t o m at ch t he exper im ent al product branching at 300 K, t he com put ed product yields agree well wit h t hose observ ed

ov er t he ent ir e exper im ent al

290−1200 K r egion. I t should be not ed howev er t hat t he predict ed near const ancy of t he product yields over t he w ide 200- 2000 K range, ~ 80% H( 2 S) + HCCO and ~ 20% CH 2 ( X3 B1 ) + CO, is due t o a large ext ent t o t he pat hway on t he excit ed surface, 3 A′; ( v ) Using t he sam e scaling fact or for t he I n t 1 t he pr oduct

dist ribut ion

evaluat ed

for

I nt 2 isom er izat ion as above,

collision- fr ee condit ions is in

good

agreem ent w it h r ecent m olecular beam m easur em ent s at a collision energy of 9.5 kcal/ m ol; 11 ( v i) Reducing t he CBS−QCI / APNO com put ed ent rance channel barr iers by 0.5 kcal/ m ol so as t o fit t he exper im ent al dat a at 300 K, t he com put ed ov erall TST rat e coefficient for t he range 200−2000 K: k( T) = 6.14 × 10 −15 × T exp( −1244

K/ T)

cm

3

m olecule

−1

−1

s ,

is

in

excellent

agreem ent

1. 28

wit h

×

t he

exper im ent al dat a ov er t he ent ir e range. The newly charact er ized r eact ion pat h on t he excit ed

3

A′ sur face account s quant it at ively for t he t oo low earlier

t heor et ical k ( T) pr edict ions at t he higher t em perat ures. 7,14 ( v ii) Of t he ab init io m et hods applied here, CBS−QCI / APNO affords t he best m at ch of t he ex per im ent al energies for t he radical product s as well as t he ent rance t ransit ion st at es.

66

Chapt er I I I : The O + C2 H 2 React ion

Re fe r e nce s ( 1) William s A.; Sm it h D. B. Chem . Revs., 1 9 7 0 , 70, 267. ( 2) Peet ers J. Bull. Soc. Chim . Belg., 1 9 9 7 , 106, 337; and r efer ences t her ein. ( 3) Peet ers J.; Schaeker s M.; Vinck ier C. J. Phy s. Chem ., 1 9 8 6 , 90, 6552. ( 4) Frank P.; Bhaskar an K. A.; Just T. H. 21t h Sym p. ( I nt .) on Com bust ion, 1 9 8 6 , 885. ( 5) Mahm ud, K.; Font ij n, A. J. Phys. Chem ., 1 9 8 7 , 91, 1918. ( 6) Peet ers J.; Vanhaelem eersch S.; Van Hoey m issen J.; Borm s R.; Verm ey len D. J. Phys. Chem ., 1 9 8 9 , 93, 3892. ( 7) Michael J. V.; Wagner A. F. J. Phys. Chem . , 1 9 9 0 , 94, 2453; and references t herein. ( 8) Boullart W.; Peet ers J. J. Phys. Chem ., 1 9 9 2 , 96, 9810. ( 9) Peet ers J.; Langhans I .; Boullart W. I nt . J. Chem . Kinet ., 1 9 9 4 , 26, 869. ( 10) Huang X.; Xing G.; Bersohn R. J. Chem . Phys., 1 9 9 4 , 101, 5818. ( 11) Capozza G. ; Segoloni E.; Leonor i F.; Volpi G. G.; Casavecchia P. J. Chem . Phys., 2 0 0 4 , 120, 4557. ( 12) Chikan V.; Leone S. R. J. Phy s. Chem . A, 2 0 0 5 , 109, 2525. ( 13) Harding L. B. J. Phys. Chem ., 1 9 8 1 , 85, 10. ( 14) Harding L.; Wagner A. F. J. Phys. Chem ., 1 9 8 6 , 90, 2974. ( 15) Fr om NI ST w eb page: ht t p: / / srdat a.nist .gov / cccbdb/ ( 16) Cicek J. J. Chem . Phys., 1 9 6 6 , 45, 4256. ( 17) Curt iss L. A.; Raghavachari K.; Redfer n P. C.; Rassolov V.; Pople J. A. J. Chem . Phys., 1 9 9 8 , 109, 7764. ( 18) Mont gom ery , Jr. J. A.; Ocht erski J. W.; Pet ersson G. A. J. Chem . Phys., 1 9 9 4 , 101, 5900. ( 19) Bohn B.; St uhl F. J. Phys. Chem ., 1 9 9 0 , 94, 8010. ( 20) Sheaffer, P. M.; Zit t el, P. F. J. Phys. Chem . A, 2 0 0 0 , 104, 10194. ( 21) Baulch D. L.; Cobos C. J.; Cox R. A.; Frank P.; Haym an G.; Just Th.; Kerr J. A. ; Murr ells T.; Pilling M. J.; Tr oe J.; Walk er R. W.; Warnat z J. J. Phys. Chem . Ref. Dat a, 1 9 9 4 , 23, 847; and refer ences t herein. ( 22) Melius C. F. unpublished, see ref 14. ( 23) Girard Y.; Chaquin P. J. Phys. Chem . A, 2 0 0 3 , 107, 10462. ( 24) Becke A. D. J. Chem . Phys. 1 9 9 3 , 98, 5648. ( 25) St ev ens P. J.; Devlin F. J.; Chablowsk i C. F.; Fr isch M. J. J. Phys. Chem ., 1 9 9 4 , 98, 11623. ( 26) Raghavachar i K.; Tr ucks G. W. ; Pople J. A.; Head- Gordon M. Chem . Phys. Let t ., 1 9 8 9 , 157, 479. ( 27) EMSL Basis Set Librar y, ht t p: / / ww w. em sl.pnl.gov / form s/ basisfor m .ht m l ( 28) Halk ier A.; Helgak er T.; Jorgensen P.; Klopper W. ; Koch H.; Olsen J.; Wilson A. K. Chem . Phys. Let t ., 1 9 9 8 , 286, 243. ( 29) Möller C.; Plesset M. S. Phys. Rev., 1 9 3 4 , 46, 618. ( 30) Head- Gordon M.; Pople J. A.; Frisch M. J. Chem . Phys. Let t ., 1 9 8 8 , 153, 503. ( 31) Mont gom ery J. A. Jr.; Fr isch M. J.; Ocht erski J. W.; Pet ersson G. A. J. Chem . Phys., 1 9 9 9 , 110, 2822. ( 32) Baboul A. G. ; Curt iss L. A.; Redfern P. C.; Raghavachar i K. J. Chem . Phys., 1 9 9 9 , 110, 7650. ( 33) Mebel A. M.; Mor okum a K.; Lin M. C. J. Chem . Phys., 1 9 9 5 , 103, 7414. ( 34) May er P. M.; Park inson C. J.; Sm it h D. M.; Radom L. J. Chem . Phys., 1 9 9 8 , 108, 604. ( 35) Werner H. J.; Know les P. J. J. Chem . Phys., 1 9 8 5 , 82, 5053. ( 36) Know les P. J.; Wer ner H. J. Chem . Phy s. Let t ., 1 9 8 5 , 115, 259. ( 37) Gonzalez C.; Schlegel H. B. J. Chem . Phys. 1 9 8 9 , 90, 2154. ( 38) Gonzalez C.; Schlegel H. B. J. Phys. Chem . 1 9 9 0 , 94, 5523.

Chapt er I I I : The O + C2 H 2 React ion

67

( 39) Fr isch M. J.; Tr uck s G. W.; Schlegel H. B. et al. Gaussian 03, Gaussian, I nc., Pit t sburgh, PA, ( 2 0 0 4 ) . ( 40) DALTON, a m olecular elect r onic st ruct ure program , wr it t en by Helgaker T.; Jensen H. J. Aa. ; Joergensen P.; Olsen J.; Ruud K.; Aagr en H.; Auer A. A. et al., Release 1.2 ( 2 0 0 1 ) . ( 41) MOLPRO is a package of ab init io pr ogram s wr it t en by Wer ner H.- J.; Know les P. J.; Schüt z M.; Lindh R.; Celani P. ; Korona T.; Rauhut G.; Manby F. R. ; Am os R. D.; Ber nhardsson A.; Ber ning A.; Cooper D. L.,; Deegan M. J. O.; Dobby n A. J. ; Eckert F; et al. ( 2 0 0 2 ) . ( 42) Hippler H.; Troe J; Wendelken H. J. J. Chem . Phys., 1 9 8 3 , 78, 6709. ( 43) Gilbert R. G.; Sm it h C. S. Theory of Unim olecular and Recom binat ion React ions; Blackwell Scient ific: Oxford, 1990. ( 44) Holbrook K.; Pilling M.; Robert son S. Unim olecular React ions, 2nd ed.; Wiley : New York , 1996. ( 45) St einfeld J. I .; Francisco J. S. ; Hase W. L. Chem ical Kinet ics and Dynam ics; Prent ice- Hall: Englew ood Cliffs, NJ, 1999. ( 46) Baer T.; Hase W. L. Unim olecular React ion Dy nam ics: Theory and Exper im ent ; Ox ford Universit y Pr ess: Oxford, 1996. ( 47) Bey er T.; Swinehar t D. F. Com m . Assoc. Com put . Machines, 1 9 7 3 , 16, 379. ( 48) St ein S. E. ; Rabinovit ch B. S. J. Chem . Phy s., 1 9 7 3 , 58, 2438. ( 49) Malick D. K.; Pet ersson G. A. ; Mont gom ery J. A. Jr . J. Chem . Phys., 1 9 9 8 , 108, 5704. ( 50) Miller W. H. J. Am . Chem . Soc. 1 9 7 9 , 101, 6810. ( 51) Peet ers J. ; Devr iendt K. 21t h Sy m p. ( I nt .) on Com bust ion, 1 9 9 6 , 1001. ( 52) Nguyen, T. L.; Vereeck en, L.; Hou, X. J. ; Nguy en, M. T. ; Peet ers, J. J. Phys. Chem . A, 2 0 0 5 , 109, 7489. ( 53) Nguyen, T. L.; Dils, B. ; Car l, S. A.; Vereecken, L.; Peet ers, J. J. Phys. Chem . A, 2 0 0 5 , 109, 9786. ( 54) Marcus R. A. J. Chem . Phys., 1 9 6 6 , 45, 2138. ( 55) Marcus R. A. J. Chem . Phys., 1 9 6 7 , 46, 959. ( 56) Mor ok um a K.; Eu B. C.; Karplus M. J. Chem . Phy s., 1 9 6 9 , 51, 5193. ( 57) Lin S. H.; Lau K. H.; Ey r ing H. J. Chem . Phys., 1 9 7 1 , 55, 5657. ( 58) NI ST web page: ht t p: / / physics.nist .gov / PhysRefDat a/ Handbook / per iodict able.ht m . ( 59) Mordaunt D. H.; Osborn D. L. ; Choi H.; Bise R. T.; Neum ark D. M. J. Chem . Phys., 1 9 9 6 , 105, 6078 ( 60) Choi H.; Mordaunt D. H.; Bise R. T. ; Tay lor T. R.; Neum ark D. M. J. Chem . Phys., 1 9 9 8 , 108, 4070. ( 61) Scot t A. P. ; Radom L. J. Phys. Chem ., 1 9 9 6 , 100, 16502.

68

Chapt er I I I : The O + C2 H 2 React ion

Cha pt e r I V: Pot e n t ia l En e r gy Su r fa ce s, Pr odu ct D ist r ibu t ion s a n d Th e r m a l Ra t e Coe fficie n t s of t h e Re a ct ion of O( 3 P) w it h C2 H 4 ( X 1 A g ) : A Com pr e h e n siv e Th e or e t ica l St u dy †

I V .1 . I n t r odu ct ion The elect rophilic addit ion react ion of t he ox y gen at om O( 3 P) w it h t he sim plest alk ene, C2 H 4 , is an at t ract iv e subj ect for bot h exper im ent al and t heoret ical st udies. There ar e som e int er est ing reasons for r einvest igat ing t his react ion: ( i) t he O( 3 P) + C2 H 4 r eact ion not only plays an im port ant role in t he C2 H 4 / O2 flam e, 13

but also in com bust ion chem ist ry in general since C2 H 4 is a k ey int erm ediat e in

t he ox idat ion of m et hane and of larger hydrocarbons; 4- 6 ( ii) it is t hought t o be also of som e im port ance in som e cases for phot ochem ical air pollut ion; 7 ( iii) t his react ion is of fundam ent al im port ance in chem ical k inet ics and challenging t o t heor et ical chem ist s because of it s com plicat ed react ion m echanism . Since t he ear ly 1950’s, Cv et anov ic and co- w ork ers 8- 13 car r ied out pioneer ing st udies of t he r eact ions of oxy gen at om s w it h olefins in t he gas phase. 8- 13 A general react ion m echanism was suggest ed and sum m ar ized in t hree m aj or st eps. 7 First , oxygen at om s in t heir t r iplet gr ound elect r onic st at e undergo an elect r ophilic addit ion ont o t he C= C bond for m ing adduct s t hat ar e vibrat ionally excit ed t r iplet biradicals corresponding t o t he spin- conserv at ion r ule. Second, t he t riplet biradicals eit her decom pose t o product s ( by H- or CH 2 - loss) or carry out int er- syst em crossing ( I SC) t o singlet biradicals. The first dir ect ev idence for im port ant H product ion from C2 H 4 + O was provided by Ravishankara et al. 14 The t riplet biradicals m ight also isom er ize by a H- m igrat ion, but t his process was found t o face a large barr ier height , 15,16 and cannot t herefor e com pet e w it h t he dir ect decom posit ion. Finally , t he singlet bir adicals produced fr om t he t r iplet adduct s upon I SC m ay eit her convert t o “ hot ” epox ides by r ing closure, or rearrange by int er nal m igrat ion of H- at om s or alky l gr oups int o “ hot ” carbony l com pounds.

These

“ hot ”

com pounds

m ay

t hen

st abilizat ion under high pressur e r eact ion condit ions

eit her 17,18

undergo

collisional

or dissociat e rapidly t o

var ious pr oduct s under low pr essur e condit ions such as in m olecular beam exper im ent s. 19 Ear lier , one of t he present aut hors used dischar ge- flow t echniques in com binat ion wit h m olecular beam m ass spect rom et ry t o m easure t he t herm al rat e coefficient s of t he O( 3 P) + †

C2 H 4 react ion at differ ent t em perat ur es under low pressur e

Thanh Lam Nguyen et al, J. Phys. Chem. A 2005, 109, 7489–7499 Chapt er I V: The O + C2 H 4 r eact ion

69

condit ions ( 0.5- 5 Tor r He) and t o det erm ine t he pr im ar y pr oduct dist ribut ions for t his r eact ion at 0.7- 5 Tor r He and T= 287- 607 K. 20,21, 23 The observ ed room t em perat ur e rat e const ant was ( 6.7 ± 1) x10 –13 cm 3 m olecule –1 s–1 , close t o t he present lit erat ure r ecom m endat ion 22 of 7.5x 10 –13 cm 3 m olecule –1 s–1 . The pr oduct dist ribut ion r esult s were: 48 ± 10 % for CH 3 + CHO, 38 ± 10 % for CH 2 CHO + H, 10 ± 5 % for H 2 CO + CH 2 ( X3 B1 ) , and 4% for H 2 CCO + H 2 ; t he observ ed yields show ed only a slight pressur e dependence and t he influence of t em perat ur e wit hin t his range w as also found t o be sm all. The result s agr eed w ell w it h t he product dist r ibut ion at room t em perat ur e r eport ed pr ev iously by Endo et al. 24 The m aj or possible pr im ary product channels of t he O( 3 P) + C2 H 4 r eact ion ar e present ed below. Wher e possible, exper im ent al r eact ion ent halpies 25 ( ∆r H( 0 K) , in kcal/ m ol) ar e given, w hile t he values in par ent heses are obt ained by us using quant um chem ical calculat ions ( see next sect ion) . O( 3 P) + C2 H 4 ( X1 Ag ) → CH 2 CHO( X2 A” ) + H( 2 S)

( –17.0)

( I V.1)

3

–5.4 ( –6.6)

( I V.2)

2

–27.9 ( –28.8)

( I V.3)

1

–84.2 ( –85.1)

( I V.4)

O( P) + C2 H 4 ( X Ag ) → CH 3 CO( X A’) + H( S)

( –23.5)

( I V.5)

1 +

–116.7 ( –117.4)

( I V.6)

–83.2 ( –83.4)

( I V.7)

–111.9 ( –110.8)

( I V.8)

–100.1 ( –100.7)

( I V.9)

3

1

1

3

1

3

1

1

3

1

2

3

1

O( P) + C2 H 4 ( X Ag ) → H 2 CO( X A1 ) + CH 2 ( X B1 ) 2

O( P) + C2 H 4 ( X Ag ) → CH 3 ( X A2 ” ) + HCO( X A’) O( P) + C2 H 4 ( X Ag ) → H 2 CCO( X A1 ) + H 2 ( X Σg ) 2

1

O( P) + C2 H 4 ( X Ag ) → CH 4 ( X A1 ) + CO( X Σ ) O( 3 P) + C2 H 4 ( X1 Ag ) → Oxirane( X1 A1 ) 3

1

3

1

1

O( P) + C2 H 4 ( X Ag ) → CH 3 CHO( X A’) 1

O( P) + C2 H 4 ( X Ag ) → CH 2 CHOH( X A’)

Som e result s of pr im ary pr oduct dist r ibut ions recent ly obt ained under low pressur e condit ions ar e collect ed in Table I V.1. I t is believ ed 7 t hat t he pr oduct s ( CH 2 CHO( X2 A”) + H( 2 S) and H 2 CO( X1 A1 ) + CH 2 ( X3 B1 ) ) pr oduced in channels ( I V.1) and ( I V.2) m ainly arise fr om t he t r iplet elect r onic surface w her eas t he ot her product s channels result fr om t he singlet elect r onic st at e. Table I V.1 shows t hat at r oom t em per at ur e t her e ar e t w o m aj or react ion channels ( I V.1) and ( I V.3) , w hich pr oduce CH 2 CHO( X2 A” ) + H( 2 S) and CH 3 ( X2 A2 ”) +

HCO( X2 A’) ,

respect ively , and cont ribut e up t o about 90% of t he product s. The r em aining 10% is m ainly account ed for by t he pr oduct s H 2 CO( X1 A1 ) + CH 2 ( X3 B1 ) , while ot her channels are m inor . At high pr essur es, P= 5- 50 bar, subst ant ial y ields of acet aldehyde and ox irane from channels ( I V.7) and ( I V.8) were observ ed. 18 I n addit ion, v iny l alcohol produced fr om channel ( I V.9) was det ect ed in solid Argon. 17 These r esult s show

70

Chapt er I V: The O + C2 H 4 r eact ion

t hat acet aldehyde, ox irane, and v iny l alcohol play a r ole as ( k ey ) int er m ediat es in t he O( 3 P) + C2 H 4 r eact ion. Table I V.1: Pr im ary pr oduct s dist r ibut ion for t he O( 3 P) + C2 H 4 ( X1 Ag ) react ion observ ed under low pressure condit ions Aut hor s, year, r eference Endo et al ( 1986) , Ref. 24 Peet ers et al ( 1988) , Ref. 23 Bley et al ( 1988) , Ref. 18 Koda et al ( 1991) , Ref. 67 Mat sui et al ( 2004) , Ref. 68 Casavecchia et al ( 2004) , Ref. 57 YT Lee et al ( 1989) , Ref. 19 C2 D 4 + O react ion

a) b)

Channel ( I V.1) 40 ± 10%

Channel ( I V.2) 10 ± 5%

Channel ( I V.3) 50 ± 10%

Channel ( I V.4)

38 ± 10%

10 ± 5%

48 ± 10%

4%

50 ± 10%

6 ± 3%

44 ± 10%

56/ 44

46 ± 15%

54 ± 15%

46/ 54

47 ± 4%

53 ± 4%

47/ 53

27 ± 6% ( 29 ± 25% )

16 ± 8%

43 ± 11%

Channel ( I V.5)

Rat io

a)

50/ 50

13 ± 3%

( 71 ± 25% )

48/ 52

1 ± 0. 5%

≈43/ 57 ( ≈30/ 70 b )

rat io of yields from t he t r iplet surface ov er t hose from t he singlet surface as follows from our t heoret ical analysis ( see t ext ) . Measured for C2 D 4 + O.

The O( 3 P) + C2 H 4 r eact ion has been ext ensiv ely st udied using quant um chem ical calculat ions. Yam aguchi et al. 26 used t he UHF/ 4- 31G lev el of t heory t o opt im ize geom et r ies for sev eral low - ly ing biradical st at es of t he r ing- opened ox ir ane ( •CH 2 CH 2 O•) . Yam aguchi 27 t hen r efined t heir previous calculat ions and ex t ensiv ely explor ed t he pot ent ial energy sur face by using t he pr oj ect ed pert urbat ion m et hod PMP2/ 6- 31G( d) / / UHF/ 6- 31G( d) . Dupuis et al. 28 charact er ized som e st at ionary point s for t he elect rophilic addit ion of O( 3 P) t o t he C= C bond of et hy lene using m ult iconfigurat ion Hart r ee- Fock calculat ions. Alt hough low er lev els of t heory w er e used at t hat t im e, t he result s obt ained from t hese calculat ions wer e im port ant and useful t o qualit at ively elucidat e t he r eact ion m echanism . 27,28 Most of t he m aj or channels on t he t riplet energy surface wer e t heor et ically inv est igat ed by Melius 15 using t he BAC- MP4 t heory and r ecent ly charact er ized in det ail by Jursic 16 using t he high lev el CBS- Q m et hod. Som e im port ant r esult s ar isen fr om t hese calculat ions ar e: ( i) t he H- abst ract ion channel fr om C2 H 4 by O( 3 P) leading t o OH( X2 Π) + C2 H 3 ( X2 A’) faces a barrier of 10.4 k cal/ m ol, m uch higher t han t hat of t he addit ion st ep ( 0.4 kcal/ m ol) , such t hat t he form er cannot com pet e w it h t he lat t er at low and fair ly high t em perat ure; ( ii) t he t r iplet biradical adduct form ed from t he addit ion st ep rapidly decom poses int o CH 2 CHO + H, facing a barr ier of

Chapt er I V: The O + C2 H 4 r eact ion

71

only 15.3 k cal/ m ol, and ( iii) t he 1,2- H m igrat ion in t his adduct faces a high energy barr ier of 28.1 k cal/ m ol, and is t herefor e not r elev ant . In

t he

singlet

elect ronic

st at e,

unim olecular

r ear rangem ent s

connect ing

acet aldehyde, hydr oxy et hy lidene ( CH 3 - C- OH) , and v iny l alcohol, follow ed by t heir t herm al decom posit ions t o var ious product s, were com put ed using G1 t heor y by Sm it h et al. 29 As far as w e are aware, prev ious t heoret ical calculat ions hav e howev er not been used t o addr ess t he pr oduct dist r ibut ions and/ or t herm al rat e coefficient s of t he O( 3 P) + C2 H 4 r eact ion. Consider ing it s im port ant r ole in com bust ion chem ist ry as well as it s fundam ent al k inet ic int er est , w e set out t o reinvest igat e t his r eact ion using

differ ent

higher

lev els

of

t heory

such

as

t he

G3, 30

CBS- QB3, 31

G2M( CC,MP2) 32 and MRCI m et hods. 33,34 We hav e const r uct ed t he pot ent ial energy surfaces of t he low est - ly ing t r iplet and singlet elect ronic st at es. Part icular at t ent ion has been paid t o t he pr oduct dist ribut ions and t herm al rat es, w hich wer e com put ed based on t he inform at ion gained fr om t he pot ent ial energy surfaces ( zer o- point - energy correct ed pot ent ial energies, harm onic vibrat ion fr equencies and r ot at ional const ant s) . We also addr essed t he int r iguingly fast t riplet → singlet I SC crossing t hat bears heav ily on t he pr oduct m ak e- up. I V .2 . Th e or e t ica l a ppr oa che s I V .2 .1 . Qu a nt u m ch e m ical ca lcu la t ions Local m inim a and t ransit ion st ruct ures ( TS) on t he pot ent ial energy sur face ( PES) wer e init ially opt im ized using densit y funct ional t heory wit h t he hy brid B3LYP35,36 funct ional in conj unct ion wit h t he 6- 311+ + G( 3df,2p) basis set . 37 Analyt ical harm onic v ibrat ion fr equencies w er e com put ed at t his level in order t o ver ify t he charact er of t he st at ionary point s locat ed ( one im aginary frequency for a TS and all r eal fr equencies for a m inim um ) . Zer o- point energies wer e used un- scaled t o corr ect t he relat ive energies. I n order t o obt ain m or e accurat e r elat iv e energies, t he G2M( CC,MP2) 32 m et hod was used t o com put e single- point elect r onic energies based on t he B3LYP/ 6- 311+ + G( 3df,2p) opt im ized geom et r ies. Addit ionally , t he CBS- QB3 31 and G3 30 m et hods wer e also used. The values com put ed at t he G2M( CC,MP2) , CBS- QB3, and G3 lev els ar e in good agreem ent w it h each ot her , wit hin 1- 2 kcal/ m ol, and wit h available exper im ent al dat a ( see Table I V.2) . I n t his paper, unless ot herw ise m ent ioned, we adopt t he av er age of t he values com put ed at t he last t hree t heoret ical lev els for t he subsequent k inet ic analyses. Our averaged values also agree w ell w it h t hose r eport ed in t he lit erat ure using t he G1 29 and CBS- Q t heor y. 16

72

Chapt er I V: The O + C2 H 4 r eact ion

Ther e are sev eral st at ionary point s of w hich t he wav e funct ions possess a t woreference charact er ( see Table I V.3) . I n t hese cases, t he m ult i- configurat ion CASSCF( 8,8) m et hod, 38,39 in com binat ion wit h t he cor r elat ion consist ent cc- pVDZ basis set , 37 was used t o re- opt im ize geom et r ies and t o perform analyt ical Hessian calculat ions. The r elev ant energies wer e t hen refined by including dynam ic elect r onic correlat ions using t he m ult i- reference int er nally cont ract ed single- and double- ex cit at ion MRCI )

33,34

configurat ion

int eract ion

m et hod

( her eaft er

denot ed

in com binat ion w it h a larger ex t ended cc- pVTZ basis set .

quadr uple cor rect ion ( Q) by Dav idson’s schem e

40

37

as The

was also included t o ov ercom e

t he size- consist ency problem in a t r uncat ed CI . For a set of const rained opt im izat ions w it h fixed C- O bond dist ances or fix ed CCO angles ( see below ) , t he CASSCF( 8,8) / cc- pVDZ m et hod was em ploy ed, and t he energies wer e t hen im pr ov ed by including dynam ic elect ronic corr elat ions using t he m ult i- r efer ence pert urbat ion t heory , CASPT2( 8,8) / cc- pVDZ. 41, 42 The DFT- B3LYP, G2M( CC,MP2) , CBS- QB3, and G3 calculat ions wer e perform ed using t he Gaussian 03 package; 43 t he CASSCF geom et r ies and vibrat ional fr equencies w er e com put ed using t he Dalt on package; 44 t he CASSCF const rained opt im izat ions, CASPT2 and MRCI energies wer e carr ied out using Molpr o 2002. 45 Table I V.2: Calculat ed r elat ive energies a) ( kcal/ m ol, T = 0 K) for various species in t he O( 3 P) + C2 H 4 ( X1 Ag ) r eact ion using different levels of t heor y. B3LYP b) G2M

Species O( 3 P) + C2 H4 ( X1 Ag ) 3

1

CH2 ( X B1 ) + H2 CO( X A1 ) CH2 ( 1 A 1 ) + H2 CO( X1 A 1 ) H( X 2 S) + H2 CCHO( X2 A″) 2

2

CBS–QB3 G3

Aver age

c)

Exp.

d)

0.0

0.0

–7.9

–7.0

–6.1

–6.8

–6.6

–5.4

3.0

2.5

1.8

2.7

2.3

3.6

–19. 7

–16.7

–17.8

–16. 5

–17. 0

–15.2

–17.4

–23. 5

–22.1

–24.4

–14.0

–15.4

–26.7

–29.7

–14. 6

–14. 9

–13. 4

CH3 ( X2 A2 ″) + CHO( X2 A’)

–33. 2

–28.8

–28.4

–29. 1

–28. 8

–27. 9

H2 CCO( X A1 ) + H2 ( X Σg )

–87. 8

–83.9

–86.3

–85. 2

–85. 1

–84. 2

HC≡COH( X1 A′) + H2 ( X1 Σg )

–50. 7

–49.3

–52.6

–51. 0

–51. 0

2

OH( X Π) + C2 H3 ( X A′) 3

0.0

0.0

–14.0

2

g)

0.0

–16.2

CO( X1 Σ+ ) + CH 4 ( X1 A1 )

CBS–Q

0.0

–13. 1

1

f)

0.0

H( X S) + CH3 CO( X A′) H( X 2 S) + CH3 ( X2 A2 ″) + CO( X1 Σ+ ) 1

G1

–25. 2

–24.0

–114.7 –117.3 3.8

8.8

–23.6

–22. 8

–117.8 –117.1 7.3

6.5

–117.4 –116.7 –117.7 10.6 7.5 8.3

7.2 –24.4

°O–CH2 –°CH2 ( A) , I n t 1 a

–29. 3

–23.6

–24.4

–24. 0 –24.0 ( –24.0)

°O–CH2 –°CH2 ( 3 A′) , I n t 1 b

–23. 2

–18.3

–17.2

–19. 3

–18. 3

–38. 4

–32.1

–32.8

–32. 2

–32. 4

–32.9

–37. 6

–32.9

–34.0

–34. 0

–33. 7

–33.6

–34. 4

–24.0

–22.7

–21. 2 –22.6 ( –25.9)

3

CH3 CHO( A) , I n t 2 3

CH2 CHOH( A) , I n t 3 1

°O–CH2 –°CH2 ( A) , I n t 4 CH3 CHO( X 1 A′) , I n t 5

–110.4 –109.8

–111.8 –110.7

Oxir ane( X1 A1 ) , I n t 6

–81. 4

–82.5

CH2 CHOH–a( X1 A′) , I n t 7 a

–99. 8

–98.5

–101.2 –100.4

CH2 CHOH–b( 1 A′) , I n t 7 b

–98. 8

–97.6

–100.3

–84.7

–82. 9 –99. 5

–110.8 –111.9 –110.8 –83. 4

–83. 2

–100.1 –100.7 –99. 1

Chapt er I V: The O + C2 H 4 r eact ion

–99.6

73

CH3 COH–a( X1 A′) , I n t 8 a

–59. 4

–59.1

CH3 COH–b( 1 A′) , I n t 8 b

–56. 9

–56.1

CH3 OCH( 1 A) , I n t 9

–42. 9

–43.1

CH3 OCH( 3 A)

–18. 8

–15.7

TS1 a( 3 A″) TS1 b( 3 A′) TS2 a( 3 A″) 3

TS2 b( A′)

–60.4

–59. 8

–59. 8

–44.0

–43. 2

–43. 5

2.8

1.0

–2.3

3.8

2.2

1.9

2.6

3.8

12.8

9.6

10.5

11.0

–59.9

0.4

1.9

10.4

4.5

13.6

10.1

11.1

11.6

–13. 8

–7.6

–9.9

–8.5

–8.7

TS4 ( A)

–4.7

–1.5

–2.4

–1.6

–1.9

TS5 ( 3 A)

–0.2

8.2

6.3

8.1

7.5

6.4

TS6 ( A)

0.1

5.6

4.1

4.7

4.8

3.7

TS7 ( 3 A)

TS3 ( 3 A) 3

3

–25. 8

–19.0

–19.1

–20. 5

–19. 6

–22.3

3

–18. 5

–14.5

–14.1

–16. 3

–15. 0

–17.9

3

–0.5

9.3

7.2

7.4

TS8 ( A) TS9 ( A) TS1 0 ( 1 A)

8.0 ( –28.4)

TS1 1 ( 1 A)

–30. 4

–22.3

–24.8

–17. 5 –21.5 ( –24.3)

TS1 3 ( 1 A)

–20. 8

–18.4

–19.3

–18. 3 –18.6 ( –24.5)

TS1 4 ( 1 A)

–44. 6

–42.2

–44.1

–43. 2

–43. 2

–43.4

–31. 1

–30.6

–31.9

–31. 5

–31. 4

–31.7 –36.5

1

TS1 2 ( A)

( –25.9)

1

TS1 5 ( A′) 1

TS1 6 ( A)

–37. 5

–35.9

–37.4

–36. 6

–36. 6

TS1 7 ( A)

e)

–41. 1

–34.7

–37.8

–34. 7

–35. 4

TS1 8 ( 1 A)

e)

–30. 7

–26.6

–25.0

–25. 2

–25. 6

TS1 9 ( 1 A)

e)

–23. 1

–18.9

–16.7

–18. 5

–18. 0

1

–33. 3

–28.8

–30.6

–29. 1

–29. 5

–29.8

1

–30. 1

–26.7

–28.0

–27. 6

–27. 4

–27.9

1

TS2 2 ( A)

–11. 5

–12.9

–14.2

–13. 9

–13. 7

TS2 3 ( 1 A)

–6.4

–7.6

–8.6

–6.5

–7.6

TS2 4 ( 1 A′)

–18. 0

–13.4

–15.6

–14. 6

–14. 5

TS2 5 ( 1 A′)

–28. 2

–25.8

–27.5

–27. 1

–26. 8

1

TS2 0 ( A) TS2 1 ( A)

a) Values in t he parent heses obt ained at t he CASSCF( 8,8) / MR–CI SD+ Q( 8,8) level. b) B3LYP st ands for t he B3LYP/ 6–311+ + G( 3df,2p) level. c) Average = ( ∆EG2M + ∆ECBS–QB3 + ∆EG3 ) / 3 d) Experim ent al values ( at T = 0 K) are t aken from t he webpage: ht t p: / / srdat a.nist . gov / cccbdb/ e) Micr ovariat ional t ransit ion st at es. f) Ref. 29 g) Ref. 16

74

Chapt er I V: The O + C2 H 4 r eact ion

Table I V.3: Com put ed t wo m ost im port ant CI - coefficient s ( C1 and C2 ) for wav e funct ions of var ious st at ionary point s using t he CASSCF( 8,8) / cc- pVDZ lev el of t heor y. St ruct ure

C1

I nt 4

C2

–0.80928624

0.56130081

TS10

0.85525651

–0.47847802

TS11

0.80533897

–0.56843161

TS12

–0.88994747

0.40236129

TS13

0.79640488

–0.57531866

I V .2 .2 . RRKM / M a st e r Equ a t ion ca lcula t ion s The chem ically act ivat ed CH 3 CHO( X1 A’) int erm ediat e can dissociat e int o CH 3 + CHO, H + CH 2 CHO or H + CH 3 CO, all t hr ee w it hout ex it barr ier . For t hese channels, var iat ional t r ansit ion st at e t heor y 46- 49 was used t o locat e t he k inet ic bot t leneck. For t his pur pose, first t he UB3LYP/ 6- 311+ + G( 3df,2p) lev el of t heor y was em ploy ed t o opt im ize geom et ries and calculat e v ibrat ion fr equencies along t he react ion coordinat e ( RC) by const rained opt im izat ions w it h fixed C- C or C- H bond lengt hs in CH 3 CHO( X1 A’) . The t ot al energies along t he RC wer e t hen refined by single- point elect ronic energy calculat ions at t he G2M lev el. Using t he r esult ing PES, t he k ( E) at ev er y posit ion along t he RC were com put ed for an int er nal CH 3 CHO energy E of 110.8 k cal/ m ol, corr esponding t o t he decrease in pot ent ial energy r elat iv e t o t he r eact ant s. The m inim al k( E) w er e found for bond dist ances of 2.7 Å for t he C- C bond, 2.6 Å for C- H in t he CH 3 gr oup and 2.7 Å for C- H in t he CHO gr oup, r espect iv ely. The charact erist ics at t hese point s along t he RCs w ill be used in t he subsequent kinet ic calculat ions. I t should be not ed t hat for t he CH 3 CHO( X1 A’) → CH 3 ( X2 A2 ” ) + CHO( X2 A’) channel, one int ernal degree of freedom of t he CH 3 group in t he CH 3 CHO m olecule and in t he var iat ional t ransit ion st ruct ures cor responds t o a hindered int er nal r ot at ion around t he C- C ax is. This m ode was pr oj ect ed out and t reat ed appr opr iat ely as a one- dim ensional hindered int ernal rot at ion hav ing an approx im at e pot ent ial energy funct ion

V=

Vo (1 − cos σ x ) 2

( I V.10)

wher e Vo is t he classical barr ier height of int ernal r ot at ion, σ = 3 t he r ot at ional sym m et ry num ber, and x t he int ernal rot at ion angle. Ther efore, t he sum and densit y of st at es are now t ak en as t he convolut ion of t he densit y of t he onedim ensional hinder ed int er nal r ot or w it h t he sum and densit y of st at es of t he vibrat ion lev els: 46,49

Chapt er I V: The O + C2 H 4 r eact ion

75

E −E ≠

α k(E) = h

∫ 0

Gv≠ ( E − E ≠ − x ) ρ hr≠ ( x )dx ( I V.11)

E

∫ ρ ( E − y )ρ v

hr

( y )dy

0

wher e ρhr is t he densit y of st at es of t he one- dim ensional hindered int ernal rot or which can be eit her dir ect ly count ed if Eint ( int er nal energy ) ≤ 10×Vo based on t he first fift y int er nal r ot at ion energy lev els gained by solv ing t he one- dim ensional Schrödinger equat ion 50, 51 or approx im at ely com put ed using an analyt ical form ula der iv ed by Kny azev 52 for classical one- dim ensional hindered int er nal rot at ions if Eint > 10×Vo . I nput param et ers r equired for t hese calculat ions ar e giv en in Table I V.4. The Bey er - Sw inehart - St ein- Rabinov it ch algor it hm 53,54 was used t o com put e t he sum and densit y of st at es in eq ( I V.11) em ploy ing a grain size of 1 cm –1 . Table I V.4: Com put ed classical bar r ier height ( Vo , in cm –1 ) for t he int er nal rot at ion of t he CH 3 gr oup ar ound t he C- C axis in CH 3 CHO, int ernal- r ot at ion const ant ( B, in cm –1 ) , harm onic vibrat ion frequency fr om t he Hessian ( in cm –1 ) and int er nal rot at ion fr equency at t he pot ent ial m inim um ( ω, in cm - 1 ) using t he UB3LYP/ 6- 31G( d 5d ,p) level for t he CH 3 group in var ious configurat ions, opt im ized at fixed C- C bond dist ances, along t he r eact ion coordinat e of t he CH 3 CHO → CH 3 + CHO channel. RC–C a) CH3 CHO 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

Vo ( cm –1 ) 404. 5 d) 63.7 44.0 31.4 21.4 15.0 11.0 8.9 7.8 6.3 5.3 4.2

Harm . freq. ( cm – 1 ) 156. 5 92.4 83.4 75.3 68.6 61.8 56.6 49.9 41.7 31.8 18.9 13.9

B ( cm –1 ) b) 6. 33 6. 09 6. 06 6. 03 6. 01 5. 99 5. 97 5. 93 5. 90 5. 89 5. 88 5. 86

ω ( cm –1 )

c)

151.8 59.1 49.0 41.3 34.0 28.4 24.3 21.8 20.4 18.3 16.7 14.9

a) C- C bond dist ance ( Å) b) B = h 2 / ( 8 π2 I hr ) , w it h I h r t aken as I CH3 ×I CHO/ ( I CH3 + I CHO) , where I CH3 and I CHO are t he m om ent s of inert ia of CH3 and CHO rot at ing around t he C–C ax is, respect ively. c)

ω = σ Vo B ,

where σ = 3 is t he rot at ional sym m et ry num ber of t he CH3 group.

d) I n good agreem ent wit h Vo = 400 cm –1 observed in experim ent ( Ref. 69) .

The product dist r ibut ions for t he O( 3 P) +

C2 H 4 r eact ion occurring on t he

( separ at e) t r iplet or singlet energy surfaces were obt ained by solv ing t he energy grained m ast er equat ions under various condit ions ( P = 5- 760 Torr , T = 2872000 K) . The Lennard- Jones collision param et ers for t he bat h gas He ar e σ = 2.55 Å and ε/ k B = 10 K while t hose for [ C2 H 4 O] species are t ak en t o be σ = 4.08

76

Chapt er I V: The O + C2 H 4 r eact ion

Å and ε/ k B = 421 K, i.e. sim ilar t o t hose of et hy lene ox ide. 55 The collision fr equency Z LJ [ M] was com put ed t o be 1.1x10 10 s –1 at 1 at m osphere and room t em perat ur e. An av erage energy t ransfer red per collision < ∆E> all of - 130 cm - 1 was chosen. 55 Addit ionally, w e com put ed t he pr im ar y product s dist ribut ion under t he collisionfr ee condit ions ( P≈0 Torr ) in m olecular beam exper im ent s ( MBE) . I n part icular, values at an int er nal energy of 12.9 kcal/ m ol abov e t he init ial r eact ant s hav e been det er m ined in order t o com par e w it h t he recent MBE r esult s of Casavecchia et al. 56- 58

I V .3 . Re sult s a nd D iscussion I V .3 .1 . Pot e n t ia l e n er gy sur f a ce s Th e t r iple t elect r onic st a t e . As first r eact ion st ep, t he O( 3 P) at om m ay eit her abst ract a H at om or add ont o a C at om of t he H 2 C= CH 2 m olecule. These processes proceed on t he t r iplet energy surface cor responding t o t he spinconser vat ion r ule. H- abst ract ion from C2 H 4 by O( 3 P) proceeds v ia t he asym m et r ic TS2 a ( 3 A” ) as well as t he sym m et r ic TS2 b( 3 A’) , w hich bot h dir ect ly cor relat e t o product s OH( X2 Π) + C2 H 3 ( of Fig. I V.1) . Bot h t r ansit ion st at es hav e a high r elat iv e energy of about 11- 12 kcal/ m ol abov e t he init ial r eact ant s and ∼10 kcal/ m ol higher t han t he addit ion t ransit ion st r uct ures, such t hat abst ract ion cannot com pet e w it h addit ion at low t o fair ly high t em perat ur es. Addit ion of O( 3 P) ont o a C at om t akes place v ia bot h TS1 a ( 3 A” ) and TS1 b( 3 A’) , w hich lie only 1.3 and 2.6 kcal/ m ol abov e t he init ial react ant s. These values agr ee w ell w it h t he Arr henius act ivat ion energy of about 2 kcal/ m ol derived from exper im ent . 20-22,59 I t should be indicat ed here t hat TS1 a is a v ery ear ly , react ant - lik e t ransit ion st at e and could not be locat ed using B3LYP; we used t he I RCMax 60 m et hod t o charact erize t his t ransit ion st at e. The addit ion of O( 3 P) v ia TS1 a form s t he t r iplet biradical •

CH 2 CH 2 O•( 3 A) ( her eaft er denot ed as I nt 1 a ) w hich lies 24.0 kcal/ m ol below t he

init ial r eact ant s, w her eas t hat v ia TS1 b leads t o I n t 1 b ( 3 A’) , ly ing 18.3 kcal/ m ol below t he react ant s. I t is expect ed t hat I n t 1 b, aft er being form ed, will undergo facile conv ersion by an int ernal r ot at ion int o I n t 1 a . Because of t he lar ge energy difference, I n t 1 a is expect ed t o com pr ise t he bulk of t he I nt 1 populat ion and t he react ions of t he I n t 1 b form ar e not st udied separat ely . St art ing at I nt 1 a, t her e ar e four possible react ion pat hways: ( i) it can dissociat e int o t he product s CH 2 CHO + H via TS3 w it h a barr ier height of 15.3 k cal/ m ol; ( ii) I n t 1 a can dissociat e int o pr oduct s CH 2 ( X3 B1 ) + H 2 CO by br eak ing t he C- C bond via TS4 , facing a bar rier of 22.1 kcal/ m ol; ( iii) I n t 1 a could undergo a 1,2- H shift ,

Chapt er I V: The O + C2 H 4 r eact ion

77

facing a high bar r ier of 31.5 k cal/ m ol ( TS5 ) and leading t o t riplet CH 3 CHO ( I nt 2 ) ; ( iv ) and finally, I nt 1 a could isom erize t o t r iplet CH 2 CHOH by 1,2- H m igrat ion ov er a high barr ier of 28.8 kcal/ m ol ( TS6 ) . Clearly, t he lower- barr ier dissociat ion r eact ions will far out r un t he isom er izat ions. Since m oreov er t he isom er izat ion TSs lie even above t he ent rance t ransit ion st at es, it is j ust ified t o neglect t hese st eps in our k inet ic analysis in t he next sect ion.

Figure I V.1: Tr iplet pot ent ial energy sur face for t he O( 3 P) + C2 H 4 r eact ion based on t he average r elat iv e energies com put ed at t he G3, CBS- QB3, and G2M lev els of t heory .

78

Chapt er I V: The O + C2 H 4 r eact ion

Tr iplet CH 3 CHO I nt 2 , if form ed, can decom pose v ia t hr ee differ ent channels, of which t he channel v ia TS7 leading t o CH 3 + CHO is predom inant , because of it s low er bar rier height of 12.8 kcal/ m ol. Tr iplet CH 2 CHOH ( I nt 3 ) , being 33.7 kcal/ m ol below t he init ial react ant s, w ill rapidly dissociat e int o product s H 2 CCHO + H wit hout an exit bar r ier. As can be seen fr om t he discussion abov e and fr om Fig. I V.1, t he pr oduct s CH 2 ( X3 B1 ) + H 2 CO and H 2 CCHO + H are found t o be m aj or and pr edom inant on t he t r iplet energy surface whereas CH 3 + CHO are expect ed t o be m inor pr oduct s wit h y ields ≤ 1% . How ev er, t he exper im ent al dat a ( Table I V.1) show t hat t he yield of t he pr oduct s CH 3 + CHO is about 40- 50% depending on t he ex per im ent al condit ions. One m ust t her efor e conclude t hat t he product s CH 3 + CHO arise from t he singlet elect ronic st at e aft er I SC from t he t r iplet surface of t he •CH 2 CH 2 O• biradical t o t he singlet surface, in agreem ent wit h t he v iews of Cvet anov ic et al. 7 Not e t hat t he I SC pr ocess at hand is not collision- induced since t he pr oduct branching rat ios are near ly insensit ive t o t he pr essure ( from 30 m Torr t o 760 Torr ) . 19 I n addit ion, t he product s CH 3 + CHO wer e observ ed in m olecular beam exper im ent s under collision- fr ee condit ions pr eviously by Lee et al. 19 and m or e recent ly by Casav ecchia et al. 56-58

Figure I V.2: Crossing seam s on t he t r iplet and singlet ener gy surfaces com put ed at t he CASPT2/ CASSCF level: ( a) for t he C- O st ret ching coordinat e and ( b) for t he CCO bending coordinat e.

Chapt er I V: The O + C2 H 4 r eact ion

79

Thus it is of int er est t o t heor et ically inv est igat e crossing seam s, which are expect ed t o lie close t o t he st at ionar y point s of bot h t he t r iplet and singlet biradicals



CH 2 CH 2 O•. Not e t hat t he rat ios of each of t he harm onic vibrat ion

fr equencies ( ex cept t he int ernal rot at ion m ode of t he CH 2 group ar ound t he C- C axis) of t he t r iplet and singlet biradicals com put ed at t he CASSCF lev el ar e all close t o unit y, indicat ing t hat bot h energy sur faces are st rongly parallel in t he harm onic vibrat ion space regions in t hese 3N- 7 dim ensions. To illust rat e crossing seam s, we chose t he C- O st ret ching and t he CCO bending coordinat es ( see Fig. I V.2) . First w e car r ied out const rained opt im izat ions at fixed C- O bond lengt hs or fix ed CCO angles using t he CASSCF( 8,8) / cc- pVDZ level; energies wer e t hen refined using t he CASPT2( 8,8) / cc- pVDZ m et hod. Fig. I V.2a shows t he addit ion pat h of O ont o a C= C carbon, cor responding t o t he C- O st ret ching m ode. Aft er passing from t he init ial react ant s t hrough TS1 a on t he t riplet surface, t he t riplet curv e along t he C- O st ret ching coordinat e appr oaches t he singlet curv e and t hey st art ov er lapping at a C- O bond lengt h of about 1.6 Å. Such crossing seam s of t he t wo energy sur faces are a space region lying close t o t he harm onic v ibrat ion region of t he t r iplet bir adical adduct . Fig. I V.2b shows t he bending pat h of t he CCO angle. The t r iplet and singlet cur ves alm ost ov er lap in t he CCO angle r ange from 110 t o 120 degrees. I t t hus t urns out t hat crossing seam s are predict ed t o ex ist

in

t he

( 3N- 7) - dim ensional •

space

regions neighbor ing

t he st at ionary

• 3

st ruct ur e of t he CH 2 CH 2 O ( A) biradical, such t hat high pr obabilit ies ar e expect ed for crossing from t he t r iplet t o t he singlet surface. I t should be em phasized t hat t he I SC process considered her e is one of a highly chem ically act ivat ed species.

80

Chapt er I V: The O + C2 H 4 r eact ion

Figure I V.3: Singlet pot ent ial energy sur face for t he O( 3 P) + C2 H 4 r eact ion based on t he average r elat iv e energies com put ed at t he G3, CBS- QB3, and G2M lev els of t heor y. The values in par ent heses ar e obt ained at t he MRCI lev el. The t r iplet ent rance part is show n by dashed lines.

Th e single t e le ct ronic st a t e. The singlet biradical •CH 2 CH 2 O• ( denot ed as I nt 4 ) , produced by I SC fr om I n t 1 a , lies 25.9 kcal/ m ol ( com put ed at t he MRCI m et hod) below t he init ial r eact ant s and only 1.9 kcal/ m ol lower t han it s t r iplet count erpart . St art ing from I n t 4 , t her e ar e t hr ee possible react ion pat hways ( see Fig. I V.3) , nam ely : ( i) a 1,2- H m igrat ion leading t o acet aldehyde CH 3 CHO ( I n t 5 ) v ia TS1 0 ; t he sm all bar r ier of 0.5 k cal/ m ol at t he CASSCF lev el disappears w hen using t he MRCI lev el, indicat ing t hat I n t 4 has a v ery short lifet im e and isom er izes spont aneously t o t he singlet CH 3 CHO; ( ii) I n t 4 r ing- closur e form ing t he ox irane ( I nt 6 ) v ia TS1 1 w it h a barr ier height of 1.6 k cal/ m ol; ( iii) I n t 4 isom er izat ion t o viny l alcohol CH 2 = CHOH ( I nt 7 a ) by a 1,2- H shift v ia TS1 2 ; t his lat t er st ep is charact er ized as bar r ier less at t he MRCI lev el. Acet aldehyde ( I nt 5 ) can isom er ize t o I nt 6 , I n t 7 a , and 1- hydr ox yet hy lidene CH 3 COH I nt 8 a v ia TS1 3 , TS1 4 , and TS1 5 , r espect iv ely. These st eps face barr iers of 86.3, 67.6, and 79.4 kcal/ m ol, r espect ively . I nt 5 can also direct ly decom pose int o differ ent product s such as CH 3 + CHO and CH 3 CO/ CH 2 CHO + H

Chapt er I V: The O + C2 H 4 r eact ion

81

via loose, var iat ional t r ansit ion st at es w it hout ex it bar riers or t o CH 2 CO + H 2 and CH 4 + CO t hrough t ight t ransit ion st ruct ures w it h ex it bar r iers. As a result , t he product pair CH 3 + CHO em erges as predom inant wher eas t he pair CH 4 + CO is predict ed as m inor ev en t hough t he lat t er pr oduct channel is t herm odynam ically t he m ost fav ored, hav ing indeed t he largest r eact ion ent halpy of –116. 7 kcal/ m ol. Ox irane ( I nt 6 ) can eit her isom er ize ( back ) t o I nt 5 , t o I nt 7 and t o singlet carbene CH 3 OCH v ia TS1 3 , TS2 2 and TS2 3 , respect iv ely, or direct ly dissociat e int o pr oduct s CH 2 ( 1 A1 ) + H 2 CO w it hout ex it barr ier . As can be seen in Fig. I V.3, t he isom er izat ion of I nt 6 back t o I nt 5 faces t he low est barr ier and, accordingly , t his st ep should predom inat e.

Sim ilar t o I n t 6 , t he I n t 7 a and I nt 8 a species

produced from t he abov e- m ent ioned st eps rapidly isom erize back t o I n t 5 v ia low ly ing t ransit ion st ruct ur es TS1 4 and TS1 5 . Dir ect H 2 - elim inat ion from I n t 7 a and I n t 8 a ar e also possible leading t o CH 2 CO + H 2 via TS2 4 and TS2 5 . The lat t er TSs however lie subst ant ially higher t han TS1 4 , TS1 5 and TS1 6 . Ov erall, t he t heoret ical result s obt ained for t he singlet energy surface indicat e t hat isom er izat ion pr ocesses t ake place v ia t ransit ion st r uct ur es t hat ar e lying low in energy , such t hat ext ensiv e int er nal r earr angem ent s of t he singlet [ C2 H 4 O] syst em should occur befor e final fragm ent at ion t o end pr oduct s at low pressur es or collisional st abilizat ion at high pr essur es.

I V .3 .2 . Pr od uct dist ribu t ion Te m per a t u r e a n d pr essur e dep e nd e nce . The part ial pr oduct dist ribut ions from t he t riplet and singlet •CH 2 CH 2 O• adduct s wer e der ived separat ely by solv ing t he appr opr iat e m ast er equat ions independent ly. The r esult s obt ained under var ious react ion condit ions ( T= 287- 2000 K and P= 5- 760 Torr ) are pr esent ed in Tables I V.5 and I V.6. They are independent

of pr essur e ov er t he range

consider ed, but vary as a funct ion of t em perat ure. Our pr edict ions agr ee well w it h t he exper im ent al observat ions ( see Table I V.1) . We perform ed som e sam ple calculat ions at higher pressur es t o search t he onset of t he fall- off region, but found only a sm all fr act ion of st abilizat ion ( < 10% ) even at 100 at m for com bust ion

t em perat ures,

T∼1500

K.

Hence,

w e w ill

focus her e on

t he

afor em ent ioned t em per at ur e and pressur e region. For t he t r iplet st at e, t he y ield of H + CH 2 CHO from I nt 1 a m ark edly decr eases from 89% at 287 K t o 35% at 2000 K, w hile t he y ield of CH 2 ( X3 B1 ) + H 2 CO increases from 11% t o 65% . The yield of collisionally st abilized t r iplet adduct is negligible even at P= 760 Torr , as t he collision frequency of 10 10 s–1 is m uch sm aller t han t he com bined rat e of t he

decom posit ion channels ( § ×10 11 s–1 at r oom t em perat ur e) and dozens of

82

Chapt er I V: The O + C2 H 4 r eact ion

collisions are required t o br ing t he adduct ’s t ot al energy below t he lev el of t he lowest - lying decom posit ion TS. I n fact , t he t r iplet biradical barely looses any of it s init ial energy during it s lifet im e. Not e how ev er t hat t he lifet im e of t he t r iplet adduct of about 8 ps should be long enough for quasi- st at ist ical dist r ibut ion of t he vibrat ion energy ov er all m odes. The fract ion of adduct s t hat r e- dissociat e back int o t he init ial r eact ant s is also sm all, < 1% . I t t urns out t hat isom er izat ion processes of t he t r iplet adduct I n t 1 a t o I n t 2 and I n t 3 cont r ibut e less t han 1% ev en at 2000 K, since TS5 and TS6 lie m uch higher in energy t han TS1 a ( see Fig. I V.1) . For t he singlet st at e, t he calculat ed y ield of collisionally st abilized [ C2 H 4 O] is sim ilarly sm all and negligible. When t he t em perat ur e incr eases fr om 287 K t o 2000 K, t he yield of CH 3 + CHO drops consider ably fr om 87% t o 67% , while t he yields of CH 3 CO + H, CH 2 CHO + H, H 2 + H 2 CCO and CH 4 + CO each increase t o

§5% .

Tables I V.5 and I V.6 show t hat H + CH 2 CHO and CH 2 ( X3 B1 ) + H 2 CO are m aj or product s from t he t riplet



CH 2 CH 2 O• adduct , wher eas t he singlet adduct yields

predom inant ly CH 3 + CHO. To com put e t he ov erall pr oduct dist r ibut ion fr om t he part ial, separat e dat a obt ained abov e, t he int ersyst em crossing bet ween t he t riplet and singlet ener gy surfaces m ust be t aken int o account , in com pet it ion wit h t he chem ically act ivat ed r eact ions of t he adduct s. Thus, infor m at ion is needed on t he I SC rat es at t he cr ossing seam s of t he t wo energy surfaces. To com put e t hese rat es, t raj ect or y dynam ic calculat ions, e.g. “ on t he fly ” nonadiabat ic dynam ics, 61 are cert ainly r equired. However, such calculat ions ar e far beyond t he scope of t his st udy. Here, w e w ill proceed in a differ ent way: m ak ing use of t he rat es of t he unim olecular r eact ions der iv ed fr om RRKM t heory , we w ill est im at e t he I SC r at e based on t he exper im ent ally observ ed ov erall pr im ar yproduct s ( collect ed in Table I V.1) .

Chapt er I V: The O + C2 H 4 r eact ion

83

Table I V.5: Com put ed prim ary pr oduct s dist r ibut ion ( % ) ev olv ing fr om t he t r iplet CH 2 CH 2 O• biradical on t he t r iplet sur face under var ious r eact ion condit ions.



P ( Tor r)

287

5

88.7

11.3

0. 0

760

88.8

11.2

0. 0

5

88.4

11.6

0. 0

760

88.5

11.5

0. 0

5

81.5

18.4

0. 1

760

81.5

18.4

0. 1

5

76.9

23.0

0. 1

760

76.9

23.0

0. 1

5

60.0

39.7

0. 3

760

60.0

39.7

0. 3

5

44.2

55.2

0. 6

760

44.2

55.2

0. 6

5

34.7

64.6

0. 7

760

34.7

64.6

0. 7

298 500 607 1000 1500 2000

H + CH2 CHO

CH2 ( X3 B1 ) + H2 CO

T ( K)

O + C2 H4

Table I V.6: Com put ed prim ary pr oduct s dist r ibut ion ( % ) ev olv ing fr om t he singlet CH 2 CH 2 O• biradical on t he singlet surface under var ious react ion condit ions.



T ( K) P ( Tor r) CH3 + CHO CH3 CO + H CH2 CHO + H H2 + H2 CCO CH4 + CO 287 298 500 607

5

86.5

4.1

0.7

4.4

4.3

760

86.9

4.0

0.6

4.3

4.2

5

86.5

4.1

0.7

4.4

4.3

760

86.8

4.1

0.6

4.3

4.2

5

85.2

4.6

0.9

4.7

4.6

760

85.4

4.6

0.8

4.7

4.5

5

84.3

5.0

1.0

5.0

4.7

760

84.5

4.9

1.0

4.9

4.7

80.1

6.4

1.8

6.1

5.6

80.1

6.4

1.8

6.1

5.6

73.6

8.5

3.4

7.9

6.6

73.6

8.5

3.4

7.9

6.6

67.1

10.5

5.2

9.9

7.3

67.1

10.5

5.2

9.9

7.3

1000 5 760 1500 5 760 2000 5 760

Our t heoret ical result s clearly show t hat t he product s CH 2 CHO + H and H 2 CO + CH 2 ( X3 B1 ) of channels ( I V.1) and ( I V.2) ar ise from t he t riplet elect ronic sur face wher eas t he ot her product channels can only r esult from t he singlet elect r onic st at e. Com bining t his w it h t he exper im ent al product yields present ed in Table I V.1, one has t o conclude t o a rat io for t ot al t riplet and t ot al singlet yields of about 45% versus 55% , w it h an uncert aint y m argin of 5% each.

84

Chapt er I V: The O + C2 H 4 r eact ion

Anot her quest ion t hat rem ains open is whet her t he crossing rat es and t heir rat io depend on int ernal ener gy. According t o pr ev ious exper im ent s carried out by us, 23 and alw ays in t he light of t he present t heor et ical findings, t he rat io for t riplet yields and singlet yields is only w eak ly dependent on t em perat ur e, it s value barely changing from 45/ 55 at 287K t o 43/ 57 at 607K. For a recent m olecular beam st udy by Casav ecchia 56–58 at a collision energy of 12.9 kcal/ m ol, t his rat io am ount s t o 43/ 57. I t t her efor e appears t hat t he ov erall effect of crossing is not ov er ly

sensit ive

to

t he

int er nal

energy

of



CH 2 CH 2 O•,

alt hough

furt her

exper im ent al st udies ov er a w ider t em perat ur e range ar e necessary t o fully set t le t his quest ion. I n t his wor k, we used t he value of 45/ 55 for t he r at io in our following calculat ions of t he rat e of t he I SC crossing as w ell as of t he overall product branching rat ios. To est im at e t he r at e of t he I SC cr ossing of t he chem ically act ivat ed



CH 2 CH 2 O• radical, w e use k inet ic Schem e 1 present ed

below :

O(3 P) + C 2 H 4



CH 2CH 2 O •( 3A)

k f,ISC k r,ISC



CH 2CH 2 O •( 1A)

k SP

k TP Products (triplet)

Products (singlet)

Scheme 1.

wher e k TP and k SP ar e t he overall disappearance rat es of t r iplet •CH 2 CH 2 O• on t he t riplet surface and of singlet •CH 2 CH 2 O• on t he singlet sur face, r espect ively ; k f,I SC and k r, I SC ar e t he forward and r ev erse rat es of I SC crossing, respect ively. At or near r oom t em perat ur e, w e hav e k TP ≈ 1.3×10 11 s –1 ( see above) w hereas k SP is found t o be very high, ≈2.7×10 13 s –1 . Alt hough t his k SP value m ay be im precise, it s large m agnit ude shows t hat all singlet adduct s, once form ed, im m ediat ely evolve int o furt her int er m ediat es ( CH 3 CHO et c.) , i.e. t hat k SP > > k r,I SC, and hence t hat t he rat e of product for m at ion t hrough t he singlet adduct is equal t o t he rat e k f,I SC of t he t r iplet

→ singlet I SC. I t t hen also follows t hat

k TP/ k f,I SC ≈ t riplet

yields/ singlet y ields = 0.45/ 0.55, w hich leads t o k f,I SC ≈ 1.6×10 11 s–1 for t he chem ically act iv at ed r adical w it h int er nal energy cont ent of roughly 30- 40 kcal/ m ol.

Chapt er I V: The O + C2 H 4 r eact ion

85

Table I V.7: Com put ed ov erall pr im ary pr oduct s dist ribut ion ( % ) as a funct ion of t em perat ur e using t he value of 45/ 55 for t he rat io of t riplet and singlet y ields. Wher e possible, exper im ent al dat a ar e also giv en in par ent heses. T ( K) H + CH2 CHO CH2 ( X3 B1 ) + H2 CO CH3 + CHO CH3 CO + H H2 + H2 CCO CH4 + CO 287

40.3 ( 39 ± 10) a)

5. 1 ( 10 ± 5)

47.8 ( 46 ± 10)

2. 2

2.4

2.3

298

40.1 ( 40 ± 10) b)

5. 2 ( 10 ± 5)

47.7 ( 50 ± 10)

2. 2

2.4

2.3

500

37.1

8. 3

47.0

2. 5

2.6

2.5

607

35.2 ( 38 ± 10) a)

10.3 ( 13 ± 5)

46.4 ( 44 ± 10)

2. 7

2.7

2.6

1000

28.0

17.8

44.1

3. 5

3.4

3.1

1500

21.7

24.8

40.5

4. 7

4.3

3.6

2000

18.5

29.1

36.9

5. 7

5.4

4.0

( 5)

( 5)

a) Ref. 23 b) Ref. 24

Figure I V.4: Pr im ary pr oduct s dist ribut ion for t he O( 3 P) + C2 H 4 r eact ion obt ained at t em perat ur es in t he range of 287- 2000K and P= 760 Torr using t he value of 45/ 55 for t he rat io of t r iplet and singlet y ields.

86

Chapt er I V: The O + C2 H 4 r eact ion

The r esult s obt ained for t he ov erall pr oduct branching rat ios are t abulat ed in Table I V.7 and plot t ed in Fig. I V.4. Fig. I V.4 shows t hat t he y ield of CH 2 ( X3 B1 ) + H 2 CO rapidly increases wit h incr easing t em per at ur es, and becom es dom inant at 2000K. At t he sam e t im e t he y ields of CH 3 + CHO and H + CH 2 CHO decrease, but t hey ar e t he m ost im port ant product s at low t em perat ur es. Ot her product s, such as H + CH 3 CO, H 2 + H 2 CCO, and CH 4 + CO, ar e rat her m inor ( ≤ 5% ) and slight ly dependent on t em perat ur e over t he w ide range considered. At t em perat ures in t he range 290- 600K, wher e exper im ent al dat a are available, our com put ed det ailed

pr oduct

dist r ibut ion

values are

in

excellent

agr eem ent

wit h

t he

exper im ent al r esult s ( see Table I V.7) and lie wit hin t he ex perim ent al error bar. At room t em perat ure and at low t o at m ospher ic pressur es, w e w ould r ecom m end t he follow ing pr oduct branching rat ios: 40 ± 5% for H + CH 2 CHO, 5 ± 3% for CH 2 ( X3 B1 ) + H 2 CO, 48 ± 5% for CH 3 + CHO, 2% for CH 3 CO + H, 2.5% for H 2 + H 2 CCO, and 2.5% for CH 4 + CO. The er ror bar for t he com put ed result s was evaluat ed by shift ing TS3 down 1 kcal/ m ol and TS4 up 1 k cal/ m ol on t he t r iplet energy surface. Table I V.8: Calculat ed m icrocanonical rat e const ant s ( s –1 ) under collision- free condit ions ( P≈0 at m ) in t he m olecular beam experim ent for init ial collision energies Ecol of 6.0 and 12.9 kcal/ m ol. React ion channel

k ( E) / s–1 Ecol = 6.0 Ecol = 12.9

°CH2 –CH2 –O°( 3 A) ( I nt 1a) → TS3 → H2 CCHO( X2 A” ) + H°( X2 S) °CH2 –CH2 –O°( 3 A) ( I nt 1a) → TS4 → CH2 ( X3 B1 ) + H2 CO( X1 A1 ) °CH2 –CH2 –O°( 3 A) ( I nt 1a) → TS5 → CH3 CHO( 3 A) ( I nt 2) °CH2 –CH2 –O°( 3 A) ( I nt 1a) → TS6 → CH2 CHOH( 3 A) ( I nt 3) °CH2 –CH2 –O°( 1 A) ( I nt 4) → TS10 → CH3 CHO( X1 A’) ( I nt 5) CH3 CHO( X1 A’) ( I nt 5) → TS10 → °CH2 –CH2 –O°( 1 A) ( I nt 4) °CH2 –CH2 –O°( 1 A) ( I nt 4) → TS11 → c- C2 H4 O( X1 A1 ) ( I nt 6) c- C2 H4 O( X1 A1 ) ( I nt 6) → TS11 → °CH2 –CH2 –O°( 1 A) ( I nt 4) °CH2 –CH2 –O°( 1 A) ( I nt 4) → TS12 → CH2 CHOH( X1 A’) ( I nt 7a) CH2 CHOH( X1 A’) ( I nt 6a) → TS12 → °CH2 –CH2 –O°( 1 A) ( I nt 3) CH3 CHO( X1 A’) ( I nt 5) → TS13 → c- C2 H4 O( X1 A1 ) ( I nt 6) c- C2 H4 O( X1 A1 ) ( I nt 6) → TS13 → CH3 CHO( X1 A’) ( I nt 5) CH3 CHO( X1 A’) ( I nt 5) → TS14 → CH2 CHOH( X1 A’) ( I nt 7a) CH2 CHOH( X1 A’) ( I nt 7a) → TS14 → CH3 CHO( X1 A’) ( I nt 5) CH3 CHO( X1 A’) ( I nt 5) → TS15 → CH3 COH( X1 A) ( I nt 8a) CH3 COH( X1 A) ( I nt 8a) → TS15 → CH3 CHO( X1 A’) ( I nt 5) CH2 CHOH( X1 A’) ( I nt 7a) → TS16 → CH3 COH( X1 A) ( I nt 8a) CH3 COH( X1 A) ( I nt 8a) → TS16 → CH2 CHOH( X1 A’) ( I nt 7a) c- C2 H4 O( X1 A1 ) ( I nt 6) → TS22 → CH2 CHOH( X1 A’) ( I nt 7a) CH2 CHOH( X1 A’) ( I nt 7a) → TS22 → c- C2 H4 O( X1 A1 ) ( I nt 6) CH3 CHO( X1 A’) ( I nt 5) → TS17 → °CH3 ( X2 A2 ” ) + °CHO( X2 A’) CH3 CHO( X1 A’) ( I nt 5) → TS18 → CH3 C°O( X2 A’) + H°( X2 S) CH3 CHO( X1 A’) ( I nt 5) → TS19 → CH2 CHO( X2 A” ) + H°( X2 S) CH3 CHO( X1 A’) ( I nt 5) → TS20 → CH2 CO( X1 A1 ) + H2 ( X1 Σg ) CH3 CHO( X1 A’) ( I nt 5) → TS21 → CH4 ( X1 A1 ) + CO( X1 Σ+ ) CH2 CHOH( X1 A’) ( I nt 7a) → TS24 → CH2 CO( X1 A1 ) + H2 ( X1 Σg ) CH3 COH( X1 A) ( I nt 8a) → TS25 → CH2 CO( X1 A1 ) + H2 ( X1 Σg )

2. 35x10 11 6. 29x10 10 0. 0 3. 46x10 7 1. 46x10 13 4. 46x10 8 3. 83x10 12 9. 23x10 9 9. 07x10 12 4. 39x10 8 1. 27x10 8 2. 01x10 10 6. 38x10 9 1. 01x10 10 1. 08x10 9 1. 88x10 11 3. 19x10 9 3. 51x10 11 5. 52x10 8 5. 53x10 6 2. 75x10 10 1. 55x10 9 2. 93x10 8 8. 03x10 8 1. 51x10 9 3. 09x10 7 7. 30x10 10

Chapt er I V: The O + C2 H 4 r eact ion

6. 05x10 11 4. 13x10 11 2. 37x10 8 1. 27x10 9 1. 38x10 13 1. 08x10 9 3. 99x10 12 2. 15x10 10 9. 31x10 12 1. 10x10 9 3. 39x10 8 4. 68x10 10 1. 15x10 10 1. 73x10 10 2. 50x10 9 3. 09x10 11 6. 11x10 9 5. 02x10 11 1. 99x10 9 2. 18x10 7 5. 77x10 10 4. 50x10 9 1. 17x10 9 1. 98x10 9 3. 96x10 9 1. 31x10 8 1. 40x10 11

87

Und e r collision- fre e con dit ion s. Let us first com pare our com put ed pr im ary product dist ribut ion w it h t hose r ecent ly obser v ed in a m olecular beam st udy by Casav ecchia and co- w ork ers. 56- 58 This ex perim ent was carr ied out at a collision energy of 12.9 kcal/ m ol. We assum e here t hat t his collision energy is convert ed t o addit ional int er nal vibrat ion energy of t he init ially form ed t r iplet biradical adduct •CH 2 CH 2 O•. Not e t hat a sim ilar av erage t herm al energy of t he r eact ant s is acquired at a t em perat ur e of about 1000 K. Micr ocanonical rat e const ant s for var ious channels in t he O( 3 P) + C2 H 4 r eact ion com put ed at an int er nal energy of 12.9 kcal/ m ol abov e t he init ial r eact ant s ar e t abulat ed in Table I V.8. We used t he value of 45/ 55 for t he rat io of t he t r iplet yields over t he singlet y ields as above. Our com put ed values are as follows ( t he ex per im ent al dat a 56- 58 ar e given in parent heses) : 28% ( 27 ± 6% ) for H + CH 2 CHO, 18% ( 16 ± 8% ) for CH 2 ( X3 B1 ) + H 2 CO, 44% ( 43 ± 11% ) for CH 3 + CHO, 3.5% ( 1 ± 0.5% ) for CH 3 CO + H, 3.5% ( 13 ± 3% ) for H 2 + H 2 CCO, and 3% for CH 4 + CO. Not e t hat t hese values differ only m arginally from t he y ields calculat ed for t he t herm al r eact ion at 1000 K. Our com put ed y ields for t he t hr ee m ost im port ant channels agree w ell wit h t hose observ ed ex perim ent ally 56- 58 except for t he ket ene +

H 2 pr oduct channel.

According t o our t heor et ical result s, H 2 + H 2 CCO are form ed from t he act iv at ed singlet s CH 3 COH and CH 3 CHO, in com pet it ion m ainly w it h dissociat ion of t he lat t er t o CH 3 + CHO, in a rat io of about 1 : 13. The exper im ent al rat io how ev er is 1 : 3.3. The r eason for t his discrepancy is not clear . Non- st at ist ical effect s on t he relat iv e rat es are unlik ely , as t he lifet im e of t he act ivat ed CH 3 COH and CH 3 CHO is

§ 10 ps ( see Table I V.8) , w hich should suffice for ergodicit y. Neit her can cent r ifugal effect s of high- J init ial adduct s explain t his, as t hese should favor

dissociat ion t o CH 3 + CHO t hr ough t he loose v ariat ional TS m uch m or e t han t he four - cent er rear rangem ent / fragm ent at ion t o H 2 CCO + H 2 t hr ough t he t ight TS2 0 . I t is also of im port ance t o evaluat e t he effect of collision energy ( Ecol ) on t he product s dist ribut ion. For t his purpose w e have com put ed product branching rat ios at Ecol = 6 kcal/ m ol ( see Table I V.8) as used in a pr ev ious m olecular beam st udy. 19 Again t ak ing a t r iplet / singlet rat io of 45/ 55, t he y ield of H + CH 2 CHO considerably dr ops as a funct ion of int er nal energy , from 36% at Ecol = 6 kcal/ m ol t o 28% at Ecol = 12.9 kcal/ m ol, w her eas t he y ield of CH 2 ( X3 B1 ) + H 2 CO increases from 10% t o 18% . The yield of CH 3 + CHO is less dependent on int ernal energy.

88

Chapt er I V: The O + C2 H 4 r eact ion

I V .3 .3 . Ove r all t h er m a l r a t e coe f ficie nt The ov erall t em perat ur e- dependent rat e coefficient k ( T) overall for t he O( 3 P) + C2 H 4 react ion can be com put ed according t o t he following expr ession:

k (T )overall = (1 − γ re ) × kTST ( T)

( I V.12)

wher e γre is t he fract ion of re- dissociat ion of t he init ial adduct s back t o t he init ial react ant s, O( 3 P) + C2 H 4 , and k TST( T) is t he rat e coefficient der iv ed fr om t ransit ion st at e t heor y. The value of γre is a funct ion of pressur e and t em perat ur e; at t he condit ions considered ( T= 287- 2000 K and P ≤ 760 Torr ) it is negligibly sm all ( all of –130 cm –1 was adopt ed.

Chapt er V: The O + C2 F4 react ion

99

V .4 . Th eor e t ica l Re su lt s a n d D iscu ssion s V .4 .1 . Pot e nt ial e ne r gy sur f ace Th e t riplet e lect r onic st a t e . According t o t he spin- conserv at ion rule, t he elect r ophilic addit ion r eact ion of t r iplet O( 3 P)

t o singlet C2 F4 t ak es place on t he

t riplet surface. Unless m ent ioned ot herw ise, t he r elat iv e energies given below wer e obt ained at t he G2M, CBS- QB3, and G3 lev els of t heory and t he averages of t hese values were used for t he kinet ic com put at ions. The O + C2 F4 react ion is init iat ed by a chain- addit ion on t he C= C double bond in C2 F4 t o form a vibrat ionally excit ed t r iplet OCF2 CF2 int erm ediat e ( denot ed hereaft er as I nt 1) . An addit ion t ransit ion st at e does not exist at t he B3LYP/ 6- 311+ G( 3df) level of t heor y. How ev er , pr ev ious exper im ent al st udies 44-46 indicat e t hat t he ov erall rat e const ant of t he O( 3 P) + C2 F4 r eact ion depends posit ively on t em perat ure w it h an Arr henius

act ivat ion

energy

of

0.6

I RCMax ( G2M( UCC,MP2) : B3LYP/ 6- 311+ G( 3df) ) , 47

±

0.2

k cal/ m ol. 46

I RCMax ( CBS- QB3) , 47

and

I RCMax ( G3) calculat ions were t hen car r ied out along t he r eact ion coordinat e wit hin a Cs sym m et r y and a

3

A” elect r onic st at e. An addit ion TS ( denot ed

her eaft er as TS1) was locat ed at an O- C bond dist ance of 2.1 Å. TS1 lies 1.1 and 0.5 kcal/ m ol in energy abov e t he init ial r eact ant s at t he G2M and G3 lev els, respect ively , while at t he CBS- QB3 level it lies 0.3 kcal/ m ol below t he react ant s, always aft er ZPE- corr ect ion. These result s indicat e t hat t his addit ion st ep has a very sm all or ev en non- ex ist ent barr ier . The average of t hese t hr ee values, 0.4 kcal/ m ol, w ill be adopt ed for com put ing t he ov erall rat e coefficient k( T) ( see below ) .

100

Chapt er V: The O + C2 F4 react ion

Figure V.1: Pot ent ial energy surface for t he O( 3 P) + C2 F4 ( X1 Ag ) react ion on t he t riplet surface const ruct ed using av erage r elat ive ener gies com put ed at t he G2M, CBS- QB3 and G3 levels of t heory .

Tr iplet OCF2 CF2 I nt 1 form ed v ia t he addit ion r eact ion m echanism above has no sym m et ry and lies 43.0 kcal/ m ol low er t han t he r eact ant s. St art ing at I n t 1 , t her e are t hr ee possible channels: ( 1) elongat ion of t he C- C bond in I n t 1 leading t o product s CF2 ( 3 B1 ) + F2 CO v ia TS2 of Cs sym m et ry and

3

A” elect ronic st at e,

present ing a barr ier of 15.5 kcal/ m ol ( see Fig. V.1) ; ( 2) loss of an F at om from t he CF2 m oiet y in I nt 1 t o form product s F + F2 CCFO v ia TS3 w it h a barr ier height of 21.8 kcal/ m ol; ( 3) a sim ult aneous 1,2 F- shift and C- C bond br eakage in I n t 1 via TS4 t o form CF3 + CFO, t he m ost exot her m ic pr oduct s on t he t r iplet sur face. TS4 is a very t ight t r ansit ion st ruct ur e and present s a huge barr ier of 49.9

Chapt er V: The O + C2 F4 react ion

101

kcal/ m ol, so t his channel cannot com pet e w it h t he form er t w o. We were not successful in locat ing a TS dir ect ly connect ing I n t 1 t o t riplet CF3 CFO as all at t em pt s always conver ged t o eit her TS3 or TS4 . O’Gara and Dailey 48 calculat ed a barr ier for 1,2- F m igr at ion in t r iplet 2,2,2- t r ifluoroet hy lidene of 50.8 kcal m ol - 1 at t he QCI SD( T) / 6- 311G( 2d,2p) / / MP2/ 6- 31G( d,p) lev el. I f a dir ect TS for t he I n t 1 → 3 CF3 CFO channel ex ist s, t his channel is expect ed t o show a sim ilar ly high barr ier and should t her efor e be negligible. Thus, it is im m ediat ely apparent from Fig. V.1 t hat t wo channels should kinet ically cont r ol t he product form at ion on t he t r iplet PES: O + C2 F4 → OCF2 CF2 → TS2 → CF2 ( 3 B1 ) + F2 CO and O + C2 F4 → OCF2 CF2 → TS3 → F + F2 CCFO, w it h t he form er clear ly expect ed t o dom inat e. Pr im ary product ion of CF2 ( 3 B1 ) ( + F2 CO) was indeed observ ed in several ex per im ent al st udies, 2,4-6, 24,49,50 wher eas F2 CCFO ( + F) form at ion has not yet been report ed. I t is of k ey im port ance t o not e her e t hat CF3 + CFO cannot be for m ed in any significant am ount s fr om t he t r iplet PES, present ed in Fig. V.1. How ev er, t hese product s were report ed t o be form ed in considerable y ields by Dodonov et al. 24 and unam biguously confirm ed as prim ary product s wit h subst ant ial y ields in our exper im ent al inv est igat ion ( see above) . The abov e st rongly indicat es t he need for a fast I SC event fr om t he t r iplet t o t he singlet PES t o describe t he ex per im ent al product y ields com plet ely.

Figure V.2: Const rained opt im izat ions for sev er al fix ed OCC angles at t he CASSCF( 8,8) / CASPT2( 8,8) / cc- pVDZ level of t heory on t he singlet and t riplet surfaces for ( a) t he O( 3 P) + C2 H 4 r eact ion, and ( b) t he O( 3 P) + C2 F4 r eact ion.

102

Chapt er V: The O + C2 F4 react ion

I t is t herefore of pr im ordial int er est t o inv est igat e t he t r iplet - t o- singlet crossing seam in t he O( 3 P) + C2 F4 r eact ion, and t o com pare it w it h t he I SC crossing in t he O( 3 P) + C2 H 4 r eact ion, described ear lier. 21 I n bot h cases, t he t r iplet → singlet crossing occurs for t he init ial- adduct OCX2 - CX2 st ruct ures and is follow ed by a fast quasi- bar r ier less subsequent ring closure on t he singlet sur face charact er ized m ainly by a decrease in t he OCC angle. As shown for t he C2 H 4 O sy st em , 21 norm al st ret ching v ibrat ions t end t o hav e a sim ilar par abolic energy pr ofile wit hout over ly affect ing t he t r iplet - singlet PES energy gap, while t he OCC bending coor dinat e w ill alt er t he geom et r ies m or e and hence be t he forem ost coordinat e affect ing t he energet ic

differences

opt im izat ions

for

bet ween

sev eral

t r iplet

fixed

CASSCF( 8,8) / CASPT2( 8,8) / cc- pVDZ

and

OCC lev el

angle 23

for

singlet

surfaces.

wer e bot h

carr ied t he

singlet

Const rained out and

at

t he

t r iplet

surfaces ( see Fig. V.2a for C2 H 4 and Fig. V.2b for C2 F4 ) . As can be seen in t hese figur es, t he C2 F4 O singlet / t r iplet sur faces over lap over a m uch w ider range of OCC angles com pared t o t he C2 H 4 O syst em , such t hat t he crossing space region in t he t it le r eact ion is m uch wider, incr easing t he lik elihood of crossing significant ly. I n addit ion, t he fluor ine at om s in C2 F4 are m uch heav ier t han t he hydrogen at om s in C2 H 4 , addit ionally enhancing surface crossing. Hence, w e expect t hat t he I SC process in t he O( 3 P) + C2 F4 react ion occurs m uch fast er t han for O( 3 P) + C2 H 4 , such t hat t he rat io of t r iplet / singlet yields for t he O( 3 P) + C2 F4 r eact ion should be sm aller t han t he value of 45/ 55 for t he O( 3 P) + C2 H 4 r eact ion, 21 ev en w hen allowing for t he short er unim olecular - r eact ion lifet im e of t he hot t r iplet CF2 CF2 O adduct ( ~ 1 ps; ESM analysis of t his w or k ) as com pared t o t hat of CH 2 CH 2 O ( ~ 8 ps

21

).

Th e sin gle t ele ct r onic st a t e . I nt ersyst em cr ossing of t he init ial t r iplet •O- CF2 •

CF2 biradical I nt 1 y ields t he singlet biradical •O- CF2 - •CF2 ( denot ed her eaft er as

I n t 1 s) . At t he low UHF/ 6- 31G( d) level of t heory , we locat ed t he I n t 1 s as a st at ionary point on t he singlet sur face, w it h r elat ive energy 1.1 kcal/ m ol above t he t riplet biradical I nt 1 . How ev er, I n t 1 s does not appear as a local m inim um at t he m ore r igorous CASSCF( 8,8) / cc- pVDZ lev el of t heory ; opt im izat ions at t his level alway s converge t o t he cyclic conform er singlet t et rafluor ine et hy lene ox ide, I n t 3 ( see Fig. V.3) . Addit ionally, const rained opt im izat ions for sev eral fix ed CCO angles in I n t 1 s at t he CASPT2/ / CASSCF lev el showed t hat t here is indeed no barr ier t o cyclizat ion, indicat ing t hat ot her pr ocesses ( e.g. a 1,2 F- shift in I n t 1 s) can cert ainly not com pet e w it h t his r ing- closur e. Therefor e, aft er I SC, t he result ing I nt 1 s w ill prom pt ly r elax t o singlet t et rafluor ine et hy lene ox ide ( I nt 3 ) . I n t 3 , C2v point gr oup and a 1 A1 elect ronic st at e, has an int ernal ener gy of 111.1

Chapt er V: The O + C2 F4 react ion

103

kcal/ m ol r elat ive t o t he init ial r eact ant s. Not e t hat t he fast r elaxat ion t o I n t 3 m ov es int erm ediat es away fr om t he t r iplet ↔ singlet crossing seam , virt ually elim inat ing t he possibilit y of r ev erse I SC back t o t he t riplet sur face.

Figure V.3: Pot ent ial ener gy surface for t he O( 3 P) + C2 F4 ( X1 Ag ) r eact ion occur r ing on t he singlet sur face const ruct ed using av er age r elat ive ener gies com put ed at t he G2M, CBS- QB3 and G3 levels of t heor y. The t riplet ent rance part is shown by dashed lines.

104

Chapt er V: The O + C2 F4 react ion

St art ing at I n t 3 , t her e are t wo accessible r eact ion channels: ( 1) decom posit ion t o product s CF2 ( X1 A1 ) + F2 CO v ia TS7 , which has Cs sym m et ry , a 1 A’ elect ronic st at e and lies 32.5 kcal/ m ol above I n t 3 ; and ( 2) a concert ed 1,2- F shift in com binat ion wit h C- O bond br eak ing t o form singlet t et r afluor ine acet aldehyde CF3 CFO I n t 4 via TS8 w it h a barr ier height of 39.5 kcal/ m ol. The form er pat hway is slight ly m or e fav orable in ener gy. I nt 4 , w it h a relat ive energy of 142.2 kcal/ m ol below t he init ial react ant s, of Cs point gr oup and 1 A’ elect r onic st at e, can react in t hree possible ways, show n in Fig. V.3: ( 1) isom erisat ion back t o I n t 3 v ia TS8 w it h a barr ier height of 70.6 k cal/ m ol; ( 2) concert ed 1,2- F m igrat ion com bined w it h C- C bond scission leading t o CF4 + CO v ia TS9 , which lies 90.1 kcal/ m ol abov e I n t 4 ; ( 3) fr agm ent at ion t o pr oduct s CF3 + CFO v ia a ( near - ) bar rier less t ransit ion st at e TS1 0 ( not show n in Fig. V.3) , w hich is locat ed 79.7 kcal/ m ol abov e I n t 4 . As t he t ransit ion st at es for decom posit ion of I n t 4 int o react ion pr oduct s lie m uch higher t han t hat for t he I n t 3 ↔ I nt 4 ( re- ) isom er izat ion ( see Fig. V.3) , t he efficiency of t he second channel of I nt 3 , abov e, will be furt her r educed. As a consequence, CF2 ( X1 A1 ) + F2 CO are ex pect ed t o be t he dom inant pr oduct s on t he singlet surface. V .4 .2 . Pr oduct dist r ib ut ion The part ial product dist ribut ions fr om t he t r iplet



CF2 CF2 O• and singlet ox irane

adduct s were der iv ed separat ely by solv ing t he appropriat e m ast er equat ions independent ly. The init ial energy dist r ibut ion of form at ion of t he t riplet •CF2 CF2 O• adduct

from

O( 3 P)

+

C2 F4

v ia

TS1

was

der iv ed

fr om

det ailed

balance

considerat ions. 51 The r esult s obt ained under various react ion condit ions ( T= 298700 K and P= 10 –3 - 1 at m ) are pr esent ed in Tables V.2 and V.3. Th e t r iple t su r fa ce . Com put at ion of t he pr oduct yields by ESM solut ion of t he m ast er equat ion for t he select ed react ion condit ions P= 10 Tor r and T= 298 K giv es

+9.5 3 89.5+−5.5 9.5 % CF2 ( B1 ) + F2 CO, and 10.5−5.5 % F2 CCFO + F, wher eas y ields of all

ot her pr oduct s are negligible ( < 1% ) . The errors on t he y ields wer e ev aluat ed by vary ing t he TS3 energy by 2 kcal/ m ol. I n fact , t he yields are found t o be invar iant ov er t he pr essure range of < 10 –3 t o 1 at m , but t o slight ly change as a funct ion of t em perat ur e ( see Table V.2) . The y ield of F2 CCFO + F incr eases from 10.5% at 298 K t o 14.5% at 700 K at t he ex pense of t he CF2 ( 3 B1 ) + F2 CO yield. The pressur e- independence reflect s t he short unim olecular lifet im e of t he “ hot ” t r iplet adduct OCF2 CF2 , com put ed t o be ≈1 ps, such t hat at pressur es below 1 at m it suffers no collision ener gy losses. Th e single t su r f a ce . The pr oduct y ields w er e com put ed by solv ing t he ME, using t he average relat iv e energies from our G2M, CBS- QB3 and G3 calculat ions. The

Chapt er V: The O + C2 F4 react ion

105

lifet im e of t he init ial “ hot ” oxirane is est im at ed t o be ≈ 1.5 ps, while it r equir es dozens of collisions t o st abilize t his adduct . As a result , t he product dist ribut ion was lik ew ise found t o be independent of pressure below 1 at m , but t o change slight ly as a funct ion of t em perat ur e. The com put ed y ield of CF3 + CFO increases from 12.9% at 298 K t o 14.4% at 700 K w her eas t he yield of CF2 ( X1 A1 ) + F2 CO decreases by about 1.8% ( see Table V.3) . For t he react ion condit ions of T= 298 K and P= 10 Torr , product s yields were com put ed t o be 86.5% CF2 ( X1 A1 ) + F2 CO, 12.9% CF3 + FCO, and 0.6% CF4 + CO, w it h an est im at ed er ror of ± 5% . Table V.2: Com put ed product s dist ribut ion ( % ) as a funct ion of t em perat ure and pressur e for t he O( 3 P) + C2 F4 ( X1 Ag ) r eact ion occurr ing on t he t r iplet surface. Product s CF2 ( 3 B1 ) + F2 CO( X1 A1 ) F( X2 P) + F2 CCFO( X2 A” ) • O- CF2 - CF2 • ( 3 A) O( 3 P) + C2 F4 ( X1 Ag ) Product s CF2 ( 3 B1 ) + F2 CO( X1 A1 ) F( X2 P) + F2 CCFO( X2 A” ) • O- CF2 - CF2 • ( 3 A) O( 3 P) + C2 F4 ( X1 Ag )

P = 10 Torr 298 K 400 K 89.6 88.7 10.4 11.3 0 0 0 0 T = 298 K 1 Torr 89.6 10.4 0 0

10 Torr 89.6 10.4 0 0

500 K 86.6 12.4 0 0

600 K 87.6 13.4 0 0

700 K 85.5 14.5 0 0

100 Tor r 89.6 10.4 0 0

1 at m 89.6 10.4 0 0

Table V.3: Com put ed pr oduct dist ribut ion ( % ) as a funct ion of t em per at ur e and pressur e for t he O( 3 P) + C2 F4 ( X1 Ag ) r eact ion occurr ing on t he singlet surface. P = 10 Torr 298 K 88.6 12.8 0. 6 0 0 0

400 K 86.2 13.2 0.6 0 0 0

CF2 ( X1 A1 ) + F2 CO( X1 A1 ) CF3 ( X2 A1 ) + CFO( X2 A’) CF4 ( X1 A1 ) + CO c- OC2 F4 ( X1 A1 ) CF3 CFO( X1 A’) O( 3 P) + C2 F4 ( X1 Ag )

T = 298 K 1 Torr 86.5 12.9 0. 6 0 0 0

10 Torr 86.5 12.9 0.6 0 0 0

106

Chapt er V: The O + C2 F4 react ion

Product s CF2 ( X1 A1 ) + F2 CO( X1 A1 ) CF3 ( X2 A1 ) + CFO( X2 A’) CF4 ( X1 A1 ) + CO c- OC2 F4 ( X1 A1 ) CF3 CFO( X1 A’) O( 3 P) + C2 F4 ( X1 Ag ) Product s

500 K 85.7 13.6 0. 7 0 0 0

600 K 85.3 14.0 0. 7 0 0 0

100 Torr 86.5 12.9 0. 6 0 0 0

700 K 84.8 14.4 0.8 0 0 0

1 at m 86.5 12.9 0.6 0 0 0

Ove r a ll

pr im a ry

pr odu ct

dist r ibut ion.

To com put e t he ov erall product

dist ribut ion, account ing for t he rat e of int ersy st em crossing bet ween t he t r iplet and singlet surfaces, one m ust know t he I SC rat es at t he m inim um in t he seam of crossing ( MSX) 52 bet w een t he t riplet and singlet surfaces. To com put e t hese rat es, t raj ect or y dy nam ic calculat ions, e.g. “ on t he fly” non- adiabat ic dynam ics, 53 are r equir ed. How ev er , such calculat ions ar e far bey ond t he scope of t his paper. Lacking accurat e dy nam ic calculat ions, in t his work w e est im at e t he ov erall product dist ribut ion by m at ching t he y ield of CF3 =

16 ±

8%

observ ed

4

exper im ent ally earlier by Donodov and by us in t his paper ( see abov e) . Overall product dist ribut ion was com put ed as a funct ion of t he CF3 y ield, vary ing from t he exper im ent al low er lim it of 8% up t o 13% ; t his lat t er upper lim it r equir es 100% int er- syst em crossing according t o our RRKM- ME r esult s. The calculat ed result s present ed in Table V.4 show t hat t he CF2 ( X1 A1 ) + F2 CO and CF2 ( a 3 B1 ) + F2 CO yields so found ar e highly sensit iv e t o t he adopt ed CF3 y ield. I ncreasing t he CF3 yield by 5% , fr om 8% t o 13% , increases t he yield of CF2 ( X1 A1 ) by 33% , from 53% t o 86% , whereas t he CF2 ( a 3 B1 ) y ield significant ly r educes from ~ 34 t o 0% . I t should be indicat ed t hat for t he considered r ange of 8- 13% CF3 , t he CF2 ( a 3 B1 ) yield can not ex ceed 35% . Table V.4 also show s t hat F2 CCFO + F ( < 4% ) and CF4 + CO ( ≈ 0.5% ) are pr edict ed t o be m inor pr oduct s, insensit iv e t o t he adopt ed yield of

CF3 ;

neit her

of t hese t wo pr oduct

channels has been observ ed

exper im ent ally . Our pr edict ed t ot al F2 CO product yield of 87- 88% is in excellent agreem ent w it h t he exper im ent al values of

84 +−711% observ ed by us and 83% ± 8%

by Dodonov et al. 4,24 Table V.4: Calculat ed ov erall pr oduct dist r ibut ion ( % ) as a funct ion of t he CF3 yield obser ved in t he ex per im ent at t he r eact ion condit ions of T= 298 K and P= 10 Torr. Singlet product s

Triplet product s 1

CF3 + CFO

CF2 ( X A1 ) + F2 CO

CF4 + CO

CF2 ( a 3 B1 ) + F2 CO

F( 2 P) + F2 CCFO

8.0

53.2

0. 3

34.5

4.0

9.0

59.9

0. 4

27.5

3.2

10.0

66.5

0. 4

20.7

2.4

11.0

73.2

0. 4

13.8

1.6

12.0

79.8

0. 5

6.9

0.8

13.0

86.5

0. 5

0.0

0.0

Assum ing a y ield of 10% for CF3 , pr oduct for m at ion cont ribut ions from t he t r iplet and singlet sur faces are predict ed t o be 20% and 80% , respect iv ely. Using t hese

Chapt er V: The O + C2 F4 react ion

107

cont r ibut ions, we der iv e an I SC crossing rat e of ≈4 × 10 12 s–1 fr om t he t r iplet t o t he singlet sur face using t he RRKM- ESM lifet im e of 1 ps found her e for t he t r iplet adduct •CF2 CF2 O•. V .4 .3 . Ove ra ll r at e coe fficien t The ov erall t em perat ur e- dependent rat e coefficient k( T) overall for t he O( 3 P) + C2 F4 react ion can be com put ed as follows:

k (T )overall = (1 − γ re ) × kTST ( T)

( V.2)

wher e k TST( T) is t he rat e coefficient derived from t ransit ion st at e t heor y and γre is t he y ield of OCF2 CF2 r e- dissociat ion back t o t he init ial r eact ant s, O( 3 P) + C2 F4 . The value of γre is a funct ion of pr essur e and t em perat ure; at t he condit ions consider ed ( T= 298- 700 K and P ≤ 1at m ) it is negligibly sm all ( see Tables V.2 and V.3) such t hat k ( T) can be com put ed dir ect ly fr om t he t ransit ion st at e t heor y expr ession:

k (T )overall = k (T )TST = α ×

QTS≠ 1 (T ) kbT × exp( − E ≠ / RT ) h QO (T )QC2 F4 (T )

( V.3)

wher e Q( T) is a com plet e part it ion funct ion, k b Bolt zm ann’s const ant , h Planck ’s const ant , R t he univ ersal gas const ant , E≠ is t he bar r ier height of 0.4 kcal/ m ol ( see higher ) for t he init ial addit ion channel, and α is t he r eact ion pat h degeneracy REWDLQHG IURP WKH V\PPHWU\ QXPEHU UDWLR

C2F4  TS1 =

4. The elect r onic part it ion

funct ion of t he O at om explicit ly includes t he t hr ee low est - ly ing elect ronic st at es ( 3 P2 ( elect ronic degeneracy g = 5) ,

3

P1 ( g= 3) , and

3

P0 ( g= 1) ) , w it h r elat ive

energies of 0.000, 0.453, and 0.649 kcal/ m ol, r espect iv ely . 54 The elect ronic degeneracy of 3 for TS1, which has a t riplet elect ronic st at e, w as duly t aken int o account . Ov erall t herm al rat e coefficient s in t he w ide r ange of t em perat ures 150- 1500 K wer e com put ed and plot t ed in Fig. V.4, t oget her w it h som e of t he av ailable exper im ent al dat a for com par ison, show ing a fav orable com par ison bet ween our com put ed k ( T) and t he av ailable experim ent al dat a 44- 46 cov ering t he range 298500 K. Our com put ed rat e const ant of 7.7× 10 –13 cm 3 m olecule –1 s–1 at room t em perat ur e is in good agr eem ent w it h t he av ailable exper im ent al values of ( 713) × 10 –13 cm 3 m olecule –1 s–1 . 3,6, 44- 46 Our k ( T) ov erall is well- r epr oduced by t he expr ession:

k (T ) = 1.64 × 10−16 × T 1.48 .

I t should be m ent ioned t hat for t he r eact ion of O( 3 P) w it h halogenat ed et hy lene syst em s ( F2 C= CXY, w her e X and Y = H, F, Cl, Br ) , a linear corr elat ion bet w een rat e coefficient s and ionizat ion pot ent ials ( I P) has been r eport ed, i.e. t he larger I P t he fast er react ion rat e. 55,56 F2 C= CF2 has t he lar gest I P, 55 and accordingly also t he

108

Chapt er V: The O + C2 F4 react ion

highest rat e coefficient for t he react ion w it h O( 3 P) . This t r end is opposit e w it h t hat in t he r eact ions of O( 3 P) wit h alk y l- subst it ut ed et hy lenes. 46

Figure V.4: Overall t her m al rat e coefficient s com put ed ( TST) at t em perat ures in t he range of 150- 1500K. Experim ent al dat a are giv en for t he purpose of com parison.

Chapt er V: The O + C2 F4 react ion

109

Figure V.5: Recom m ended react ion m echanism for t he O( 3 P) +

C2 F4 ( X1 Ag )

react ion. React ion ener gies and ov erall product dist r ibut ion ar e also giv en.

110

Chapt er V: The O + C2 F4 react ion

V .5 . Con clusion s The O( 3 P) + C2 F4 ( X1 Ag ) react ion was inv est igat ed exper im ent ally using dischargeflow

t echniques

and

m olecular - beam - sam pling

t hreshold- ionizat ion

m ass

spect rom et ry, as w ell as t heoret ically using var ious high lev els of quant um t heory followed by st at ist ical rat e RRKM - Mast er Equat ion analyses. I n t he exper im ent al st udy, we obser ved t he m aj or pr im ary react ion product s t o be F2 CO ( w it h co- product CF2 , eit her t r iplet or singlet ) and CF3 ( w it h FCO or CO +

F co- pr oduct s) , in a r at io of ca. 0.84: 0.16 wit h er ror m argins § ±0.08, t hus confir m ing t he r esult s of Dodonov et al. 4 The

com put at ional

r esult s

show

t hat

t he

observ ed

product

dist ribut ion

necessit at es a non- adiabat ic react ion m echanism involving fast t riplet → singlet int ersyst em crossing of t he init ial F2 C- CF2 O adduct at a rat e of ca. 4 × 10 12 s –1 , wit h t he m aj or it y of t he product s result ing from subsequent react ions on t he singlet surface, t o giv e an overall pr oduct dist r ibut ion pr esent ed in Fig. V.5. This non- adiabat ic r eact ion m echanism is sim ilar t o t hat of t he O( 3 P) + C2 H 4 ( X1 Ag ) react ion. 21 Our com bined exper im ent al and t heor et ical r esult s show t hat bot h singlet and t riplet

CF2

can

be produced

in

t he

O( 3 P)

+

C2 F4 r eact ion,

but

w it h

a

preponderance of t he singlet gr ound st at e, t he yield of t r iplet CF2 being predict ed t o be at m ost 35% and possibly only a few % . Finally, overall t herm al TST rat e coefficient s wer e com put ed for t em perat ur es in t he range of 150- 1500K; t hey can be expr essed as

k (T ) = 1.64 × 10−16 × T 1.48 . The

k( T) result s, der ived ent ir ely fr om first principles, ar e in agr eem ent wit h t he available exper im ent al dat a.

Chapt er V: The O + C2 F4 react ion

111

Re fe r e nce s ( 1) Chase M. W. Jr. NI ST- JANAF Them ochem ical Tables, Fourt h Edit ion, J. Phys. Chem . Ref. Dat a, Monograph 1 9 9 8 , 9, 1- 1951. ( 2) Heick len J. Adv. Phot ochem . 1 9 6 9 , 7 , 57 and refer ences cit ed t her ein. ( 3) Young R. A.; Blauer J.; Bow er R.; Lin C. L. J. Chem . Phy s. 1 9 8 8 , 88, 4834. ( 4) Dodonov A. F.; Zelenov V. V.; Kuk ui A. S. Sov. J. Chem . Phys. 1 9 9 0 , 6, 3368. ( 5) Koda S. Chem . Phys. Let t . 1 9 7 8 , 55, 353; ( 6) Koda S. J. Phys. Chem . 1 9 7 9 , 83, 2065. ( 7) Yam aguchi K. ; Yabushit a S. ; Fueno T.; Kat o S. ; Morokum a K. Chem . Phys. Let t . 1 9 8 0 , 70, 27. ( 8) Dupuis M.; Wendoloski J. J.; Takada T.; Lest er W. A. Jr . J. Chem . Phys. 1 9 8 2 , 76, 481. ( 9) Melius C. F. BAC- MP4 m et hod ( see Ref. 14) . ( 10) Fueno T.; Tak ahara Y.; Yam aguchi K. Chem . Phy s. Let t . 1 9 9 0 , 167, 291. ( 11) Jursic B. S. Theochem 1 9 9 9 , 492, 85. ( 12) Endo Y.; Tsuchiya S.; Yam ada C.; Hir ot a E.; Koda S. J. Chem . Phys. 1 9 8 6 , 85, 4446 and see r eferences t her ein. ( 13) Bley U.; Dransfeld P.; Him m e B.; Koch M.; Tem ps F.; Wagner H. G. 22 t h Sy m p. ( I nt .) Com b. 1 9 8 8 , 997. ( 14) Schm olt ner A. M.; Chu P. M.; Br udzynsk i R. J. ; Lee Y. T. J. Chem . Phys. 1 9 8 9 , 91, 6926 and see references t herein. ( 15) Koda S.; Endo Y.; Tsuchiy a S.; Hirot a E. J. Phys. Chem . 1 9 9 1 , 95, 1241. ( 16) Knyazev V. D.; Ar ut y unov V. S.; Vedeneev V. I . I nt . J. Chem . Kine. 1 9 9 2 , 24, 545. ( 17) Abou- Zeid O. K.; McDonald J. D. J. Chem . Phys. 1 9 9 8 , 109, 1293. ( 18) Quandt R.; Min Z.; Wang X.; Bersohn R. J. Phys. Chem . A 1 9 9 8 , 102, 60. ( 19) Min Z.; Wong T. H.; Quandt R.; Bersohn R. J. Phy s. Chem . A 1 9 9 9 , 103, 10451. ( 20) Oguchi T. ; I shizak i A.; Kak ut a Y.; Mat sui H.; Miy oshi A. J. Phys. Chem . A 2 0 0 4 , 108, 1409 and see Ref. 14 t her ein. ( 21) Nguyen T. L.; Ver eeck en L.; Hou X. J.; Nguy en M. T.; Peet ers J. J. Phys. Chem . A 2 0 0 5 , 109, 7489. ( 22) Boullart W.; Dev r iendt K.; Borm s R and Peet ers J. J. Phys. Chem . 1 9 9 6 , 100, 998. ( 23) Nguyen, T. L.; Dils, B. ; Car l, S. A.; Vereecken, L.; Peet ers, J. J. Phys. Chem . A 2 0 0 5 , 109, 9786. ( 24) Dodonov A. F. ; Zelenov V. V.; Kuk ui A. S., Sov. J. Chem . Phys. 1 9 9 1 , 7, 1089. ( 25) Becke A. D. J. Chem . Phys. 1 9 9 3 , 98, 5648. ( 26) St ev ens P. J.; Devlin F. J.; Chablowsk i C. F.; Fr isch M. J. J. Phys. Chem ., 1 9 9 4 , 98, 11623. ( 27) EMSL Basis Set Librar y, ht t p: / / ww w. em sl.pnl.gov / form s/ basisfor m .ht m l ( 28) Gonzalez C.; Schlegel H. B. J. Chem . Phys. 1 9 8 9 , 90, 2154. ( 29) Gonzalez C.; Schlegel H. B. J. Phys. Chem . 1 9 9 0 , 94, 5523. ( 30) Mebel A. M.; Mor okum a K.; Lin M. C. J. Chem . Phys., 1 9 9 5 , 103, 7414. The t ot al energy of a m olecular syst em is com put ed as follows: E[ G2M( UCC,MP2) - a] = E[ CCSD( T) / 6- 311G( d) / / B3LYPL] + { E[ MP2/ 6311+ G( 3df) / / B3LYPL] - E[ MP2/ 6- 311G( d) / / B3LYPL] } + ZPE[ B3LYPL] , where B3LYPL st ands her e for t he geom et r ies opt im ized at t he B3LYP/ 6- 311+ G( 3df) level of t heory . E[ G2M( UCC,MP2) - b] = E[ CCSD( T) / 6- 311+ G( d) / / B3LYPL] + { E[ MP2/ 6311+ G( 3df) / / B3LYPL] - E[ MP2/ 6- 311+ G( d) / / B3LYPL] } + ZPE[ B3LYPL] . ( 31) Mont gom ery J. A. Jr.; Fr isch M. J.; Ocht erski J. W.; Pet ersson G. A. J. Chem . Phys. 1 9 9 9 , 110, 2822.

112

Chapt er V: The O + C2 F4 react ion

( 32) Curt iss L. A.; Raghavachari K.; Redfer n P. C.; Rassolov V.; Pople J. A. J. Chem . Phys., 1 9 9 8 , 109, 7764. ( 33) Lee T. J.; Tay lor P. R. I nt . J. Quant . Chem . Sym p. 1 9 8 9 , 23, 199. ( 34) Fr isch M. J.; Tr uck s G. W.; Schlegel H. B. et al. Gaussian 03, Gaussian, I nc., Pit t sburgh, PA, ( 2 0 0 3 ) . ( 35) DALTON, a m olecular elect r onic st ruct ure program , wr it t en by Helgaker T.; Jensen H. J. Aa. ; Joergensen P.; Olsen J.; Ruud K.; Aagr en H.; Auer A. A. et al., Release 1.2 ( 2 0 0 1 ) . ( 36) MOLPRO is a package of ab init io pr ogram s wr it t en by Wer ner H.- J.; Know les P. J.; Schüt z M.; Lindh R.; Celani P. ; Korona T.; Rauhut G.; Manby F. R. ; Am os R. D.; Ber nhardsson A.; Ber ning A.; Cooper D. L.,; Deegan M. J. O.; Dobby n A. J. ; Eckert F. Et al. ( 2 0 0 2 ) . ( 37) Gilbert R. G.; Sm it h C. S. Theory of Unim olecular and Recom binat ion React ions ( Blackwell Scient ific, Ox ford, 1 9 9 0 ) . ( 38) Baer T.; Hase W. L. Unim olecular React ion Dy nam ics: Theory and Exper im ent ( Ox ford Universit y Pr ess, Oxford, 1 9 9 6 ) . ( 39) Bey er T.; Swinehar t D. F. Com m . Assoc. Com put . Machines, 1 9 7 3 , 16, 379. ( 40) St ein S. E. ; Rabinovit ch B. S. J. Chem . Phy s., 1 9 7 3 , 58, 2438. ( 41) Holbrook K.; Pilling M.; Robert son S. Unim olecular React ions, 2nd edit ion ( Wiley, New Yor k, 1 9 9 6 ) . ( 42) St einfeld J. I .; Francisco J. S. ; Hase W. L. Chem ical Kinet ics and Dynam ics ( Prent ice- Hall, Englewood Cliffs, NJ, 1 9 9 9 ) . ( 43) Hippler H.; Troe J; Wendelken H. J. J. Chem . Phys., 1 9 8 3 , 78, 6709. ( 44) Herr on J. T. ; Huie R. E. J. Phys. Chem . Ref. Dat a 1 9 7 3 , 2, 467. ( 45) Gershenzon Y. M.; Moin F. B.; Yur kev ich Y. P. Kinet . Cat al. 1 9 7 5 , 16, 1192. ( 46) Cv et anov ic R. J. J. Phys. Chem . Ref. Dat a 1 9 8 7 , 16, 261. ( 47) Malick D. K.; Pet ersson G. A.; Mont gom ery J. A. Jr. J. Chem . Phys. 1 9 9 8 , 108, 5704. ( 48) O’Gara J E.; Dailey W. P. J. Am . Chem . Soc. 1 9 9 4 , 116, 12016. ( 49) Ty erm an W. J. R. Trans. Farad. Soc. 1 9 6 9 , 65, 163. ( 50) Hsu D. S. Y. ; Lin M. C. Chem . Phys. 1 9 7 7 , 21, 235. ( 51) Forst W. Theory of Unim olecular React ions ( Academ ic Pr ess, New York , 1973). ( 52) Dunn K. M.; Mor ok um a K. J. Chem . Phys. 1 9 9 5 , 102, 4904. ( 53) Klein S.; Bearpark M. J.; Sm it h B. R.; M; Robb M. A.; Oliv ucci M.; Bernardi F. Chem . Phys. Let t . 1 9 9 8 , 292, 259. ( 54) ht t p: / / physics.nist .gov / PhysRefDat a/ Handbook / per iodict able.ht m . ( 55) Lee S. Y.; Yoo H. S.; Kang W. K.; Jung K. H. Chem . Phys. Let t . 1 9 9 6 , 257, 415. ( 56) Yi H. J.; Jee Y. J.; Lee K. W. ; Jung K. H. Chem . Phys. Let t . 2 0 0 0 , 327, 325. ( 57) Kraka E.; Konk oli Z.; Crem er D.; Fow ler J.; Schaefer H. F. J. Am . Chem . Soc. 1 9 9 6 , 118, 10595. ( 58) Buckley T. J. ; Johnson R. D.; Huie R. E.; Zhang Z.; Kuo S. C.; Klem m R. B. J. Phys. Chem . 1 9 9 5 , 99, 4879.

Chapt er V: The O + C2 F4 react ion

113

Cha pt e r VI : A Qu a n t u m Ch e m ica l a n d St a t ist ica l Ra t e St u dy of t h e Re a ct ion of O( 3 P) w it h Alle n e : O- a ddit ion a n d H a bst r a ct ion Ch a n n e ls †

V I .1 . I n t r odu ct ion Chem ical processes occurr ing in com bust ion and flam es go t hr ough com plex react ion net w orks norm ally consist ing of several hundreds and ev en t housands of coupled elem ent ary r eact ions. Charact er izing t he dom inant elem ent ar y react ions is v er y im port ant t o underst and t he ov erall r eact ion m echanism s, wit h t he ult im at e goal of opt im izing t he com bust ion pr ocess. I t is well k nown t hat sm all unsat urat ed hydr ocarbons such as C2 H 2 , C2 H 4 , C3 H 4 , and C3 H 6 are cr ucial int erm ediat es

in

hydrocarbon- fueled

flam es. 1,2

Maj or,

if

not

dom inant ,

consum pt ion pat hway s of t hese sm all unsat urat ed hydrocarbons are t heir react ions wit h t r iplet ground st at e oxygen at om s. These r eact ions also play an im port ant role in t he generat ion of polycyclic arom at ic hydr ocarbons ( PAH) and soot precursors. I n previous papers, t he r eact ions of C2 H 2 , 3 C2 H 4 , 4 and C2 F4 5 w it h O( 3 P)

w er e

t heor et ically

charact er ized

using

high- lev el

st at e- of- t he- art

calculat ions. Our predict ed react ion m echanism s, product dist ribut ions, and t herm al rat e coefficient s are in good agreem ent wit h t he available exper im ent al dat a. I n t his paper, we address t he r eact ion of allene w it h t r iplet oxygen at om . The O( 3 P) + allene react ion was ext ensiv ely st udied exper im ent ally. 6−15 Var ious react ion channels leading t o different pr im ary product s were obser v ed and are sum m ar ized below ; w her e possible, t he exper im ent al react ion ent halpies16 at 0 K are indicat ed, wit h com put ed values at t he CBS- QB3 level given in parent heses for t he purpose of com parison ( kcal/ m ol) . O( 3 P) + CH 2 = C= CH 2 → CO +

1

−119.6 ( −119.6)

( VI .1)

−24.4 ( −23.6)

( VI .2)

O( P) + CH 2 = C= CH 2 → H + CH 2 = C −CHO

( −14.0)

( VI .3)

O( 3 P) + CH 2 = C= CH 2 → cyclopr opanone

( −99.9)

( VI .4)

( −90.0)

( VI .5)

( −120.4)

( VI .6)

−24.0 ( −24.9)

( VI .7)

−77.6 ( −77.5)

( VI .8)

−12.9 ( −12.8)

( VI .9)

3

C2 H 4

3

O( P) + CH 2 = C= CH 2 → CH 2 + H 2 CCO 3





3

O( P) + CH 2 = C= CH 2 → allene ox ide 3

O( P) + CH 2 = C= CH 2 → acrolein 3





O( P) + CH 2 = C= CH 2 → C2 H 3 + HC O 3

O( P) + CH 2 = C= CH 2 → C2 H 2 + H 2 CO 3





O( P) + CH 2 = C= CH 2 → H 2 C −C≡CH + OH †

Thanh Lam Nguyen et al, J. Phys. Chem. A 2006, 110, 12166-12176. Chapt er VI : The O + allene react ion

115

To our k nowledge, t he H- abst ract ion channel ( VI .9) has not been consider ed befor e; our pr edict ions ( see below ) suggest it play s an im port ant r ole in hight em perat ur e condit ions. Even t hough r eact ion channel ( VI .1) is a spin- forbidden process, all ear lier exper im ent s7−12 showed CO plus singlet et hene t o be m ost im port ant pr oduct s for t he t it le r eact ion. This channel was believed 7- 12 t o occur v ia an addit ion m echanism of t riplet ox ygen on t he cent ral carbon in allene and t o go t hr ough v ibrat ionally hot , long- liv ed, singlet cyclopropanone for m ed aft er int ersyst em crossing ( I SC) fr om t he st art ing t riplet t o t he low est - ly ing singlet surfaces. Pr im ary k et ene form at ion ( channel VI .2) w as det ect ed in m at rix st udies of t he r eact ion bet w een allene and t r iplet oxygen at om s produced in t he phot olysis of ozone. 9 The product s H • + allenylox y r adical fr om channel ( VI .3) and



C2 H 3 + HC• O fr om channel ( VI .7) are t hought t o ar ise from t he O- at om

addit ion st ep on t he t erm inal carbon. 10 Furt herm or e, in exper im ent al m at rix st udies9 cyclopr opanone, allene ox ide, and acrolein w ere st abilized and observ ed as final product s. How ev er, only a sm all am ount of acrolein was det ect ed in x enon m at r ices, in good agr eem ent w it h an ear lier st udy in t he gas phase. 7 Finally, t race am ount s of form aldehyde, lik ely arising fr om channel ( VI .8) , were observ ed. 9 Therm al rat e coefficient s for t he O( 3 P) + allene → product s were m easured at low and m oderat e t em perat ures ( T < 900 K) . 13−15 From t hese dat a, t he ex per im ent al Arr henius act ivat ion energy was der iv ed t o be about 1.85 ± 0.2 k cal/ m ol; 6 at room

t em perat ur e,

recom m ended.

a

rat e

of

ca.

1.3

×

10



cm 3

m olecule −1

s −1

was

6

There are t w o published t heor et ical st udies of t he O( 3 P) + allene r eact ion. Chiu and Abidi 17 explained t he spin- forbidden r eact ion channel ( VI .1) using selfconsist ent t heory, while Lest er and cowork ers18 invest igat ed t he O- at om at t ack on t he cent ral and t er m inal carbon at om s in allene using t he CASSCF( 6,6) / DZP level of t heory . The lat t er aut hors com put ed t he bar r ier height s for t hese t wo st eps t o be 15 and 17 kcal/ m ol, r espect iv ely. These values ov erest im at e t he exper im ent al dat a by ~ 14 kcal/ m ol, pr im ar ily because t he CASSCF m et hod does not include dy nam ic elect ron cor relat ions. From t he abov e w e see t hat t he m echanism of t he O( 3 P) + allene r eact ion appears t o be v er y com plicat ed, consist ing of parallel and coupled r eact ion channels as well as I SC processes. I n t his cont ext , high- lev el quant um chem ical calculat ions in com binat ion wit h accurat e st at ist ical k inet ic analysis ar e necessary in order t o bet t er under st and t he react ion m echanism s as well as t he form at ion of t he m aj or pr oduct s of t he t it le react ion.

116

Chapt er VI : The O + allene r eact ion

V I .2 . M e t hodolog y V I .2 .1 . Qu a nt u m Ch e m ica l Ca lcu la t ion s Geom et ries of st at ionar y point s on t he t r iplet and singlet surfaces were opt im ized at t he hybr id densit y funct ional B3LYP/ 6- 311G( d,p) level of t heory , 19,20 followed by analyt ical fr equency calculat ions at t he sam e lev el t o ver ify t he st at ionary point s locat ed ( one im aginary frequency for a t ransit ion st ruct ur e and all posit iv e fr equencies for a m inim um ) . I nt rinsic react ion coordinat e ( I RC) 21,22 calculat ions wer e also perform ed at t his level t o est ablish t he cor rect connect ions bet ween t he react ion int erm ediat es, in som e cases enhanced by G2M single- point calculat ions on t he B3LYP geom et r ies t o refine t he pot ent ial energy cur ves. To obt ain m or e accurat e r elat iv e energies for t he r elevant int erm ediat es, TS and pr oduct s, t he com plet e basis- set m odel chem ist ry CBS- QB3 23 was used. Table VI .1 shows t hat t he CBS- QB3 result s ar e in good agr eem ent w it h t he available exper im ent al dat a; t he discrepancy com par ed t o exper im ent is about 1- 2 kcal/ m ol. CBS- QB3 w orks fair ly w ell for equilibr ium st ruct ures where wav efunct ions com m only hav e singlereference charact er. For t ransit ion st ruct ures, wher e t he breaking of old bonds and t he form at ion of new bonds occur sim ult aneously , t he wav efunct ions ar e m or e lik ely t o hav e m ult i- refer ence charact er or suffer from near degeneracy effect s; i.e. t he non- dynam ic elect ronic correlat ions becom e im port ant ; t his is oft en indicat ed by a high spin- cont am inat ion. I n t his work, w e used t he CASSCF24,25 m et hod t o analyze t he w av efunct ions for k inet ically im port ant m inim a and t ransit ion st at es showing a high spin- cont am inat ion in t he singlereference t reat m ent . The configurat ion int eract ion ( CI ) v ect ors from t hese com put at ions ar e present ed in Table VI .2 and show t hat t he highest CI coefficient s, i.e. t he cont ribut ions of t he m ost im port ant configurat ion, are > 0.9 for all species consider ed, indicat ing t hat t he HF- configurat ion is dom inant and hence t hat t he single- reference m et hod CBS- QB3 is expect ed t o y ield reasonable result s. To inv est igat e t he pr ox im it y or possible cr ossing of t he t r iplet and singlet pot ent ial energy curv es as a funct ion of t he OCC angle in • OCH 2 • CCH 2 species, or t he CCC angle in



CH 2 C( = O) CH 2 • species, we used t he CASSCF( 8,8) / cc- pVDZ

level of t heory t o opt im ize geom et ries at each fix ed OCC ( or CCC) angle, w hile ot her coordinat es in



OCH 2 • CCH 2 ( or



CH 2 C( = O) CH 2 • ) species w ere allowed t o

relax freely . Energies w ere subsequent ly refined em ploy ing t he CASPT2( 8,8) / ccpVDZ lev el 26 based on t he CASSCF reference wav e funct ion, t hereby t ak ing dynam ic elect ron corr elat ions int o account .

Chapt er VI : The O + allene react ion

117

The DFT- B3LYP and CBS- QB3 calculat ions w er e perform ed using t he Gaussian 03 package, 27 while t he CASSCF and CASPT2 calculat ions used t he Dalt on 28 and Molpr o 29 packages. Table VI .1: Tot al energy ( Hart rees) including ZPE cor rect ions calculat ed at t he CBS- QB3 lev el of t heor y and cor r esponding relat iv e energy ( kcal/ m ol) for var ious species in t he O( 3 P) + CH 2 = C= CH 2 react ion. Wher e possible, dat a is also giv en deriv ed fr om exper im ent - based 0 K ent halpies in t he lit erat ure.

Relat ive energy

O( P) + CH2 = C= CH2

−191.407952

HCO + C2 H3

−191.447709

0.0 −24.9

0. 0 −24. 0 ± 1.6

C2 H2 + H2 CO

−191.531528

−77.5

−77. 6 ± 0. 5

1

−191.598601

3

CO +

C2 H4

Ex pt l.

d

Tot al ener gy

Species

−119.6 −119. 6 ± 0.4

−191.445622

−23.6

−24. 4 ± 1. 0

HCCO( X A" ) + CH3

−191.453521

−28.6

−29. 2 ± 1. 4

OH + H2 CCCH( C2 v , 2 B1 )

−191.428384

−12.8

−12. 9 ± 1. 5

H + CO + C2 H3

−191.424757

−10.5

−9. 0 ± 1. 1

1

H2 CC + H2 CO

−191.462174

−34.0

3

H2 CC + H2 CO

−191.386907

3

CH2 + H2 CCO 2

CO +

3

C2 H4

−191.492672

13.2 −53.2

CO +

3

CH3 CH

−191.483104

−47.2

CO +

1

CH3 CH

−191.479133

−44.7

H + H2 CCHCO

−191.457397

−31.0

H + H2 CCCHO( Cs, 2 A" ) H + H2 CCCHO( Cs, 2 A′)

−191.430238

−14.0

−191.423338

−9.7

H + CH3 CCO

−191.441714

−21.2

H + HCCCH2 O

−191.386875

H2 + H2 CCCO

−191.530180

13.2 −76.7

I nt 1a( C2v , 3 B2 )

−191.517546

−68.8

I nt 1b( C2v , 3 B1 )

−191.486270

−49.1

−191.446695

−24.3

I nt 2a( Cs, A" ) : t rans O–CH2 – C= CH2 I nt 2b( Cs, 3 A′)

−191.445006

−23.3

−191.437097

−18.3

I nt 3: CH3 –C( = O) –CH

−191.496225

−55.4

I nt 4

−191.483514

−47.4

I nt 5: cyc- CH2 OC – CH2

−191.451344

−27.2

I nt 6: • CH2 –O– • C= CH2

−191.447832

−25.0

3





I nt 2a( Cs, A" ) : cis O–CH2 – C= CH2 3













−191.502742

−59.5





I nt 7: CH2 – CHCHO

−191.489004

−50.9

I nt 8: • CH2 –CH2 – • C= O

−191.506178

−61.6

I nt 9: t rans- CH3 – • CH– • CO

−191.522602

−71.9

I nt 9: cis- CH3 – CH– CO

−191.522458

−71.9

I nt 10: cis- CH3 –C–CHO

−191.497396

−56.1

I nt 10: t rans- CH3 –C–CHO

−191.494944

I nt 7: CH2 – CHCHO



1

I nt 11( C2 v , A1 )

118



−54.6 −67. 0

Chapt er VI : The O + allene r eact ion

a

I nt 12: cyclopropanone

−191.567162

I nt 13: allene oxide

−191.551301

−99. 9 −90. 0

I nt 15: m et hyl ket ene

−191.599289

−120.1

I nt 16: t rans- acrolein

−191.599743

−120.4

I nt 16: cis- acrolein

−191.596458

TS1a( Cs, 3 A" ) TS1b( Cs, 3 A′)

−191.406607

−118.3

0.8; 0. 9

c

3. 5

c

1.2; 1. 4

c

3. 1

c

TS2a( Cs, A" ) TS2b( Cs, 3 A′)

−191.406043

TS3a( Cs, 3 A" )

−191.396312

7. 3

TS3b( Cs, A’)

−191.392722

TS4

−191.434020

9. 6 −16. 4

TS5

−191.434847

−16. 9

TS6

−191.425495

−11. 0

TS7

−191.424484

−10. 4

TS8

−191.428845

−13. 1

TS9

−191.427161

−12. 1

TS10

−191.433200

−15. 8

TS13

−191.417779

−6. 2

TS14

−191.417175

−5. 8

TS15

−191.451302

−27. 2

TS16

−191.446861

−24. 4

TS17

−191.441421

−21. 0

TS18

−191.444961

−23. 2

TS20

−191.482837

−47. 0

TS22

−191.457033

3

3

TS23 TS24

−30. 8 −43. 3

−191.455750

−30. 0 −65.5

TS31

c

b

TS32

−191.503085

−59. 7

TS33

−191.509124

−63. 5

TS34

−191.482013

−46. 5

TS35

−191.465968

−36. 4

TS37

−191.456414

−30. 4

TS38

−191.484775

−48. 2

a

Der iv ed from t he r elat ive ener gy of –68.8 kcal/ m ol for t riplet oxyally l and a t r iplet - singlet energy gap for ox yallyl of ca. 1.8 k cal/ m ol. 40 b Der iv ed from a relat ive energy of −67.0 kcal/ m ol for singlet biradical oxy allyl and a barr ier height of 1.5 kcal/ m ol com put ed at UB3LYP/ 6- 311G( d, p) level of t heory . c Obt ained from I RCMax ( CBS- QB3: B3LYP) calculat ions. d Mainly t aken from t he web- page: ht t p: / / srdat a.nist .gov/ cccbdb/ ; all values were obt ained at 0 K: ∆H0 f ( O) = 58. 98 k cal/ m ol; ∆H0 f ( C2 H2 ) = 54.48 ± 0.2 kcal/ m ol; ∆H0 f ( H) = 51.63 kcal/ m ol; ∆H0 f ( CH2 = C= CH2 ) = 47.4 ± 0.3 kcal/ m ol; ∆H0 f ( CH2 ( X3 B1 ) ) = 93.2 ± 0. 5 kcal/ m ol; ∆H0 f ( CO) = −27.2 kcal/ m ol; ∆H0 f ( OH) = 8. 84 ± 0. 1 k cal/ m ol; ∆H0 f ( H2 CO) = −25.1 ± 0.1 kcal/ m ol; ∆H0 f ( C2 H4 ) = 14. 6 ± 0.1 k cal/ m ol; ∆H0 f ( CH3 ) = 35.8 ± 0.1 kcal/ m ol; ∆H0 f ( C2 H3 ) = 73.0 ± 0.8 kcal/ m ol; ∆H0 f ( HCO) = 9. 95 ± 0.1 kcal/ m ol; ∆H0 f ( H2 CCCH) = 85.2 ± 1 k cal/ m ol; 37 ∆H0 f ( H2 CCO) = −10.66 ± 0.3 kcal/ m ol; ∆H0 f ( HCCO) = 42. 0 ± 1 kcal/ m ol;

Chapt er VI : The O + allene react ion

119

Table VI .2: Com put ed CI - v ect or for som e select ed species in t he O( 3 P) + allene react ion using t he CASSCF( 8,8) / cc- pVDZ lev el of t heory

Species I nt 1a I nt 1b I nt 2a TS1a TS3a TS3b TS4 TS7 TS9 TS13

< S2 > 2. 195 2. 195 2. 195 2. 305 2. 325 2. 421 2. 068 2. 387 2. 368 2. 354

1st 0. 95444 0. 95449 0. 96834 0. 95398 0. 94147 0. 93246 0. 96240 0. 95584 0. 95132 0. 93555

2nd –0.13858 –0.12587 –0.16814 –0.18176 –0.16319 –0.15773 –0.08233 –0.22748 –0.10976 –0.17545

CI - vect or 3rd 0. 11159 0. 11765 –0.08921 –0.15110 –0.15954 0. 14422 –0.08017 –0.07431 –0.10390 –0.13529

4t h 0.11134 0.11751 0.07510 –0.07923 0.10251 –0.14239 –0.07804 –0.05498 –0.08989 0.11237

5t h –0.10375 –0.10928 0.05159 –0.05388 0.08544 0.08808 –0.07562 –0.05963 0.11226

V I .2 .2 . RRKM / M a st e r Equ at ion ca lcu la t ion s Product dist r ibut ions as a funct ion of t em perat ur e and pr essure ( P = 1 at m , T = 300- 2000 K) for t he O( 3 P) + allene react ion pr oceeding on t he t r iplet and singlet

surfaces íFRQVLGHUHG DVEHLQJDGLDEDWLFíZHUHVHSDUDWHO\REWDLQHGE\VROXWLRQ of t he w eak - collision m ast er equat ion using Gillespie’s exact st ochast ic sim ulat ion m et hod, 30−32 explained in det ail in our earlier paper 33 and in t he chapt er I I . The Lennard−Jones collision param et ers for bat h gas He are σ = 2.55 Å and ε/ k B = 10 K. 34 Since no collision param et ers for [ C3 H 4 O] are available in t he lit erat ure, t he v alues σ = 4.08 Å and ε/ k B = 421 K ar e est im at ed based on t hose of et hy lene

ox ide C2 H 4 O. 34 Thus, t he collision fr equency Z LJ [ M] was est im at ed at §×10 10 s −1 at 1 at m ospher e and room t em perat ur e.

V I .3 . Result s a n d discu ssion V I .3 .1 . Pot e n t ial e ne r gy sur f ace s As first react ion st ep occurr ing on t he t r iplet surface, t he t r iplet O- at om can eit her at t ack on t he cent ral or t he t erm inal carbon at om , or abst ract an H- at om from allene. We w ill discuss t hese t hree react ion pat hways separat ely . Unless m ent ioned ot herw ise t he r elat iv e energies obt ained at t he CBS- QB3 lev el of t heor y w er e used in t he follow ing discussion. Figur es VI .1 and VI .2 show part of t he t r iplet pot ent ial energy surface for t he react ion of t r iplet ox ygen at om w it h allene, each st art ing from a different addit ion channel: O- at t ack on t he cent ral carbon ( Figur e VI .1) or O- addit ion on t he t erm inal carbon ( Figure VI .2) . Transit ion st at es connect ing t hese t wo regions of t he PES ar e show n in bot h figur es. The singlet PES is draw n in Figure VI .3. Dur ing t he discussion w e w ill indicat e t he dom inant pat hways; t he relat iv e im port ance of

120

Chapt er VI : The O + allene r eact ion

a react ion channel is det erm ined not only by t he bar r ier height of t he TS, but also by t he r igidit y of t he TS, i.e. it s "looser" or " t ight er" ent ropic charact er ist ics, as well as by t he available excess energy in chem ically act iv at ed unim olecular react ions. I nform at ion on t he r elat iv e r igidit y of TS and on t he effect of chem ical act ivat ion w er e obt ained in t he Mast er Equat ion analyses discussed lat er, and is used qualit at iv ely in t his sect ion.

Figure VI .1: Sect ion of t he t r iplet pot ent ial ener gy surfaces st art ing fr om Oaddit ion on t he cent ral carbon at om in CH 2 = C= CH 2 . Maj or pat hway is indicat ed in bold. Tw o H- abst ract ion channels of t he O + C3 H 4 react ion are also indicat ed. The dashed lines in t he cent er connect t o t he PES sect ion of Figur e VI .2

Chapt er VI : The O + allene react ion

121

Figur e VI .2: Sect ion of t he t r iplet pot ent ial ener gy surfaces st art ing at t he Oaddit ion on a t erm inal carbon at om in allene. Maj or pat hways are indicat ed in bold. The dashed lines in t he cent er connect t o t he PES sect ion of Figur e VI .1

122

Chapt er VI : The O + allene r eact ion

Figure VI .3: Singlet pot ent ial ener gy surfaces for t he O( 3 P) +

CH 2 = C= CH 2

react ion. I nt ersyst em crossing fr om t r iplet t o singlet sur face ( I SC) is indicat ed by arr ows, The r eact ion channels on t he t r iplet sur face leading t o t he I SC st art ing int erm ediat es are show n by dashed lines. Maj or pat hways ar e indicat ed in bold.

O- a t om a t t a ck on t h e ce nt r a l ca r bon . Addit ion of t he O- at om on t he cent ral carbon at om can occur on t wo differ ent elect r onic- st at e surfaces, 3 A" and 3 A′, v ia TS1 a ( 3 A" )

and

TS1 b ( 3 A′)

leading

t o adduct s I n t 1 a ( 3 B2 )

and

I n t 1 b ( 3 B1 ) ,

respect ively ( see Figure VI .1) . These st eps face bar riers of 0.9 and 3.5 kcal/ m ol height , r espect iv ely . I RC calculat ions im posing a Cs sym m et ry indicat e t hat TS1 a ( 3 A" ) and TS1 b ( 3 A′) cor relat e t o I nt 1 b ( 3 B1 ) and I n t 1 a ( 3 B2 ) , r espect iv ely ( see Figur e VI .4) , such t hat t he t wo curv es,

3

A" and

3

A′, hav e t o cross

som ew her e, ar ound a C- O bond dist ance of 1.7 Å. I n realit y , t he react ion proceeds on sur faces w it hout sym m et ry const r aint s, and t hese t wo curv es avoid crossing w hen r elaxat ion of sym m et ry from Cs t o C1 is allow ed.

Chapt er VI : The O + allene react ion

123

Figur e VI .4: G2M/ / B3LYP/ 6- 311G( d,p) calculat ions for t he O- addit ion on t he cent ral carbon at om wit h im posed Cs sym m et ry , 3

generat ing t w o different ,

3

crossing elect r onic st at e surfaces, A″ and A′

Tr iplet biradicals I n t 1 a ( 3 B2 ) and t he first elect ronically excit ed st at e I n t 1 b ( 3 B1 ) lie 68.8 and 49.1 kcal/ m ol, r espect iv ely, below t he init ial react ant s. The m ain differences bet w een t hese t wo st at es is t he locat ion of t he t w o unpaired elect r ons, so it is int er est ing t o br iefly discuss t he shape of t he t wo part ly single occupied m olecule orbit als ( SOMO) in I nt 1 b , especially as t hey change t he geom et r y of I n t 1 b com pared t o I n t 1 a . One SOMO orbit al of I nt 1 b, form ed by a com binat ion of t w o Px ( C) orbit als locat ed at t w o differ ent carbon cent ers ( see Figur e VI .5) , is perpendicular t o t he m olecular plane of sym m et r y, such t hat t he elect r on is delocalized along t he CCC backbone. This result s in t he calculat ed C- C bond dist ance of 1.393 Å, i.e., closer t o a double t han a single C- C bond lengt h. The second SOMO orbit al lies in t he m olecular plane of sym m et r y and is locat ed

124

Chapt er VI : The O + allene r eact ion

at t he O- at om . So, t he C- O bond dist ance elongat es t o 1.333 Å, ~ 0. 1 Å longer t han a norm al > C= O double bond. At t em pt s t o locat e TS from t his elect ronically excit ed I n t 1 b leading t o ( elect ronically excit ed) isom ers or dir ect ly t o product s, including e.g. CH 3 + HCCO( A2 A′) , were unsuccessful. By analogy w it h t he I n t 1 a react ions, t he r elevant barr iers ar e expect ed t o lie higher t han t he ent rance TS, such t hat re- dissociat ion back t o t he init ial r eact ant s would be t he dom inant fat e. Also, m any of t he isom erisat ion pat hways for I nt 1 b w ould m erge wit h t he pat hways of I nt 1 a due t o a lowering/ changing of t he sym m et ry around t he TS, such t hat t he k inet ic behav ior is essent ially t he sam e as if int er nal conv ersion ( I C) t o t he low er elect r onic surface first t ook place. As a r esult , I n t 1 b is m odeled t o undergo int er nal conv ersion t o t he low er elect ronic surface ( I n t 1 a ) ev en at t he highest t em perat ur es, since t he r e- dissociat ion v ia TS1 b back t o t he init ial

react ant s at t he av erage int ernal energy ( k ( < E> ) § 10 s−1 wit h < E> = 98 kcal/ m ol at 2000 K, i.e. 49 kcal/ m ol abov e t he init ial r eact ant s) is m uch slow er t han t he I C, w it h rat e est im at ed at 10 12 s−1 . 35,36

Figure VI .5: SOMO orbit als for I n t 1 a and I nt 1 b . Som e k ey bond dist ances are giv en in Angst röm

Chapt er VI : The O + allene react ion

125

Tr iplet biradical I nt 1 a belongs t o t he C2v point group. I t s first SOMO orbit al is sim ilar t o t he first SOMO in I n t 1 b , but t he shape of t he second SOMO orbit als ar e com plet ely different ( see Figure VI .5) . I n I nt 1 a it is form ed by a linear com binat ion of Px ( C) , Px ( C) , and Px ( O) orbit als locat ed at t hree differ ent at om ic cent ers. As a consequence, bot h unpaired elect rons in I n t 1 a are delocalized ov er t he m olecule, and t he C- O bond lengt h, 1.266 Å, is closer t o a double bond t han in I n t 1 b while t he C- C bond lengt h, 1.466 Å, is closer t o a single t han a double C- C bond. St art ing at I n t 1 a , t here ar e five different react ion pat hway s, nam ely : ( i) CH 2 - loss via TS4 t o y ield t r iplet m et hy lene plus k et ene, needing t o overcom e a barrier of 52.4 k cal/ m ol height ; ( ii) a 1,3 H- shift v ia TS5 t o isom er ize t o I nt 3 , facing a barr ier of 51.9 kcal/ m ol; ( iii) Rearrangem ent t o I n t 4 v ia TS6 w it h a barr ier of 57.8 kcal/ m ol by a 1,3 H- shift from a carbon t o t he ox ygen at om ; ( iv ) ring- closure leading t o I n t 5 v ia TS7 , ov er a barr ier of 58.4 k cal/ m ol; ( v ) and finally, re- dissociat ion back t o t he init ial react ant s, w hich however cannot com pet e wit h t he form er r eact ions because of it s high barr ier of 69.7 kcal/ m ol. I nt 3 , ly ing 55.4 kcal/ m ol below t he init ial react ant s, eit her br eak s t he C- C bond t o y ield m et hy l plus k et enyl radicals v ia TS8 , or isom er izes back t o I nt 1 a v ia TS5 . The barr ier height s for t he form er and lat t er pat hways ar e 42.3 and 38.5 k cal/ m ol, respect iv ely. Alt hough TS8 lies 3.8 kcal/ m ol higher t han TS5 , t he for m er is m uch looser, such t hat t hese t w o pat hways should be com pet it iv e. I nt 4 , 47.4 kcal/ m ol below t he init ial r eact ant s, decom poses m ore easily t o hy drox y l plus propargy l radical v ia t he looser TS9 t han it s r ear rangem ent back t o I nt 1 a v ia t he t ight er TS6 . The decom posit ion v ia TS9 faces a bar r ier of 35.3 k cal/ m ol. I n t 5 w ill undergo t he r ing- opening process via TS1 0 w it h a sm all barr ier of 11.4 kcal/ m ol, leading t o I n t 2 a . This int erm ediat e is m or e closely relat ed t o addit ion react ions on t he cent ral carbon ( see Figure VI .2) and it s subsequent r eact ions ar e discussed in t he next sect ion. I n sum m ary, t he O- addit ion on t he cent ral car bon at om in allene result s init ially in t he form at ion of bot h t he t r iplet biradical adduct s, I nt 1 a and I nt 1 b, of which t he lat t er rapidly carr ies out an I C pr ocess t o y ield I n t 1 a . When t he spin is conserv ed t hroughout t he r eact ion, decom posit ion of I nt 1 a int o t r iplet m et hylene plus k et ene is expect ed t o be dom inant . Ot her decom posit ion channels could cont r ibut e, but are expect ed t o play a very m inor role. I t should be m ent ioned t hat t he product ket ene was exper im ent ally obser ved earlier. 9 React ions follow ing an int ersyst em crossing t o t he singlet surface are discussed in a lat er sub- sect ion.

126

Chapt er VI : The O + allene r eact ion

O- a t om a t t a ck on a t er m in a l ca r b on . I RC calculat ions show t hat t he Oaddit ion on t he t erm inal carbon at om in allene proceeds t hr ough t wo differ ent t ransit ion st ruct ur es, TS2 a ( 3 A" ) and TS2 b ( 3 A′) , giv ing rise t o t r iplet bir adical adduct s I n t 2 a ( 3 A" ) and I n t 2 b ( 3 A′) , r espect iv ely. The barrier height s com put ed for t hese t w o st eps ar e 1.4 and 3.1 kcal/ m ol.

Figure VI .6: SOMO orbit als for I n t 2 a and I nt 2 b

I n t 2 b ( 3 A′) has a relat ive energy of −18.3 kcal/ m ol w it h r espect t o t he init ial react ant s and belongs t o a Cs point group. Bot h SOMO orbit als carry ing t he t w o unpair ed elect r ons in I n t 2 b lie in t he m olecular plane of sy m m et ry. One orbit al is locat ed on t he O- at om , while t he ot her is on t he cent ral carbon at om ( see Figur e VI .6) . As for I n t 1 b, t he elect r onically excit ed I nt 2 b can eit her undergo an int ernal conv ersion ( I C) t o t he lower- ly ing elect ronic st at e ( I nt 2 a ) , re- dissociat e t o t he original react ant s, or undergo furt her unim olecular r eact ions ( dissociat ion or isom er isat ion) . As seen for I n t 2 a , t he TS for t hese lat t er pr ocesses all break t he sy m m et ry, m erging t he A’ and A" surfaces such t hat t hese r eact ions are sim ilar t o r eact ions aft er I C. While w e cannot a prior i quant ify t he rat e of I C, it is expect ed t o be dom inant at low t o m oderat e t em perat ur es ( T ”  .  wher eas r e- dissociat ion is expect ed t o becom e favorable only at very high

t em perat ur es ( T •. 7KHFDOFXODWHGHQHUJ\- specific rat e coefficient at t he

Chapt er VI : The O + allene react ion

127

average energy for t he r e- dissociat ion st ep of I n t 2 b sharply increases as a funct ion of t em perat ure, for exam ple fr om 5.2×10 9 s −1 at T = 1000 K, passing t hrough 3.3×10 10 s−1 at T = 1500 K, and t o 1.2×10 11 s −1 at T = 2000 K. Ther efore, t he I C pr ocess should st rongly com pet e w it h t he r e- dissociat ion, except at t he highest t em perat ur es. I nt 2 a, −24.3 kcal/ m ol r elat ive t o t he init ial r eact ant s, possesses a 3 A" sym m et ry in t he Cs point gr oup. I n I n t 2 a , one SOMO orbit al lies in t he m olecular sym m et ry plane at t he cent ral car bon at om while t he ot her SOMO orbit al is perpendicular t o t he m olecular plane and locat ed on t he O- at om ( see Figur e VI .6) , result ing in an addit ional st abilizat ion of 6 k cal/ m ol for I n t 2 a

com par ed t o I n t 2 b. The

chem ically act ivat ed t r iplet adduct I n t 2 a can elim inat e an H- at om t o giv e r ise t o alleny lox y radical + H product s. This pat hway goes t hr ough a loose t ransit ion st ruct ur e TS1 3 aft er clear ing a bar r ier of 18.1 kcal/ m ol height . I nt 2 a can eit her proceed t hrough a r ing- opening process leading t o I n t 5 v ia TS1 0 , or concert edly undergo a 1,2 H- m igrat ion and a r ing- opening t o I n t 7 v ia TS1 4 . The bar r ier height s are 8.5 and 18.5 kcal/ m ol, respect ively . I nt 5 , −27.2 kcal/ m ol relat iv e t o t he init ial react ant s, will rear range t o I nt 1 a v ia TS7 by r ing- opening. This pat hway

needs

to

overcom e

a

barr ier

of

16.8

kcal/ m ol,

and

leads

to

int erm ediat es already discussed ear lier ( see Figur e VI .1) . St ill, rearr angem ent of I nt 5 back t o I nt 2 a v ia TS1 0 is m or e fav orable, in v iew of t he lower bar rier of 11.4 kcal/ m ol. I nt 7 , w it h r elat ive energy 59.5 kcal/ m ol below t he init ial react ant s, has four different accessible react ion pat hw ays, nam ely: ( i) H- elim inat ion t o yield H • + CH 2 = CH−• CO via TS1 6 , needing t o overcom e a barrier of 35.1 kcal/ m ol; ( ii) C- C bond r upt ure t o pr oduce • C2 H 3 + HC• O v ia TS1 7 aft er clear ing a bar r ier of 38.5 k cal/ m ol height ; ( iii) rearrangem ent by 1,2- H m igrat ion t o I n t 8 v ia TS1 5 w it h a barr ier of 32.3 kcal/ m ol; and finally , ( iv ) a 1,3 H- shift t o giv e r ise t o I nt 9 v ia TS1 8 , t he bar rier height of w hich is 36.3 k cal/ m ol. Next , I n t 9 , −71.9 kcal/ m ol com pared t o t he init ial react ant s, can eit her decom pose t o t r iplet CH 3 CH plus CO v ia TS2 3 by C- C bond r upt ure or t o H • + CH 2 = CH−• CO v ia TS2 4 by H- loss. These pat hways face barr iers of 28.6 and 41.9 kcal/ m ol, respect iv ely. I n t 9 could undergo a 1,2 H- shift leading t o I nt 8 t hr ough TS2 2 aft er clearing a high barr ier of 41.1 kcal/ m ol. Because TS2 3 is t he low est and loosest TS, t he I nt 9 → TS2 3 → 3 CH 3 CH + CO channel is expect ed t o be t he dom inant decom posit ion channel of I nt 9 . I n t 8 , −61.6 kcal/ m ol relat iv e t o t he init ial react ant s, will im m ediat ely break t he single C- C bond v ia a v ery low - ly ing TS2 0 t o y ield t r iplet et hy lene plus CO. This st ep faces a low barr ier of only 14.6 kcal/ m ol, indicat ing t hat rearrangem ent back t o I nt 7 or I nt 9 is unlik ely .

128

Chapt er VI : The O + allene r eact ion

I n sum m ary , t he O- addit ion on t he t erm inal carbon at om in allene giv es r ise t o t he v ibrat ionally excit ed, t r iplet biradical adduct s I nt 2 a and I n t 2 b. Under condit ions of low t o m oderat e t em perat ur es, I n t 2 b w ill rapidly undergo an I C process t o I n t 2 a , w her eas at higher t em perat ur es I n t 2 b re- dissociat es back t o t he init ial r eact ant s in com pet it ion w it h t he I C process. I n t 2 a , in t ur n, will m ost ly decom pose t o alleny lox y plus hydr ogen at om . The I n t 2 a → I n t 5 → I nt 1 a → 3

CH 2 + k et ene pat hway is also expect ed t o play an im port ant r ole, indicat ing t hat

t he t riplet sur face for t he O- at t ack on t he t erm inal and cent ral car bon at om s cannot be div ided int o independent sit e- specific regions. The product s alleny loxy radical plus hydrogen at om w er e obser ved in m olecular beam experim ent s by Lee and cow ork ers, 10 while k et ene was det ect ed in an exper im ent al m at rix st udy. 9 The I n t 2 a → TS1 4 → I n t 7 st ep cannot com pet e w it h t he t wo form er channels; as a r esult , ot her product s ( including CO +

3

C2 H 4 and CO +

3

CH 3 CH) produced

from I n t 7 on t he t r iplet PES in Figur e VI .2 can only be very m inor. This result reconfirm s t he exper im ent al st udies, 7−11 which concluded t hat t he pr oduct CO should be form ed from a singlet int erm ediat e aft er an I SC process fr om t r iplet t o singlet sur faces. Th e H - a bst r a ct ion . The O- at om can abst ract an H- at om in allene y ielding hydrox y l plus pr opargy l radical; t he r eact ion has a calculat ed exot herm icit y of 12.8 kcal/ m ol, in excellent agreem ent w it h t he exper im ent al ∆r H( 0 K) v alue of −12.9 kcal/ m ol. 16,37 This pat hway can pr oceed via t w o t ransit ion st ruct ur es of different elect ronic st at es, TS3 a ( 3 A″) and TS3 b( 3 A′) , cor r elat ing direct ly w it h t he hydrox y l radical w it h a

2

Π sym m et ry ( see Figur e VI .1) . The bar r ier height s for

t hese channels are 7.3 and 9.6 kcal/ m ol, respect ively , about 5- 6 k cal/ m ol higher in energy t han t he O- addit ion t ransit ion st ruct ures. Consequent ly, t he Habst ract ion channel cannot com pet e w it h any of t he O- addit ion channels at low t o m oderat e t em perat ur es, but m ay play a r ole at com bust ion t em perat ur es. To t he best of our k now ledge, all exper im ent al st udies for t he t it le r eact ion w er e car r ied out at eit her low and m oderat e t em perat ures ( T < 900 K) 13−15 or at m oderat e collision energies in m olecular beam exper im ent s. 10 As a consequence, t he product s hy droxy l and propargy l radical w er e never observ ed. I t is of int erest t o know how im port ant t he H- abst ract ion flux is com pared t o t he t ot al chem ical react ion flux at higher t em perat ur es. For t his purpose, t he fract ion of Habst ract ion flux as a funct ion of t em perat ur e is com put ed using eq ( VI .10) based on TST considerat ions, and plot t ed in Figure VI .7.

Chapt er VI : The O + allene react ion

129

FH −abstraction =

PH −abs PO −add −cen + PO −add −ter + PH −abs

( VI .10)

w it h ≠ PH − abs = κ TS 3a × QTS≠ 3a × exp( − ETS 3a / RT ) + κ TS 3b × QTS 3b × exp( − ETS 3b / RT ) ,

PO −add −cen = QTS≠ 1a × exp( − ETS 1a / RT ) + QTS≠ 1b × exp( − ETS 1b / RT ) , and ≠ PO −add −ter = QTS≠ 2 a × exp( − ETS 2 a / RT ) + QTS 2 b × exp( − ETS 2 b / RT )

w here QTS is t he com plet e part it ion funct ion of t he given TS, R is t he univ ersal gas const ant , ETS is t he energy of t ransit ion st ruct ure TS r elat iv e t o t he init ial react ant s, and κTS3 is t he one- dim ensional t unneling corr ect ion for H- abst ract ion, w hich is com put ed by assum ing an asym m et ric Eckart pot ent ial. 38,39 I t should be m ent ioned t hat t he r eact ion pat hway degeneracy derived fr om t he sym m et ry num bers in t he part it ion funct ion for rot at ion is equal t o 4 for each of t he t hree react ion channels. This is obv ious for t he H- abst ract ion ( 4 equivalent hy drogens) and st ill easy t o see for t he addit ion t o t he t er m inal carbons ( CH 2 groups) : t her e are t w o of t hese carbons, and at t ack can com e from t wo sides, ex act ly equiv alent for each of t he lobes of t he π- bond. For t he cent ral C, t her e ar e 4 equivalent dir ect ions of at t ack, t wo for each of t he t w o π- bonds it part icipat es in.

130

Chapt er VI : The O + allene r eact ion

Figure VI .7: Calculat ed fract ions of t he H- abst ract ion and t he sit e- specific Oaddit ion r eact ion flux es as a funct ion of t em perat ur e ( not corr ect ed for redissociat ion)

As can be seen in Figur e VI .7, t he fract ion of t he H- abst ract ion flux depends st rongly on t em perat ur e, i.e. ∼0% at 300 K, 4% at 900K, and r ising t o ∼35% at 2000 K, indicat ing t hat t he H- abst ract ion r eact ion channels can cont r ibut e significant ly at com bust ion t em perat ur es. I n cont rast , t he fract ion of O- addit ion on t he cent ral carbon at om st eeply decreases w it h incr easing t em perat ur es, from 75% at 300 K t o 35% at 2000 K. Nonet heless, it rem ains t he m ost im port ant react ion

flux

in

m uch

of

t he

relevant

t em perat ur e

range.

Tem perat ur e-

dependence of t he fract ion of t he O- addit ion on t he t erm inal carbon is m ore com plicat ed. First , it sharply rises wit h t em per at ur e and reaches a m axim um of

Chapt er VI : The O + allene react ion

131

45%

ar ound T= 800 K, t hen alm ost

linear ly decreases w hen t em perat ur e

increases furt her. St ill, t hr oughout t he 300 - 2000 K r ange, t his channel cont r ibut es 30- 45% . Not e t hat t ak ing int o account t he var iat ional char act er of t he low- bar rier t ransit ion st at es for addit ion should ev en incr ease som ew hat t he calculat ed cont r ibut ion of H- abst ract ion at high t em perat ur es. Th e low e st - lyin g sing le t su r face. The CBS- QB3 result s for t he t riplet sur face and st at ist ical- kinet ics predict ions of t he pr oduct dist ribut ion ( see next sect ions) show t hat CO pr oduct ion fr om t he t riplet sur faces has a very sm all yield ( < 2% ) under all relevant condit ions. Yet , all experim ent al st udies7−12 find CO t o be t he m ost im port ant pr oduct . Hence, t he t it le react ion cannot t ak e place only on t he t riplet sur faces, but m ust also inv olv e t he ( low est - ly ing) singlet surface follow ing an I SC pr ocess.

Figur e VI .8: Pot ent ial energy curv es around t he crossing seam for oxyally l as relevant for O- at t ack on t he cent ral carbon at om . Squar es: ener gies of t r iplet I nt 1 a opt im ized at fix ed C- C- C angles; Circles: ener gies of singlet wav efunct ions calculat ed on t he opt im ized t r iplet geom et r ies ( i.e. v ert ical t ransit ion from t r iplet t o singlet surface) ; Tr iangles: ener gies of singlet I nt 1 1 opt im ized at fix ed C- C- C angles, show ing a negligible bar rier t o cyclizat ion .

132

Chapt er VI : The O + allene r eact ion

For t he O- addit ion on t he cent ral carbon at om , a crossing r egion bet w een t r iplet and singlet surfaces is expect ed in t he v icinit y of t he equilibr ium st ruct ure of t riplet biradical adduct I n t 1 a , because t he t r iplet - singlet energy gap of oxy ally l is very sm all, 1- 2 kcal/ m ol as alr eady com put ed by Coolidge et al. 40 at t he MRCI level of t heory . To inv est igat e t he pr ox im it y of t he t r iplet and singlet pot ent ial energy cur v es, w e exam ine t he energies of ox yally l as a funct ion of t he CCC angle in t he 90- 140° range, i.e. t he degr ee of freedom in w hich t he t r iplet ( I nt 1 a ) and singlet ( I n t 1 1 ) oxyally l species differ m ost and wher e facile cyclizat ion t o cyclopropanone m ight dist ort t he PES m ost . We carried out geom et r y opt im izat ions at several fix ed CCC angles for a t r iplet wavefunct ion, while allow ing t he r em aining 3N- 7 int ernal coor dinat es in oxyally l t o relax fr eely ; we t hen also used t he sam e geom et r ies t o calculat e t he energies for a singlet wavefunct ion, allow ing us t o exam ine t he energy differ ence for a vert ical t ransit ion from t r iplet t o singlet surface. The obt ained r esult s plot t ed in Figure VI .8 show t hat t he t r iplet and singlet cur ves ar e close, w it hin a few k cal/ m ol, for all CCC angles. Tr iplet oxyally l is predict ed t o be t he ground st at e, w it h a t r iplet singlet energy split t ing of only 1 kcal/ m ol, in good agreem ent w it h t he r eport ed MRCI r esult . 40 Therefor e, in addit ion t o react ions of t r iplet ox yally l I n t 1 a on t he t riplet surface, I n t 1 a has a fair pr obabilit y t o undergo an int ersyst em crossing process t o v ibrat ionally excit ed singlet ox y ally l. Ox yally l geom et r ies re- opt im ized for singlet wavefunct ions at fix ed CCC bond angles were also calculat ed t o visualize t he geom et r ic relaxat ion of t he singlet int erm ediat es aft er t he v ert ical I SC ev ent ( see Figur e VI .8) . As expect ed, cyclizat ion t owards cyclopr opanone is found t o hav e only a sm all bar r ier. The r elat ive im port ance of sur face crossing versus react ions on t he t r iplet sur face depends on t he t em perat ur e, wit h t he lat t er m or e fav orable at higher t em perat ur es, and t he form er pr edom inant at low t o m oderat e t em perat ur es. For exam ple, at 300 K rat e for decom posit ion of I n t 1 a is est im at ed t o ~ 2×10 9 s−1 , m uch slow er t han rat e of I SC ( est im at ed at ca.

10 11 s−1 ) , w hile decom posit ion at t he av erage energy at 2000 K ( r at e §×10 11 s−1 )

is m uch fast er t han I SC. Accurat e dy nam ic calculat ions are cert ainly requir ed in order t o quant ify t hese t wo com pet ing r eact ion st eps, but t hese ar e out side t he scope of t he pr esent w ork. The subsequent r eact ions of t he singlet int erm ediat es t hus form ed ar e show n in Figure

VI .3.

Singlet

ox yally l

I nt 1 1

will

undergo

ring- closure

to

singlet

cyclopr opanone, I nt 1 2 , wit h an energy of −99.9 k cal/ m ol r elat iv e t o t he init ial react ant s ( see Figure VI .3) . This st ep goes t hr ough TS3 1 and faces a sm aller barr ier of 0.4 kcal/ m ol obt ained at t he CASSCF level, 41 and t hat t he react ion is

Chapt er VI : The O + allene react ion

133

ev en bar r ier less when dynam ic elect ron corr elat ion w as t aken int o account using t he CASPT2 m et hod. 41 Hence, t he I n t 1 1 → I n t 1 2 st ep has a v ery sm all t o nonex ist ent bar r ier, and in any case w ill be v er y r apid giv en t he chem ical act ivat ion of I n t 1 1 ar ising aft er t he I SC pr ocess. I n t 1 1 could also undergo r ing- closur e leading t o singlet allene oxide, I nt 1 3 . How ev er, t his pat hway goes v ia a t ight er TS3 2 w it h a higher barr ier of ~ 7 kcal/ m ol, and so cannot com pet e wit h t he I nt 1 1 → TS3 1 → I nt 1 2 r out e. Under high- pr essure condit ions, singlet cyclopr opanone I n t 1 2 can be t herm ally st abilized; it was observ ed in a m at r ix st udy. 9 At low pressur es, I n t 1 2 w ill eit her decom pose t o y ield CO + C2 H 4 v ia TS3 3 or isom er ize t o m et hy lk et ene I n t 1 5 v ia TS3 4 or t o 2- propanal I n t 1 6 v ia TS3 5 . The barr ier height s for t hese t hree pat hways ar e 36.4, 53.4 and 63.5 kcal/ m ol, r espect ively . As, m or eover, TS3 3 is looser t han t he ot her t wo, t his r esult s in pr edom inance for t he I n t 1 2 → TS3 3 → CO + C2 H 4 channel, in good agreem ent w it h t he exper im ent al obser vat ions. 7−12 Not e t hat because bot h TS3 1 and TS3 3 lie far below t he init ial react ant s and only slight ly abov e t he init ial singlet int erm ediat e I n t 1 1 , t he chem ically act ivat ed I nt 1 1 w ill isom er ize in less t han a ps t o I n t 1 2 w hich in t ur n w ill decom pose equally pr om pt ly t o end pr oduct s ( or be collisionally st abilized at v ery high pressur es) , such t hat in any case t he probabilit y of occur rence of a r ev erse I SC of singlet I n t 1 1 back t o t r iplet I n t 1 a w ill be st rongly r educed. We also carr ied out an analogous CASPT2/ / CASSCF st udy of t he crossing seam around t he equilibr ium st ruct ur e of t r iplet biradical adduct oxyallene I nt 2 a ( see Figur e VI .9) , w here t he C- C- O bond angle was found t o be t he m ost significant difference bet w een t he singlet and t r iplet surfaces. The v ert ical energy gap bet ween t r iplet and singlet surfaces, calculat ed fr om t he energies of t r iplet biradical • O−CH 2 −• C= CH 2 geom et r ies ( I nt 2 a ) opt im ized for fix ed CCO bond angles v ersus t he energies of singlet wav efunct ions on t hese sam e geom et r ies, is only a few kcal/ m ol for bond angles abov e 100°. The prox im it y of singlet and t riplet PES enhances t he probabilit y of an I SC ev ent allowing t he form at ion of singlet int erm ediat es. Energies of re- opt im ized singlet



O−CH 2 −• C= CH 2 int erm ediat es

( I nt 1 3 ) as a funct ion of t he CCO bond angle ( see Figure VI .9) show a negligible barr ier t o cyclizat ion yielding allene ox ide. Not e t hat t he lifet im e of t r iplet oxyallene is so short , i.e. ∼10 ps at 300K and quickly reducing t o ∼1 ps at 1000K, t hat an I SC pr ocess is lik ely t o occur only at low t o m oderat e t em perat ures ( < 1000 K) . As a result , decom posit ions of I n t 2 a w it h spin- conser vat ion ar e expect ed t o be t he principal pat hways at com bust ion t em perat ur es. Singlet allene ox ide form ed fr om t he fast cyclizat ion react ion aft er I SC, can eit her be st abilized

under high- pr essur e condit ions í DVREVHUYHG LQWKHPHQWLRQHGPDWUL[ VWXG\9 í

134

Chapt er VI : The O + allene r eact ion

or isom er ize on t he singlet surface at lower pr essures. I nt 1 3 can undergo r ingopening t o I n t 1 1 , followed by rapid r earrangem ent t o I n t 1 2 and subsequent decom posit ion t o product s CO + C2 H 4 . The I nt 1 3 → TS3 2 → I n t 1 1 pat hway faces a barr ier of 30.3 kcal/ m ol, i.e., only half t he 59.6 kcal/ m ol barr ier t o I nt 1 3 → TS3 7 → I n t 1 6 . As a result , t he first pat hway , y ielding CO + C2 H 4 , is dom inant under all relev ant t em perat ur es.

Figure VI .9: Pot ent ial ener gy curv es ar ound t he crossing seam for ox yallene as relevant for t he case of O- at t ack on a t erm inal carbon at om . Squar es: energies of t riplet I nt 2 a opt im ized at fixed C- C- O angles; Circles: energies of singlet wavefunct ions calculat ed on t he opt im ized t riplet

geom et ries ( i.e.

vert ical

t ransit ion fr om t riplet t o singlet surface) ; Triangles: ener gies of singlet I nt 1 3 opt im ized at fixed C- C- O angles, show ing a negligible bar r ier t o cyclizat ion.

I n sum m ary , crossing seam s wer e locat ed for int erm ediat es form ed in bot h t he cent ral and t erm inal O- addit ions t o allene; t he int ersyst em crossings from t r iplet t o t he low est - ly ing singlet sur face can lead t o v ibrat ionally hot cyclopr opanone

Chapt er VI : The O + allene react ion

135

and allene ox ide int erm ediat es. The cont ribut ion of t hese I SC pr ocesses is sensit iv e t o t he r eact ion t em perat ure and is predict ed t o be im port ant m ost ly at low t em perat ures. The hot singlet int erm ediat es will decom pose ent irely int o CO + C2 H 4 under low - pr essure condit ions, but becom e t herm ally st abilized at high pressur es. The pr edict ed product s singlet cy clopropanone, allene ox ide, CO, and C2 H 4 w er e all observ ed exper im ent ally . 9,10 V I .3 .2 . Qu a n t ifica t ion of t he product dist r ib ut ion The overall product dist ribut ion of t he t it le r eact ion can be obt ained from t he w eight ed sum m at ion of t he product dist r ibut ions for t he differ ent , sit e- specific ent rance channels. These channel- specific pr oduct dist r ibut ions, as well as t he cont r ibut ions of t he differ ent O- addit ion and H- abst ract ion channels t o t he overall rat e coefficient , ar e funct ions of t em perat ure ( see e.g. Figure VI .7) . I n t he follow ing sect ion, we w ill discuss product dist r ibut ions pr edict ed for t he different ent rance channels as proceeding on an adiabat ic t riplet sur face. Not e t hat t he ent ire t riplet PES is included in all t he m ast er equat ion analyses, i.e. bot h r egions show n in Figur es 1 and 2 ar e included in t he k inet ic react ion schem e at all t im es. Table VI .3: Calculat ed product dist r ibut ion at P = 1 at m as a funct ion of t em perat ur e for t he O( 3 P) + CH 2 = C= CH 2 r eact ion occurr ing on t he t r iplet surface aft er O- addit ion on t he cent ral carbon ( see Figure VI .1)

Product s 3

[ OC3 H4 ]

a

300 K

500 K

650 K

800 K

1000 K

1500 K

2000 K

26.4

8.5

3.5

1.4

0.5

0.1

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0 88.5

O + C3 H4 3

71.6

87.1

90.6

91.5

91.4

89.9



0.6

1.8

2.6

3.1

3.4

3.5

3.4

1.4

2.6

3.3

4.0

4.7

6.5

8.1



CH2 + H2 CCO CH3 + HC•CO •

OH + H2 C CCH

a

Sum of yields of collisionally st abilized t riplet OC3 H4 species.

O- a t om a t t ack on t he ce n t r a l ca r bon . Various product branching rat ios com put ed at P= 1 at m as a funct ion of t em perat ure ar e t abulat ed in Table VI .3. As can be seen, t r iplet m et hylene plus ket ene are t he m aj or pr oduct s wit h v er y

high y ields, ca. 90% at T •  . $OO RWKHU SURGXFWV VKRZ PLQRU \LHOGV 7KHVH result s ar e a direct consequence of t he low - ly ing I n t 1 a → TS4 → 3 CH 2 + H 2 CCO channel, wit h TS4 a loose t ransit ion st ruct ure. At low t em perat ur es, t he y ield of st abilized t r iplet ox yally l becom es im port ant , ca. 26% at 300 K. I n cont rast , it s y ield

136

is

negligible

at

higher

t em perat ur es.

We

perform ed

Chapt er VI : The O + allene r eact ion

som e

sam ple

calculat ions at higher pressur es t o locat e t he onset of t he falloff region for a " com bust ion" t em perat ur e, T = 1500 K, but found only a sm all fract ion of st abilizat ion for t riplet ox yally l ( < 8% ) ev en at P= 100 at m . This result was expect ed as t he lifet im e of t riplet ox yally l is est im at ed t o be so short ( ~ 5 ps at T= 1500 K) , and giv en t hat it requir es m any collisions t o br ing a v ibrat ionally hot t riplet

oxy ally l

below

t he

lowest - ly ing

decom posit ion

t ransit ion

st ruct ure.

Therm ally st abilized t r iplet ox yally l is pr edict ed t o t hen undergo an I SC st ep, follow ed by r ing- closur e t o lead t o singlet cyclopropanone. Table VI .4: Calculat ed product dist r ibut ion at P =

1 at m as a funct ion of

t em perat ur e for t he O( 3 P) + CH 2 = C= CH 2 r eact ion occurring on t he t r iplet surface aft er O- addit ion on a t erm inal carbon ( see Figure VI .2)

Product s 3

[ OC3 H4 ]

a

O + C3 H4 CH2 = C•CHO + •H 3

CH2 + H2 CCO CH2 = CH•CO + •H •



300 K

500 K

650 K

800 K

1000 K

1500 K

2000 K

14.5

4.1

1.6

0.7

0.2

0.0

0.0

0.2

0.4

0.6

1.4

2.0

68.1

77.7

82.2

21.3

11.8

7.4 5.5

0.0

0.1

40.1

49.9

56.8

62.5

37.8

36.9

31.8

26.7

5.4

5.6

5.7

5.7

1.6

1.7

1.8

4.5

5.1

HC O + C2 H3

1.1

1.3

1.4

1.5

3

0.4

0.4

0.4

0.4

0.4

0.3

0.2

0.3

0.7

0.9

0.8

0.7

0.4

0.2

0.8

0.8

0.8

0.6

0.4

0.7

0.6

0.5

0.4

0.3

CO + C2 H4 • CH3 + HC•CO •

OH + H2 C•CCH

CO + a

3

CH3 CH

0.5 0.8

0.7 0.8

Sum of y ields of collisionally st abilized t riplet OC3 H4 species.

O- a t om at t a ck on a t e rm ina l car b on . Various product dist ribut ions obt ained at P= 1 at m as a funct ion of t em perat ure ar e present ed in Table VI .4. The product s CH 2 = C• −CHO + H • and 3 CH 2 + H 2 CCO clearly dom inat e. While t he form er product s arise from O- addit ion on a t erm inal carbon ( see Figur e VI .2) , t he lat t er result from O- at t ack on t he cent ral carbon ( see Figure VI .1) . At T= 300 K, t he y ields of t hese t w o product s are alm ost equiv alent . The yield of pr oduct s CH 2 = C• −CHO + H • sharply r ises w it h increasing t em perat ur es, from 40% at 300 K up t o 82% at 2000 K, w her eas t he yield of pr oduct s

3

CH 2 +

H 2 CCO dr ops sharply w it h

increasing t em perat ur es, fr om 38% at 300 K down t o 7% at 2000 K. At 1 at m and at room t em perat ur e, a 15% y ield is calculat ed for st abilized t r iplet oxyally l, w hich aft er I SC and r ing- closur e gives rise t o singlet cyclopropanone.

Howev er, t his y ield becom es v ery m inor ( ” DWKLJKWHPSHUDWXUHV 7• K) even at P = 100 at m . Not e t hat t he y ield of CH 2 = CH−• CO + H • is m inor, ca.

Chapt er VI : The O + allene react ion

137

5% , and alm ost t em perat ur e- independent , indicat ing t hat t he exper im ent ally observ ed



C3 H 3 O species10 cor responds t o t he m or e abundant CH 2 = C• −CHO

isom er form ed by O- addit ion on t he cent r al car bon. As alr eady m ent ioned, singlet cyclopropanone and/ or singlet allene ox ide form ed from I SC pr ocesses should alm ost ent irely decom pose t o y ield CO + C2 H 4 , except at v ery high pressures. Ot her product s from t he singlet PES ar e predict ed t o be all v er y m inor , w it h y ields less t han 5% . Not e t hat CO + C2 H 4 can be form ed from bot h addit ion sit es at low t em perat ur es. At higher t em perat ur es CO + C2 H 4 will only be form ed from t riplet ox yally l following O- at t ack on t he cent r al carbon. Consider ing t hat t he fr act ion of O- addit ion on t he cent ral carbon has a negat iv e t em perat ur e- dependence ( see Figur e VI .7) , t he y ield of CO + C2 H 4 is predict ed t o be considerably reduced when t em perat ure increases subst ant ially . A quant it at ive predict ion of t he ov erall product branching rat ios spanning all pot ent ial energy surfaces is at t his t im e difficult due t o t he need for accurat e dy nam ic calculat ion on t he rat e of t he I SC process. For t hese syst em s such calculat ions ar e ext rem ely dem anding and beyond our cur rent com put at ional resources. An alt er nat iv e appr oach used successfully in ear lier w or k, 4,5 w here t he rat io of I SC cr ossing versus on- surface unim olecular r eact ions is calibrat ed against exper im ent al m easurem ent s, is not possible her e due t o t he cur rent lack of sufficient ly com plet e exper im ent al pr oduct dist ribut ion dat a.

V I .3 .3 . Ov e r a ll t he r m al r at e coe f ficien t The overall t em per at ur e- dependent rat e coefficient k( T) ov erall for t he O( 3 P) + CH 2 = C= CH 2 r eact ion can be com put ed according t o t he follow ing ex pression:

k (T )overall = k (T )TST + k (T )TST + k (T )TST O − add − cen O − add − ter H − abs

( VI .11)

w here k ( T) TST is t he rat e coefficient deriv ed fr om m ult i- st at e t ransit ion st at e t heor y :

k (T )OTST−add −cen =

kbT h

×

(1 − γ TS 1a ) × QTS≠ 1a × exp( − ETS 1a / RT ) + (1 − γ TS 1b ) × QTS≠ 1b × exp( − ETS 1b / RT ) QOQC H 3

4

,

k (T )OTST−add −ter =

k bT h

×

(1 − γ TS 2 a ) × QTS≠ 2 a × exp( − ETS 2 a / RT ) + (1 − γ TS 2 b ) × QTS≠ 2 b × exp( − ETS 2 b / RT ) QOQC H 3

4

, and

138

Chapt er VI : The O + allene r eact ion

k (T )TST = H − abs

kbT h

×

κ TS 3a × QTS≠ 3a × exp( − ETS 3 a / RT ) + κ TS 3b × QTS≠ 3b × exp( − ETS 3b / RT ) QOQC H 3

4

wher e Q( T) is a com plet e part it ion funct ion, k b is Bolt zm ann’s const ant , h is Planck’s const ant , R is t he universal gas const ant , ETS is t he int ernal energy of a t ransit ion st ruct ur e relat iv e t o t he init ial react ant s, κ is t he one- dim ensional t unneling cor r ect ion com put ed by assum ing an asym m et r ic Eckart pot ent ial, 38,39 and γre is t he fract ion of r e- dissociat ion of t he init ial adduct s back t o t he init ial react ant s, funct ion of t em perat ur e and pr essure ( see Tables VI .3 and VI .4) . At low t em perat ur es, r e- dissociat ion is negligible and t he value of γre is close t o 0. Abov e 1000 K, re- dissociat ion becom es non- negligible, but it s cont r ibut ion is t he result of a com plicat ed com pet it ion bet w een t he r e- dissociat ion r eact ions, furt her isom er isat ion

and I C/ I SC pr ocesses ( I C for re

quant ificat ion of γ

I nt 1 b

and I nt 2 b) .

Accurat e

t her efor e again r equires dynam ic calculat ions, which ar e

beyond t he scope of t his paper. We t her efore lim it ed ourselv es t o t he calculat ion

of k ( T) overall at T ”  . IRU WKHVH ORZHU WHPSHUDWXUHV H[SHULPHQWDO GDWD DUH available for com par ison. The rot at ional sym m et r ies for allene and t he t ransit ion st at es are 4 and 1, respect iv ely, such t hat t he r eact ion pat h degeneracy is 4 for each channel, as alr eady discussed ear lier. The elect r onic part it ion funct ion of t he O at om explicit ly includes t he t hr ee low est - ly ing elect ronic st at es ( 3 P2 w it h elect r onic degeneracy g= 5,

3

P1 w it h g= 3, and

3

P0 w it h g= 1) , w it h relat iv e

energies of 0.0000, 0.4525, and 0.6490 kcal/ m ol, respect iv ely. 42 Also, t he elect r onic degeneracy of 3 for t he t ransit ion st r uct ur es, hav ing a t r iplet elect ronic st at e, is duly t ak en int o account .

Chapt er VI : The O + allene react ion

139

Figur e VI .10: Ov erall t herm al rat e coefficient s for t he O( 3 P) + allene react ion: com par ison bet w een ex per im ent al dat a and t he cur rent t heor et ical pr edict ions

The rat e predict ions are plot t ed in Figure VI .10 and can be w ell- r epr oduced by a m odified Arr henius equat ion 1.60 × 10 −17 × T2.05 × exp ( −90/ T) cm 3 m olecule −1 s −1 ; t he m ost recent exper im ent al dat a are also show n for com par ison. Our k( T) result s ar e in near - per fect agreem ent w it h t he exper im ent al values obt ained by Cvet anov ic and cowork ers 6,14 ov er t he ent ir e r ange 300 t o 600 K, w hile t hey ar e ov er est im at ions com par ed t o m easur em ent s by At k inson 13 and by Alek sandrov . 15 At r oom t em perat ure, our pr edict ed rat e coefficient is 1.4 × 10 –12 cm 3 m olecule −1 s −1 , in excellent agreem ent w it h t he recom m ended value of 1.3 × 10 –12 cm 3 m olecule −1 s−1 in t he lit erat ure, 6 but is an ov er est im at ion by ca. 30% of t he 1.0 ×

140

Chapt er VI : The O + allene r eact ion

10 –12 cm 3 m olecule −1 s−1 m easurem ent by At k inson. 13 I ncr easing r elat iv e energies for all O- addit ion and H- abst ract ion t ransit ion st ruct ures on t he t r iplet surface by 0.1 kcal/ m ol w ill bring k( T) values slight ly dow n t o bet ween t he exper im ent al k( T) dat a obser ved by Nip 14 and by At k inson. 13 Theor et ical k( T) values are quit e sensit iv e t o t he accuracy of barr ier height s, especially at low t em per at ur es: at room t em perat ure a difference in bar rier height by 0.5 kcal/ m ol alt ers t he com put ed k ( T) values by a fact or of 2.3. Furt her ex per im ent al st udies ar e recom m ended for a m or e pr ecise det erm inat ion of t he rat e coefficient .

V I .4 . Con clu din g r e m a r k s I n t he present t heor et ical st udy , t he low est - lying t r iplet and singlet pot ent ial energy surfaces for t he O( 3 P) +

CH 2 = C= CH 2 r eact ion w er e charact er ized

uniform ly using t he high lev el quant um chem ical CBS- QB3 m et hod. RRKM- Mast er Equat ion calculat ions, t o evaluat e pr im ary pr oduct dist r ibut ion for each of t hese surfaces and t o qualit at iv ely predict t he ov erall m aj or product s, were carried out using t he exact st ochast ic sim ulat ion m et hod. I n addit ion, ov erall t herm al rat e coefficient s w er e det er m ined in t he 200- 1000 K range using m ult i- st at e t ransit ion st at e t heory . A num ber of im port ant result s em erge fr om t his st udy and can be sum m ar ized as follows: ( i) The O( 3 P) + CH 2 = C= CH 2 r eact ion is confir m ed t o occur dom inant ly , but not exclusively v ia an elect rophilic addit ion m echanism as t he first react ion st ep. The predict ed m aj or product s from t hese addit ion react ions are CO + C2 H 4 , •

3

CH 2 +



H 2 CCO, and CH 2 = C −CHO + H . CO + C2 H 4 ar e m ainly pr oduced fr om t he lowest ly ing singlet surface following an I SC pr ocess, and can be form ed from bot h addit ion channels; t hese result s confirm ear lier exper im ent al observat ions.

3

CH 2

+ H 2 CCO can likew ise be form ed from O- at t ack on t he cent ral and t erm inal carbons, whereas CH 2 = C• −CHO + H • ar e near ly exclusiv ely product s of O- addit ion on a t erm inal carbon; ( ii) The H- abst ract ion r eact ion proceeds on t w o elect ronic sur faces, 3 A″ and 3 A′, and result s in OH( X2 Π) + H 2 C• −C≡CH pr oduct s, predict ed t o be im port ant at high t em perat ur es. The cont ribut ions of H- abst ract ion t o t he ov erall product form at ion are est im at ed t o be ca. 35% at 2000 K; ( iii) The ent rance barr ier height s and r eact ion ent halpies com put ed at t he CBSQB3 lev el of t heory ar e in bet t er agr eem ent wit h available exper im ent al dat a com pared t o prev ious t heoret ical r esult s;

Chapt er VI : The O + allene react ion

141

( iv ) The lack of accurat e dy nam ic calculat ions for I SC rat es and/ or of av ailable exper im ent al product branching rat ios prohibit s us from quant it at iv ely predict ing t he ov erall pr im ary pr oduct dist ribut ion for t he t it le r eact ion. Howev er , t he present st udy elucidat es t he react ion m echanism and qualit at ively pr edict s t he m aj or product s. ( v ) The com put ed ov er all TST r at e coefficient for t he range 200- 1000 K: k ( T) = 1.60 × 10 −17 × T2.045 × ex p( −90.5 K/ T) cm 3 m olecule −1 s−1 , is in excellent agreem ent wit h t he exper im ent al dat a in lit er at ur e.

142

Chapt er VI : The O + allene r eact ion

Re fe r e nce s ( 1) Glassm an I . Com bust ion, 2 nd ed. ; Academ ic press: Flor ida, 1987. ( 2) Gardiner W. C., Jr. Com bust ion Chem ist ry ; Spr inger - Ver lag: New York , 1984. ( 3) Nguy en T. L; Vereecken L.; Peet ers J. J. Phy s. Chem . A 2 0 0 6 , in pr ess. ( 4) Nguy en, T. L.; Ver eeck en, L.; Hou, X. J.; Nguy en, M. T. ; Peet ers, J. J. Phys. Chem . A 2 0 0 5 , 109, 7489. ( 5) Nguy en, T. L. ; Dils, B.; Car l, S. A.; Vereecken, L.; Peet ers, J. J. Phys. Chem . A 2 0 0 5 , 109, 9786. ( 6) Cv et anov ic, R. J., J. Phy s. Ref. Dat a 1 9 8 7 , 16, 261. ( 7) Hav el, J. J., J. Am . Chem . Soc. 1 9 7 4 , 96, 530. ( 8) Lin, M. C. ; Short ridge R. G.; Um st ead M. E., Chem . Phys. Let t . 1 9 7 6 , 37, 279. ( 9) Singm ast er, K. A.; Pim ent el, G. C., J. Mol. St ruc. 1 9 8 9 , 194, 215. ( 10) Schm olt ner, A. M.; Huang, S. Y.; Br udzynsk i, R. J.; Chu, P. M.; Lee, Y. T., J. Chem . Phys. 1 9 9 3 , 99, 1644. ( 11) Xing G.; Huang X.; Wang X.; Bersohn R., J. Chem . Phys. 1 9 9 6 , 105, 488. ( 12) Herbr echt sm eier, V. P.; Wagner, H. G., Ber . Bunsenges. Phys. Chem . 1 9 7 2 , 76, 517. ( 13) At k inson, R.; Pit t s, J. N. , Jr., J. Chem . Phy s. 1 9 7 7 , 67, 2492. ( 14) Nip, W. S.; Singlet on, D. L.; Cv et anovic, R. J., Can. J. Chem . 1 9 7 9 , 57, 949. ( 15) Aleksandrov, E. N.; Ar ut yunov, V. S.; Kozlov, S. N., Kinet . Cat al. 1 9 8 0 , 21, 1327. ( 16) Fr om NI ST w eb page: ht t p: / / srdat a.nist .gov / cccbdb/ ( 17) Chiu, Y. N.; Abidi, M. S. F. A., J. Phys. Chem . 1 9 8 2 , 86, 3288. ( 18) Ham m ond, B. L.; Huang, S. Y. ; Lest er, W. A., Jr.; Dupuis, M., J. Phys. Chem . 1 9 9 0 , 94, 7969. ( 19) Becke A. D. J. Chem . Phys. 1 9 9 3 , 98, 5648. ( 20) St ev ens P. J.; Devlin F. J. ; Chablowsk i C. F.; Fr isch M. J. J. Phys. Chem . 1 9 9 4 , 98, 11623. ( 21) Gonzalez C.; Schlegel H. B. J. Chem . Phys. 1 9 8 9 , 90, 2154. ( 22) Gonzalez C.; Schlegel H. B. J. Phys. Chem . 1 9 9 0 , 94, 5523. ( 23) Mont gom er y J. A. Jr.; Frisch M. J. ; Ocht erski J. W.; Pet ersson G. A. J. Chem . Phys. 1 9 9 9 , 110, 2822. ( 24) Werner H. J.; Know les P. J. J. Chem . Phys. 1 9 8 5 , 82, 5053. ( 25) Know les P. J.; Wer ner H. J. Chem . Phy s. Let t . 1 9 8 5 , 115, 259. ( 26) Celani P.; Wer ner, H. J., J. Chem . Phys. 2 0 0 0 , 112, 5546. ( 27) Fr isch M. J.; Tr ucks G. W.; Schlegel H. B. et al. Gaussian 03, Gaussian, I nc., Pit t sburgh, PA, ( 2 0 0 4 ) . ( 28) DALTON, a m olecular elect ronic st ruct ure program , writ t en by Helgaker T.; Jensen H. J. Aa. ; Joergensen P.; Olsen J.; Ruud K.; Aagr en H.; Auer A. A. et al., Release 1.2 ( 2 0 0 1 ) . ( 29) MOLPRO is a package of ab init io program s wr it t en by Werner H.- J.; Know les P. J.; Schüt z M.; Lindh R.; Celani P.; Kor ona T.; Rauhut G.; Manby F. R. ; Am os R. D.; Ber nhardsson A.; Ber ning A.; Cooper D. L.,; Deegan M. J. O.; Dobby n A. J. ; Eckert F; et al. ( 2 0 0 2 ) . ( 30) Gillespie D. T. J. Com put . Phy s. 1 9 7 6 , 22, 403. ( 31) Gillespie D. T. J. Phys. Chem . 1 9 7 7 , 81, 2340. ( 32) Gillespie D. T. J. Com put . Phy s. 1 9 7 8 , 28, 395. ( 33) Ver eeck en L.; Huy ber echt s G.; Peet ers J. J. Chem . Phy s. 1 9 9 7 , 106, 6564. ( 34) Hippler H.; Troe J; Wendelken H. J. J. Chem . Phys. 1 9 8 3 , 78, 6709. ( 35) Klessinger M. ; Michl J. Ex cit ed St at es and Phot ochem ist ry of Organic Molecules; VCH: New York , 1995. ( 36) Haas Y.; Klessinger M.; Zilberg S. Chem . Phy s. 2 0 0 0 , 259, 121 and references t herein.

Chapt er VI : The O + allene react ion

143

( 37) Rot h, W. R.; Hoff, H.; Horn, C. Chem . Ber. 1 9 9 4 , 127, 1781. ( 38) Eckart C., Phy s. Rev. 1 9 3 0 , 35, 1303. ( 39) Johnst on H. S.; Heicklen, J., J. Phys. Chem . 1 9 6 6 , 66, 532. ( 40) Coolidge, M. B. ; Yam ashit a, K. ; Mor ok um a, K.; Borden, W. T., J. Am . Chem . Soc. 1 9 9 0 , 112, 1751. ( 41) Hess, B. A., Jr.; Eckart , U. ; Fabian, J., J. Am . Chem . Soc. 1 9 9 8 , 120, 12310. ( 42) NI ST web page: ht t p: / / physics.nist .gov/ PhysRefDat a/ Handbook / per iodict able.ht m .

144

Chapt er VI : The O + allene r eact ion

Cha pt e r VI I : A Th e or e t ica l Re - in v e st iga t ion of t h e O( 3 P) + C6 H 6

Re a ct ion:

Qu a n t u m

Ch e m ica l

and

St a t ist ica l

Ra t e

Ca lcu la t ion s †

V I I .1 . I n t roduct ion Benzene is recognized as a key int erm ediat e in alm ost all hydr ocarbon flam es, 1,2 play ing an im port ant r ole as t he " first arom at ic ring" in t he for m at ion of polycy clic arom at ic hydr ocarbons ( PAHs) and soot . I t is hy pot hesized t o be for m ed m ainly by a com binat ion of t wo pr opargy l r adicals, 3,4 or by degradat ion of higher arom at ics 5 if present in t he fuel such as in curr ent gasolines t o increase t he oct ane rat ing. 6 Under fuel- lean t o m oderat ely fuel- r ich com bust ion condit ions, one of t he m aj or benzene consum pt ion pat hways is it s react ion wit h t r iplet ground st at e oxygen at om s. 4,5,7- 10 Ther efor e, elucidat ing t he r eact ion m echanism and predict ing t he product branching rat ios of t he elem ent ary C6 H 6 + O react ion ov er w ide t em per at ur e and pr essure ranges is of key im port ance t o our underst anding of t he overall react ion m echanism s of hydr ocarbon flam es as well as t o t he opt im izat ion of com bust ion processes. The react ion of benzene w it h t riplet O- at om can in pr inciple r esult in var ious

SULPDU\ SURGXFWV SUHVHQWHG EHORZ ZLWK UHDFWLRQ HQWKDOSLHV DW  . +0 r ) as obt ained

in t his wor k

exper im ent al ent halpies, O( 3 P) + c- C6 H 6 3

at 11

t he CBS- QB3

t heory.

Wher e possible,

reduced t o 0 K, are also giv en in par ent heses.

c- C6 H 5 O• + H • •

lev el of

–15.8 ( –14.6 ± 3) kcal/ m ol

( VI I .1)



O( P) + c- C6 H 6

c- C5 H 5 + CHO

–7.3 ( –6.5 ± 2) kcal/ m ol

( VI I .2)

O( 3 P) + c- C6 H 6

c- C6 H 5 • + • OH

12.2 ( 10.2 ± 2) kcal/ m ol

( VI I .3)

–101.5 ( –101.9 ± 1) kcal/ m ol

( VI I .4)

3

Phenol

3

c- C5 H 6 + CO

–73.7 ( –74.0 ± 1) kcal/ m ol

( VI I .5)

3

c- C6 H 4 + H 2 O

–28.0 ( –30.8 ± 4) kcal/ m ol

( VI I .6)

3

Benzene- ox ide / Ox epin

–56.6 / –57.2 kcal/ m ol

( VI I .7)

3

2,4- / 2,5- Cyclohexadienone

–82.5 / –84.0 kcal/ m ol

( VI I .8)

3

But adienyl- ket ene

–54.2 k cal/ m ol

( VI I .9)

O( P) + c- C6 H 6 O( P) + c- C6 H 6 O( P) + c- C6 H 6 O( P) + c- C6 H 6 O( P) + c- C6 H 6 O( P) + c- C6 H 6

Except for t he H- abst ract ion r eact ion channel ( VI I .3) , all channels proceed v ia elect r ophilic O- addit ion ont o a C- at om form ing an init ial t riplet , v ibrat ionally excit ed, biradical adduct ( t r iplet oxybenzene) . While t he spin- conserv ing r eact ion †

Thanh Lam Nguyen et al, J. Phys. Chem. A (2006), submitted. Chapt er VI I : The O + C6 H 6 React ion

145

channels ( VI I .1) , ( VI I .2) , and ( VI I .3) occur on a t riplet elect ronic st at e surface, t he rem aining channels, ( VI I .4) −( VI I .9) , occur v ia a spin- forbidden m echanism , inv olv ing int ersyst em crossing ( I SC) of t he init ial t riplet ox ybenzene t o eit her singlet phenol or singlet benzene- ox ide. These “ hot ” int erm ediat es, possessing high int er nal energies by chem ical act ivat ion, are eit her deact ivat ed by m ult iple collisions w it h t he bat h gas, or isom er ize t o ot her singlet isom ers, and/ or decom pose t o var ious product s on t he singlet elect ronic st at e surface. React ion channel ( VI I .1) , producing phenoxy plus hydrogen radicals, was first det ect ed in t he crossed m olecular beam exper im ent s by Lee and co- work ers 12 and r ecent ly confirm ed in t he m ult i- collision work by Font ij n et al, 13 w hile recent t heor et ical calculat ions carr ied out at t he high CBS- QB3 lev el of t heory also support ed phenox y + H form at ion. 14 Bot h t he exper im ent al 12,13 and t heoret ical 14 wor k showed t his channel t o be m aj or. Alt hough t here is no ex per im ent al ev idence y et for r eact ion channel ( VI I .2) form ing c- C5 H 5 • +



CHO,

earlier

t heor et ical calculat ions predict ed t hat t his channel could be im port ant at high t em perat ur es. 14 Howev er , our pr esent work will show t his pr oduct channel not play ing any r ole under all relevant condit ions. React ion channel ( VI I .3) , producing pheny l plus hy dr ox y l r adicals v ia direct H- abst ract ion, is endoergic by ca. 12 kcal/ m ol. Ther efor e, at T ”  . LW FDQQRW FRPSHWH ZLWK WKH H[RWKHUPDO 2addit ion/ elim inat ion m echanism s. 15 How ev er, OH radical pr oduct ion was det ect ed in a cr ossed m olecular beam experim ent at a high collisional energy of 16.5 kcal/ m ol. 16 This product channel was t heor et ically pr edict ed t o play an im port ant

role at higher t em perat ures ( T •. 17

The pr oduct of singlet phenol ar ising from t he r eact ion channel ( VI I .4) is well est ablished. 12,18- 21 Phenol was ev en found under collision- free condit ions in crossed m olecular beam ex perim ent s, 12,21 due t o t he very long lifet im e of t he nascent , chem ically act ivat ed singlet int erm ediat es. 12 The product ion of CO, v ia react ion channel ( VI I .5) , is a subj ect of som e cont r ov ersy. Ear lier, Sloane 21 observ ed CO as a m aj or pr oduct in his crossed m olecular beam exper im ent . I n cont rast w it h t his finding, Lee and co- w ork er s12 observ ed CO as a v ery m inor product in a sim ilar ex per im ent , but at higher collision energies. I n t wo r ecent m ult i- collision exper im ent s, t he y ield of CO if produced was est im at ed t o be less t han 5% . 13,22 The product ion of H 2 O, from t he react ion channel ( VI I .6) , has not been report ed so far. I n t he pr esent w or k it is predict ed t o be negligible. Ot her singlet product s such as benzene ox ide, cyclohex adienone, and but adieny l- k et ene arising from t he channels ( VI I .7) - ( VI I .9) , have recent ly been det ect ed in t he λ ≥ 280 nm phot olysis of benzene/ ozone m ixt ur es in an argon m at r ix at 12 K, 23 wher e ozone phot olysis was assum ed t o lead t o t riplet gr ound st at e O- at om s.

146

Chapt er VI I : The O + C6 H 6 React ion

Therm al rat e coefficient s for t he O( 3 P) + C6 H 6 r eact ion hav e been m easured over

a wide t em perat ur e range ( 298 K ” 7   . 15,22,24- 29 From t hese dat a, t he Arr henius act ivat ion energy w as der iv ed t o be about 4- 5 kcal/ m ol; at room t em perat ur e, a rat e coefficient of ( 1.7 o 0.3) × 10



cm 3 m olecule −1 s−1 is

recom m ended. 30 Ther e ar e t wo published t heor et ical st udies relevant t o t he t it le react ion. Barckholt z et al. 17 charact erized t he O- addit ion and H- abst ract ion channels using t he B3LYP/ 6- 311+ G( d, p) / / B3LYP/ 6- 31G( d) level of t heory and t hese aut hors com put ed t herm al r at e coefficient s using conv ent ional t ransit ion st at e t heory . 17 Howev er, t heir com put ed B3LYP barr ier height of 0.2 kcal/ m ol 17 is 4 kcal/ m ol lower t han t he ex per im ent al Ar rhenius act ivat ion energy. Mor eov er, subsequent decom posit ion and/ or isom er izat ion st eps of t he init ial oxybenzene adduct were not considered. Hodgson et al. 14 charact er ized t he t r iplet elect ronic st at e surface in det ail using t he high- lev el CBS- QB3 t heor y and also did RRKM calculat ions. Howev er, neit her t he H- abst ract ion channel, nor t he singlet elect ronic st at e surface and it s crossing seam s w it h t he t riplet surface w er e invest igat ed. Mor eov er, as w ill be det ailed below , t her e are som e int er est ing and m aj or differences bet w een our pr esent result s and t heirs, w it h an im port ant im pact on t he predict ed react ion m echanism . Not e t hat t he t r iplet pot ent ial ener gy surface for t he t it le r eact ion w as briefly rev iew ed by Nguy en and cowor ker s, 31 m ainly based on t he r esult s obt ained by Hodgson et al. 14 As seen above, t he O( 3 P) + C6 H 6 r eact ion m echanism appears t o be rat her com plicat ed, com pr ising am ong ot hers m ult i- st ep isom er izat ion/ decom posit ion processes and also non- radiat iv e t ransit ions from t r iplet t o singlet elect ronic st at e surfaces. I n addit ion, t he y ields of t he var ious pr im ar y pr oduct s are lik ely t o be very sensit iv e t o t em perat ure and pr essure. I t is wort h not ing t hat in t he var ious kinet ic m odeling st udies of t he ox idat ion of benzene by O differ ent react ion channels were adopt ed as dom inant ; som e aut hors32,33 assum ed t hat t he t it le react ion m ainly occurs via t he channels ( VI I .1) and ( VI I .3) , while ot hers34,35 consider ed t he channels ( VI I .3) and ( VI I .4) as predom inant . Ther efor e, high- level quant um chem ical calculat ions on all relevant react ion surfaces, in com binat ion w it h st at e- of- t he- ar t st at ist ical kinet ic analyses ov er wide t em perat ur e and pressure ranges, appear t o be in order for elucidat ing t he det ailed m echanism s and predict ing t he pr oduct dist ribut ions of t he fundam ent al and im port ant O + C6 H 6 r eact ion.

Chapt er VI I : The O + C6 H 6 React ion

147

Table VI I .1: Calculat ed t ot al ( Hart rees) and r elat ive energies ( kcal/ m ol) at t he CBS- QB3 lev el of t heor y for var ious species in t he O( 3 P) + C6 H 6 r eact ion. Wher e possible, r eact ion ent halpies available in lit er at ur e ar e given for t he purpose of com parison. Relat iv e energy Species

Tot al ener gy

O( 3 P) + C6 H6 ( D 6h , X1 Ag )

c

G2M

d

–306.777373 0.0

C6 H5 ( C2v , X2 A1 ) + OH( C  ,

2



C6 H5 O( C2v , 2 B1 ) + H( 2 S) 2

CBS- QB3

e

0.0

–306.757997 12.2

10.2 ± 2

–306.802512 –15. 8

2

Expt l.

–12.4

–14.6 ± 3

C6 H5 O( C2v , B2 ) + H( S)

–306.765710 7.3

C5 H4 CHO( Cs, 2 A" ) + H( 2 S)

–306.771079 3.9

C5 H5 ( C2v , 2 B1 ) + CHO( Cs, 2 A')

–306.789026 –7.3

H2 O( C2v , 1 A1 ) + c- C6 H4 ( C2v , X1 A1 )

–306.822073 –28. 0

–30.3

–30.8 ± 4

CO( C  , X1

–306.894742 –73. 7

–79.3

–74.0 ± 1

–306.939197 –101.5

–101.5

–101. 9 ± 1

S3( Cs, X A') , 2, 4- cyclohexadienone

–306.908870 –82. 5

–85.4

S5( Cs, X1 A') , form ylcyclopent adiene

–306.891902 –71. 9

S6( Cs, X1 A') , benzene oxide

–306.867639 –56. 6

–59.2

S7( Cs, X1 A') , oxepin

–306.868507 –57. 2

–56.9

–306.863795 –54. 2

–57.4

CO( C  , X

+

1 +

CO( C  , X1

+

) + c- C5 H6 ( C2v , X1 A1 ) 3

) + c- C5 H6 ( C2v , B2 )

–6.5 ± 2

–306.800353 –14. 4

) + CH2 CHCHCHCH( Cs, 3 A" ) –306.774023 2.1

Com plex ( C6 H5 …OH) ( Cs, 3 A" ) 3

–306.761097 10.2

T1( Cs, A')

–306.800481 –14. 5

T1ex( Cs, 3 A" )

–306.792161 –9.3

T2( Cs, 3 A')

–306.804970 –17. 3

T3( Cs, 3 A')

–306.829625 –32. 8

3

T4a( Cs, A')

–306.780665 –2.1

T4b( C1 , 3 A)

–306.778352 –0.6

T5a( Cs, 3 A" )

–306.798926 –13. 5

T5b( C1 , 3 A)

–306.798554 –13. 3

T6( Cs, 3 A" )

–306.781042 –2.3

3

T7( C2v , B2 )

–306.818483 –25. 8

T8a( Cs, 3 A" )

–306.800559 –14. 5

T8b( Cs, 3 A" )

–306.787060 –6.1

T9( Cs, 3 A" )

–306.763075 9.0

T10( Cs, 3 A')

–306.839692 –39. 1

T11( C2v , 3 A2 )

–306.802390 –15. 7

T12a( C1 , 3 A)

–306.820464 –27. 0

T12b( C1 , 3 A)

–306.819961 –26. 7

T13( C1 , 3 A)

–306.797957 –12. 9

S1( Cs, 1 A') 1

S1ex ( Cs, A" ) S2( Cs, X1 A') , phenol 1

1

S8a( C1 , X A) , BDK- Zcc S8b( C1 , X1 A) , BDK- Zt c

b

2. 7

a

5. 1

a

–306.868183 –57. 0

S8c( Cs, X1 A') , BDK- Zt t

–306.871059 –58. 8

S8d( C1 , X1 A) , BDK- Zct

–306.866692 –56. 0

S8e( C1 , X1 A) , BDK- Ect

–306.867720 –56. 7

148

Chapt er VI I : The O + C6 H 6 React ion

S8f( C1 , X1 A) , BDK- Ecc

–306.865347 –55. 2

S8g( Cs, X1 A') , BDK- Et c

–306.869990 –58. 1

S8h( Cs, X1 A') , BDK- Et t

–306.872493 –59. 7

1

S9( C1 , X A)

–306.857648 –50. 4

–53.8

S10( Cs, 1 A')

–306.841789 –40. 4,–45.9 a

–47.2

S11( C2v , X1 A1 ) , 2,5- cy clohexadienone

–306.911196 –84. 0

–86.8

S12( C1 , 1 A)

–306.783625 –3.9

TS1( Cs, 3 A')

–306.770887 4.1

TS1ex( Cs, 3 A" )

–306.770443 4.3

TS2( Cs, 3 A')

–306.783803 –4.0

TS2ex( Cs, 3 A" )

–306.751011 16.5

TS3a( C2v , 3 B1 )

–306.758837 11.6

TS3b( C2 v , 3 B2 )

–306.757750 12.3

TS4a( C1 , 3 A)

–306.749022 17.8

TS4b( C1 , 3 A)

–306.745267 20.1

3

TS5a( C1 , A)

–306.781166 –2.4

TS5b( C1 , 3 A)

–306.779028 –1.0

TS6a( C1 , 3 A)

–306.775659 1.1

TS6b( C1 , 3 A)

–306.773975 2.1

3

TS7( C1 , A)

–306.763235 8.9

TS8( C1 , 3 A)

–306.778034 –0.4

TS9( C1 , 3 A)

–306.764713 7.9

TS10( Cs, 3 A')

–306.753974 14.7

3

TS11( C1 , A)

–306.747670 18.6

TS12( C1 , 3 A)

–306.790838 –8.4

TS13( C1 , 3 A)

–306.784614 –4.5

TS14( C1 , 3 A)

–306.764659 8.0

3

TS15( C1 , A)

–306.751117 16.5

TS16( C1 , 3 A)

–306.769091 5.2

TS17( C1 , 3 A)

–306.747877 18.5

TS18( Cs, 3 A" )

–306.759626 11.1

3

TS19a( C1 , A)

–306.742090 22.1

TS19b( C1 , 3 A)

–306.722446 34.5

TS20a( C1 , 3 A)

–306.752822 15.4

TS20b( C1 , 3 A)

–306.752377 15.7

3

TS21( C1 , A)

–306.783307 –3.7

TS22( C1 , 3 A)

–306.753419 15.0

TS23( C1 , 3 A)

–306.795665 –11. 5

TS24( C1 , 3 A)

–306.777538 –0.1

1

TS25( C1 , A)

–306.827748 –31. 6

–32.8

TS26( C1 , 1 A)

–306.799422 –13. 8

–15.1

TS27( C1 , 1 A)

–306.840663 –39. 7

–42.2

TS28( C1 , 1 A)

–306.849189 –45. 1

–47.5

TS29( C1 , A)

–306.839050 –38. 7

–41.4

TS30( C1 , 1 A)

–306.840235 –39. 4

–42.6

TS31( C1 , 1 A)

–306.812372 –22. 0

–25. 7

TS32( C1 , 1 A)

–306.799597 –13. 9

–17.5

TS33( C1 , A)

–306.796027 –11. 7

–15.4

TS34( C1 , 1 A)

–306.793461 –10. 1

TS35( C1 , 1 A)

–306.782399 –3.2

1

1

Chapt er VI I : The O + C6 H 6 React ion

149

TS36( C1 , 1 A)

–306.794171 –10. 5

TS37( C1 , 1 A)

–306.761740 9.8

TS38( Cs, 1 A')

–306.859856 –51. 8

1

TS39( C1 , A)

–306.782332 –3.1

TS40( C1 , 1 A)

–306.848593 –44. 7

TS41( C1 , 1 A)

–306.794653 –10. 8

TS42( C1 , 1 A)

–18.1 –52.6

–306.778665 –0.8 1

TS_S8a_S8b( C1 , A)

–306.861991 –53. 1

TS_S8b_S8c( C1 , 1 A)

–306.863994 –54. 4

TS_S8c_S8d( C1 , 1 A)

–306.863722 –54. 2

TS_S8a_S8d( C1 , 1 A)

–306.857467 –50. 3

1

TS_S8e_S8f( C1 , A)

–306.859641 –51. 6

TS_S8f_S8g( C1 , 1 A)

–306.859732 –51. 7

TS_S8g_S8h( C1 , 1 A)

–306.864186 –54. 5

TS_S8e_S8h( C1 , 1 A)

–306.862324 –53. 3

a

Obt ained at t he CASPT2( 8,8) / cc- pVDZ/ / CASSCF( 8,8) / cc- pVDZ level of t heory. BDK st ands for but adieny l- k et ene; t he E/ Z present for t rans- and cis- configurat ions, according t o t he or ient at ion of t he m olecule about t he cent ral C= C bond. The “ t ” and “ c” st and for t rans and cis, respect iv ely. While t he first charact er refer s t o t he orient at ion of t he C= C bond adj acent t o t he C2 H3 group, t he second refers t o t he orient at ion of t he C= C bond adjacent t o t he ket ene group. c Our result s. d Xu and Lin, ref 61. e Mainly t aken from t he w eb- page: ht t p: / / srdat a.nist . gov/ cccbdb/ ; all v alues w ere obt ained at 0 K: ∆H0 f ( O) = 58.98 k cal/ m ol; ∆H0 f ( H) = 51. 63 kcal/ m ol; ∆H0 f ( CO) = −27. 2 kcal/ m ol; ∆H0 f ( OH) = 8.84 kcal/ m ol; ∆H0 f ( HCO) = 9.95 kcal/ m ol; ∆H0 f ( H2 O) = −57.10 kcal/ m ol; ∆H0 f ( c- C5 H6 ) = 36.19 kcal/ m ol; ∆H0 f ( c- C6 H6 ) = 24.0 k cal/ m ol; ∆H0 f ( c- C6 H5 ) = 84.3 kcal/ m ol; ∆H0 f ( c- C6 H5 OH) = −18.93 ± 1 k cal/ m ol, derived from ∆H298 f ( c- C6 H5 OH) = − 23.03 kcal/ m ol and ∆E( t herm al correct ion- TC) = 4.1 k cal/ m ol com put ed at t he CBS- QB3 level; ∆H0 f ( c- C5 H5 ) = 67 ± 2 k cal/ m ol, derived from ∆H298 f ( c- C5 H5 ) = 63.5 ± 1 k cal/ m ol 6 4 and ∆E( TC) = 3.5 k cal/ m ol; ∆H0 f ( o- C6 H4 ) = 109.3 ± 4 kcal/ m ol, derived from ∆H298 f ( oC6 H4 ) = 105. 9 ± 3.3 k cal/ m ol 65 and ∆E( TC) = 3.4 kcal/ m ol; ∆H0 f ( c- C6 H5 O) = 16.8 ± 3 kcal/ m ol, derived from ∆H298 f ( c- C6 H5 O) = 12.9 ± 1.5 kcal/ m ol 66 and ∆E( TC) = 3.9 kcal/ m ol; b

V I I .2 . M e t h odology V I I .2 .1 . Qu a n t u m Ch e m ica l Ca lcu la t ions Geom et ries of st at ionar y point s on t he t riplet and singlet surfaces wer e opt im ized at t he hybr id densit y funct ional B3LYP/ 6- 311G( d,p) lev el of t heory, 36,37 follow ed by analyt ical fr equency calculat ions at t he sam e level t o v er ify t he st at ionary point s locat ed ( one im aginary fr equency for a t ransit ion st ruct ur e and all posit iv e fr equencies for a m inim um ) . I nt r insic react ion coordinat e ( I RC) 38,39 calculat ions wer e also perform ed at t his lev el t o est ablish t he corr ect connect ions bet ween t he react ion int erm ediat es. To obt ain m or e accurat e relat iv e energies, t he com plet e basis set m odel chem ist ry CBS- QB3 40 was used. Table VI I .1 shows t hat t he CBSQB3 result s are in good agreem ent wit h t he available exper im ent al dat a, t he discrepancy being only about 1- 2 kcal/ m ol.

150

Chapt er VI I : The O + C6 H 6 React ion

For som e st at ionary point s ( for exam ple, singlet biradicals such as S1 , S1 e x, and S1 0 ) , w hose wav e funct ion possesses a m ult i- refer ence charact er ( or near degeneracy effect s) , w e used t he CASSCF( 8,8) / cc- pVDZ lev el of t heor y 41,42 t o r eopt im ize

geom et ries.

Energies

CASPT2( 8,8) / cc- pVDZ lev el

43

wer e

subsequent ly

r efined

em ploy ing

t he

based on t he CASSCF r efer ence wav e funct ion,

t hereby t ak ing dy nam ic elect r on cor relat ions int o account . To inv est igat e t he pr ox im it y or possible ov er lap of t he t r iplet and singlet pot ent ial energy sur faces at or near t he geom et r y of t he init ial t r iplet oxy benzene adduct , and so t o explor e possible non- radiat iv e t ransit ions t o singlet phenol or benzene ox ide ( see furt her ) , we em ploy ed t he CASPT2( 8,8) / cc- pVDZ lev el based on t he CASSCF r efer ence w av e funct ion t o com put e single- point energies at each point along I RC( t r iplet UB3LYP) opt im ized geom et r ies. The DFT- B3LYP and CBS- QB3 calculat ions wer e perform ed using t he Gaussian 03 package, 44 w hile t he CASSCF and CASPT2 calculat ions used t he Dalt on 45 and Molpr o 46 packages; all opt im ized st at ionary - point geom et r ies, energies, harm onic vibrat ion fr equencies, and r ot at ional const ant s are giv en in t he Support ing I nform at ion w hich can be found at our w eb- sit e: ht t p: / / ar r henius.chem .k uleuv en.ac.be/ labpeet er s/ labpeet ers_en.ht m l. To sim plify t he not at ion for st at ionary point s, in t his paper T, S, and TS st and for a t r iplet m inim um , singlet m inim um , and t ransit ion st r uct ure, r espect ively .

V I I .2 .2 . RRKM / M a st e r Equ a t ion ca lcu la t ion s Product dist r ibut ions as a funct ion of t em perat ur e and pressur e ( P = 10 –4 - 10 5 Torr, T = 300- 2000 K) for t he O( 3 P) + C6 H 6 r eact ion proceeding on t he t riplet and

singlet surfaces í FRQVLGHUHG DV EHLQJ DGLDEDWLF í ZHUH VHSDUDWHO\ REWDLQHG E\ solut ion of t he w eak - collision m ast er equat ion using Gillespie’s exact st ochast ic sim ulat ion m et hod, 47−49 explained in det ail in our ear lier paper 50 and in t he chapt er I I . The Lennard−Jones collision param et ers for bat h gas Ar ar e σ = 3.47 Å and ε/ k B = 114 K. 51 Since no collision param et ers for [ C6 H 6 O] ar e available in t he lit erat ur e, t he values σ = 5.92 Å and ε/ k B = 410 K ar e est im at ed based on t hose of t oluene C6 H 5 CH 3 . 51 Thus, t he collision frequency Z LJ [ M] was est im at ed at

§×10 10 s −1 at 1 at m ospher e and r oom t em perat ure. An average energy t ransfer red per collision < ∆E> all of −200 cm −1 was adopt ed. A “sink” was used t o collect t herm alized int erm ediat es st abilized by deact ivat ing collision w it h t he bat h gas, locat ed at 8 kcal/ m ol below t he low est - ly ing decom posit ion t ransit ion st ruct ur e, so as t o ensure t hat collisional r eact iv at ion is negligible.

Chapt er VI I : The O + C6 H 6 React ion

151

Schem e VI I .1: The var ious react ion pat hways for t he O + C6 H 6 r eact ion on t r iplet PES. Bold ar rows indicat e dom inant react ion rout es.

152

Chapt er VI I : The O + C6 H 6 React ion

Schem e VI I .2: The var ious react ion pat hways for t he O + C6 H 6 react ion on t he lowest - lying singlet PES st art ing at phenol or benzene ox ide, w hich ar e form ed by an I SC pr ocess from t he init ial t r iplet PES t o t he singlet PES. Bold arr ows indicat e dom inant react ion rout es.

Chapt er VI I : The O + C6 H 6 React ion

153

Figure VI I .1: The m ost im port ant r out es for t he O + C6 H 6 r eact ion on t riplet PES, as com put ed at t he CBS- QB3 level of t heory .

154

Chapt er VI I : The O + C6 H 6 React ion

Figure VI I .2: The B3LYP- opt im ized geom et r ies for som e k ey st at ionar y point s on t riplet PES.

Chapt er VI I : The O + C6 H 6 React ion

155

V I I .3 . Re su lt s a n d discussion V I I .3 .1 . Pot e nt ia l en e rgy su rf a ce s According t o t he spin- conser vat ion r ule, t he r eact ion of t r iplet O- at om w it h benzene can init ially proceed on t he t r iplet elect ronic st at e surface via eit her elect r ophilic O- addit ion ont o a C- at om or by dir ect H- abst ract ion. We will discuss t hese t wo r eact ion pat hways separat ely. Unless m ent ioned ot herw ise, t he CBSQB3 energy v alues w ill be used in t he follow ing discussions. All t he various r eact ion pat hways on t he t r iplet and singlet elect ronic st at e surfaces ( PES) ar e present ed in React ion Schem es VI I .1 and VI I .2. Howev er , only t he m ain, rat e- det erm ining react ion rout es t hat effect iv ely cont r ol t he react ion m echanism are collect ed and present ed in Figure VI I .1 for t he t r iplet PES and in Figures VI I .7 and VI I .8 for t he singlet PES. The opt im ized geom et r ies for all k ey st at ionary point s on t he t r iplet PES are show n in Figur e VI I .2. O- a ddit ion on t o a C- a t om . As m ent ioned above, t he elect r ophilic at t ack of Oat om ont o a benzene C- at om leading t o t he vibrat ionally excit ed • C6 H 6 O• t r iplet biradical adduct , followed by it s decom posit ion and/ or isom er izat ion on t he t r iplet PES, was alr eady t heor et ically charact erized at t he CBS- QB3 lev el by Hodgson et al. 14 I nt er est ingly, alt hough w e used t he sam e lev el of CBS- QB3 t heory as Hodgson et al, 14 our result s show im port ant differ ences, wit h im pact on t he react ion m echanism . First ly, t he CBS- QB3 t ot al energy of benzene as com put ed by Hodgson et al. 14 using Gaussian 98 was - 231.79069 Hart rees ( see Table 3 in ref 14) , while our calculat ions using bot h Gaussian 98 and 03 generat e a value of –231.78974 Hart r ees, about 0.6 kcal/ m ol higher. The reason for t his difference is unclear. Secondly and m or e im port ant ly, t he 3 A" elect ronic st at e t riplet biradical •

C6 H 6 O• adduct denot ed as T1 e x in t his wor k was charact er ized by Hodgson

et al. 14 as t he lowest - energy init ial t riplet adduct , and, failing an I RC analysis, er roneously assum ed by t hese aut hors t o connect t o t he ot her t r iplet isom ers and t o t he m ain product s phenox y + H, all in t heir ground st at es. However , we new ly ident ified anot her elect ronic st at e, 3 A’, of t r iplet • C6 H 6 O• , denot ed here as T1 , as t he t r ue ground st at e, 5.2 kcal/ m ol low er in energy t han T1 e x . Our I RC calculat ions show t hat bot h T1 and T1 e x connect direct ly w it h t he init ial react ant s O + C6 H 6 , v ia TS1 and TS1 e x , respect iv ely. Analogous O- ont o- carbon addit ion m echanism s in parallel on 3 A’ and 3 A" elect ronic st at e surfaces hav e been charact er ized ear lier for t he r eact ions of O- at om s wit h C2 H 2 , 52 C2 H 4 , 53 and C3 H 4 . 54 As a r esult , for t he case at hand, react ion pat hways on t he lowest - ly ing t r iplet PES rat her st art fr om t he t riplet 3 A’ • C6 H 6 O• ground- st at e adduct T1 inst ead of t he excit ed 3 A" T1 e x st at e of t he O( 3 P) + C6 H 6

156

TS1 e x

Hodgson et al. 14 Thirdly , Hodgson et al. 14 suggest ed 3 •

[ C6 H 6 O• ] T1 e x

TS3

3

Chapt er VI I : The O + C6 H 6 React ion

[ C6 H 5 CHO]

TS1 0

c- C6 H 5 +

CHO r out e ( see Figur e 4 in ref 14) t o be significant at higher

t em perat ur es, as according t o t hem t he

3 •

[ C6 H 6 O• ]

3

TS3

[ C6 H 5 CHO] st ep

faces a bar r ier height of 7 kcal/ m ol, only 1.6 k cal/ m ol higher t han t he 3 [ • C6 H 6 O• ] c- C6 H 5 O• + H • st ep ( see Figur e 4 in r ef 14) . How ev er , our I RC

TS2

calculat ions ( see Figur e VI I .3) show t hat t he

3 •

[ C6 H 6 O• ] T1

3

[ C6 H 5 CHO] T5

isom er izat ion is a t wo- st ep process and goes t hr ough a t riplet int erm ediat e T4 , which lies about 1- 2 k cal/ m ol below t he init ial react ant s ( see Schem e VI I .1) . But •

C6 H 6 O• T1 m ust under go a r ing- closure t o lead t o T4 v ia TS4 , facing a ver y high

barr ier of 32.3 kcal/ m ol, about 22 kcal/ m ol higher t han t hat for t he T1 •

TS2



c- C6 H 5 O + H pat hway ( see Schem e VI I .1) . I f form ed, T4 w ould t hen rapidly conv ert t o T5 in a nearly bar r ier - free st ep. Since t he first react ion st ep, T1 TS4 •

T4 , needs t o ov ercom e a v er y high bar rier, t he O( 3 P) + C6 H 6

CHO pr oduct channel cannot com pet e wit h O( 3 P) + C6 H 6

c- C5 H 5 • +

c- C6 H 5 O• + H • under

any relevant com bust ion condit ions. Not e, m or eover, t hat no ex per im ent al ev idence support s t he form at ion of c- C5 H 5 • + • CHO pr oduct s.

Figure VI I .3: I RC( B3LYP/ 6- 311G( d,p) ) calculat ions for t he T1 TS5

TS4

T4

T5 r out e.

Chapt er VI I : The O + C6 H 6 React ion

157

Figure VI I .4: The highest singly occupied m olecular orbit als ( SOMO) for t he t w o key t riplet • C6 H 6 O• adduct s and t he pr oduct phenoxy radical.

158

Chapt er VI I : The O + C6 H 6 React ion

We w ill now discuss t he m ost im port ant react ion pat hways on t he t riplet PES present ed

in

Figur e

VI I .1,

while

t he

ot her

rout es in

Schem e

VI I .1

are

energet ically unfavorable so as t o be unim port ant and w ill t her efore not be consider ed. Figur e VI I .1 shows t hat O- at t ack ont o a C- at om in benzene can t ak e place v ia bot h TS1 and TS1 e x leading t o t he v ibrat ionally excit ed • C6 H 6 O• adduct biradicals T1 ( 3 A’ st at e) and T1 e x ( 3 A" st at e) , respect ively , facing barr ier height s of 4.1 and 4.3 k cal/ m ol. I t is of int er est t o not e t hat TS1 is slight ly earlier t han TS1 ex , i.e. t he C- O bond dist ance of 1.917 Å in TS1 is slight ly longer t han t hat of 1.884 Å in TS1 e x ( see Figure VI I .2) . To obt ain m or e pr ecise energies for TS1 and TS1 e x , we carried out I RCMax ( CBS- QB3: B3LYP) 55 calculat ions, generat ing t he values of 4.7 and 4.8 kcal/ m ol, r espect iv ely. These values are in excellent agreem ent kcal/ m ol.

w it h

t he exper im ent al Arrhenius act ivat ion

15,22,24-29

energy

of

ca.

4- 5

T1 and T1 e x , bot h belonging t o t he Cs point group but hav ing

dist inct elect r onic st at es, lie –14.5 and –9.3 k cal/ m ol below t he init ial r eact ant s, respect ively . Not e t hat opt im izat ion of • C6 H 6 O• wit hout im posing a Cs sym m et ry always conv erges t o T1 . I t is of int er est and im port ant t o char act er ize t heir t w o key highest , singly occupied m olecular orbit als ( SOMOs) , w hich m ake T1 ’s st ruct ur e differ ent from T1 e x ’s. Bot h SOMOs of T1 hav e an A' sym m et ry , result ing in a 3 A'elect ronic st at e for T1 ( see Figur e VI I .4) . While t he fir st SOMO is for m ed by a com binat ion of four π orbit als locat ed at four differ ent at om cent ers, t he second SOMO m ainly concent rat es at t he O- at om . I n cont rast , t wo SOMOs of T1 e x hav e dist inct sym m et r ies, an A" for t he first SOMO and an A' for t he second, r esult ing in a

3

A" elect r onic st at e for T1 e x . While t he first SOMO is

com plet ely locat ed at t he O- at om cent er and is perpendicular t o t he m olecular sym m et ry plane, t he second is generat ed by a com binat ion of t hr ee differ ent π orbit als. T1 e x , once form ed, eit her can pr eferably re- dissociat e back t o t he init ial react ant s v ia TS1 e x wit h a low barr ier height of 13.6 kcal/ m ol or could decom pose by a H- elim inat ion v ia TS2 e x •

2

leading t o excit ed pr oduct s c-



C6 H 5 O ( A B2 ) + H , facing a higher barr ier of 25.8 kcal/ m ol, or t hirdly could carry out an non- radiat ive int ernal conv ersion ( I C) pr ocess t o T1 , t he lowest - ly ing t riplet adduct . Clearly , t he second pat hway cannot com pet e wit h t he first . Hence, t he fat e of T1 e x is expect ed t o depend m ainly on it s rat e of r e- dissociat ion and on t he I C rat e. Our RRKM rat e coefficient s for re- dissociat ion av eraged ov er t he int ernal energy dist r ibut ions, as a funct ion of t em perat ur e, show ed t hat t his rat e sharply incr eases w it h t em perat ure, i.e. fr om 1.7× 10 8 s –1 at 300 K t hrough 8.6× 10 10 s–1 at 1000 K t o 1.2× 10 12 s–1 at 2000 K. Lack of accurat e dynam ic calculat ions pr ohibit ed us from gaining an accurat e I C rat e, w hich can only be

Chapt er VI I : The O + C6 H 6 React ion

159

est im at ed t o be r oughly 10 11 s –1 . 56,57 Therefor e, it is reasonable t o conclude t hat

re- dissociat ion lik ely pr edom inat es at high t em perat ur es ( T • .  ZKHUHDV

t he I C is likely t o be fast er at low t em perat ur es ( T ”. ,Qt he int erm ediat e range, 800 K < T < 1500 K, t hese t w o rout es ar e com pet it iv e. The lowest - ly ing t r iplet



C6 H 6 O• adduct T1 can eit her undergo H- elim inat ion v ia

TS2 t o y ield ground st at e pr oduct radicals phenox y + H, facing a low bar r ier height of 10.5 kcal/ m ol, or it can r e- dissociat e back v ia TS1 t o t he init ial react ant s, aft er clear ing a higher bar r ier of 18.6 kcal/ m ol. As TS2 is furt herm or e looser t han TS1 , t his should result in t he dom inance of t he H- elim inat ion st ep. A t hird possible pat hway of t r iplet T1 is non- r adiat iv e t ransit ion t o t he singlet biradical • C6 H 6 O• S1 , follow ed by isom er izat ion of S1 t o singlet phenol and/ or benzene ox ide ( see discussions below ) . I t is of int erest t o br iefly discuss t he t wo low est - ly ing elect ronic st at es ( X2 B1 and A2 B2 ) of t he phenoxy r adical form ed in t he r eact ions described abov e, as it is com m on as a key int er m ediat e in ar om at ic flam es. 4,5,58 The gr ound st at e X2 B1 of phenoxy lies 23.1 kcal/ m ol ( ca. 1 eV) lower in energy t han t he first ex cit ed st at e A2 B2 .

The single occupied m olecular orbit al ( SOMO) in t he X2 B1 st at e is form ed

by a com binat ion of four π orbit als, locat ed at four different cent ers and all being perpendicular t o t he m olecular sym m et ry plane, while t he SOMO in t he A2 B2 st at e is com plet ely locat ed at t he O- at om cent er and lies in t he m olecular sym m et ry plane ( see Figur e VI I .4) . This r esult s in t he C- O bond lengt h of 1.251 Å in t he X2 B1 st at e being closer t o a CO double bond lengt h, whereas t he lengt h of 1.321 Å in t he A2 B2 st at e is closer t o a CO single bond. I n t he O + C6 H 6 r eact ion, t he yield of t he excit ed A2 B2 st at e is very sm all and est im at ed t o be less t han 5% of t he t ot al am ount of phenoxy radical. I f t he A2 B2 st at e is form ed, it is expect ed t o rapidly perform an I C process t o t he X2 B1 st at e. I n flam es, t he m aj or phenox y consum pt ion pat hway is well est ablished t o be it s pyr olysis t o y ield c- C5 H 5 + CO. 58 I n sum m ary , t he O- addit ion ont o a C- at om in benzene y ields bot h T1 and T1 e x . At low t em perat ur es, T1 ex pr eferably undergoes an I C process t o T1 , whereas re- dissociat ion of T1 e x back t o t he init ial react ant s is pr edom inant at high t em perat ur es. T1 w ill m ainly elim inat e H t o yield t he phenox y radical, which t herefor e can be predict ed as m aj or product s of t he O- addit ion m echanism on t he t riplet PES, in good agr eem ent w it h exper im ent s. 12,13 H - a bst ra ct ion by O- a t om . The O- at om can also at t ack an H- at om in benzene in an H- abst ract ion m echanism . The at t ack t ak es place in t he m olecular sym m et ry plane; t he t wo t ransit ion st ruct ures TS3 a and TS3 b ar e planar and belong t o a C2v point group ( see Figure VI I .1) , in good agr eem ent wit h

160

Chapt er VI I : The O + C6 H 6 React ion

observ at ions on t he rot at ional energy of t he OH product in a crossed m olecular beam ex per im ent . 16 The H- abst ract ion can occur on bot h t he

3

B1 and

3

B2

elect r onic st at e surfaces, which direct ly cor relat e t o hydr oxy l radical hav ing a Π elect r onic st at e. The pat hway on t he 3 B1 surface goes t hr ough TS3 a r epresent ing a barr ier of 11.6 kcal/ m ol and leads t o a weak com plex C6 H 5 …HO, bound by only 2 kcal/ m ol relat iv e t o t he separat ed pheny l and hydr ox yl radicals, and as such readily decom posing t o t hese product s. The pat hway on t he 3 B2 elect ronic st at e passes over TS3 b t o direct ly y ield pr oduct s c- C6 H 5 • + • OH, aft er clear ing a bar r ier height of 12.3 kcal/ m ol, only 0.1 kcal/ m ol abov e t he product s. Ther efor e, bot h TS3 a and TS3 b are loose and v er y lat e, i.e. pr oduct - lik e. Not e t hat t he Habst ract ion m echanism was earlier charact er ized by Barckholt z et al 17 using t he B3LYP/ 6- 31G( d) lev el. Howev er, only t he 3 B1 channel O + c- C6 H 6 C6 H 5



+



c-

TS3 a

OH was consider ed, giv ing ar ise t o an under est im at ion for t he H-

abst ract ion rat e. I t is of int er est t o evaluat e t he role and cont r ibut ion of t he chem ical flux for t he H- abst ract ion pat hw ay s as com pared t o t he O- addit ion channels. For t his purpose, w e evaluat ed t he t em perat ur e- dependent rat e coefficient s k( T) for bot h m echanism s using conv ent ional t ransit ion st at e t heory ( see eqs ( VI I .10) and ( VI I .11) below) . Since t he O + c- C6 H 6

c- C6 H 5 • +



OH react ion channel faces a

barr ier of ca. 12 kcal/ m ol, it is expect ed t o play a r ole only at high t em perat ur es. We t her efore lim it t his branching rat io evaluat ion t o t he T = 1000- 2000 K range, wher e we can furt her m ak e t he assum pt ion t hat t he O- addit ion r out e on t he 3 A" surface is only m arginal. The lat t er r out e m ay indeed be neglect ed safely abov e 1500 K, w here r e- dissociat ion of T1 e x should predom inat e ov er t he I C pr ocess ( see above) , but in t he T = 1000- 1500K range it s neglect should ent ail som e underest im at ion of t he ov erall O- addit ion fract ion.

FractionH −abs =

k (T ) H −abs , 1 k (T ) H −abs + k (T )TS O − add

( VI I .10)

FractionO − add =

1 k (T )TS O − add , 1 k (T ) H −abs + k (T )TS O − add

( VI I .11)

wit h

k (T ) H −abs =

≠ ≠ ≠ ≠ kbT κ TS 3a × QTS 3 a × exp( − ETS 3a / RT ) + κ TS 3b × QTS 3b × exp( − ETS 3b / RT ) × , h QOQC6 H6

( VI I .12) and

1 k (T )TS O − add =

≠ ≠ kbT QTS 1 exp( − ETS 1 / RT ) , × h QOQC6 H 6

Chapt er VI I : The O + C6 H 6 React ion

( VI I .13)

161

wher e QX is t he com plet e part it ion funct ion of t he given X, including t he r ot at ional sym m et ry num ber, k b is Bolt zm ann’s const ant , h is Planck’s const ant , R is t he universal gas const ant , E

TS

is t he energy of t ransit ion st ruct ure TS r elat iv e t o t he

init ial r eact ant s, and κTS is t he one- dim ensional t unneling correct ion for Habst ract ion, w hich is com put ed by assum ing an asym m et r ic Eckart pot ent ial. 59,60 I t should be m ent ioned t hat t he r eact ion pat hway degeneracy der iv ed from t he sym m et ry num bers in t he part it ion funct ion for rot at ion is equal t o 12 for t he Oaddit ion and 6 for t w o H- abst ract ion channels. This is obv ious for t he Habst ract ion ( 6 hydr ogens) w it h t ransit ion st ruct ur es being planar and also easy t o see for t he addit ion t o C- at om s: t here ar e six of t hese carbons, and at t ack can com e from t wo sides, ex act ly equivalent for each of t he lobes of t he π- bond.

Figure VI I .5: Mult i- St at e TST- com put ed cont r ibut ions of t he H- abst ract ion and Oaddit ion flux es as a funct ion of t em perat ur e in t he T = 1000- 2000 K r ange.

The com put ed r esult s present ed in Figure VI I .5 show t hat t he fract ion of t he Habst ract ion flux depends st rongly on t em perat ur es, i.e. ∼8% at 1000 K, ∼32% at 1500K, and rising t o ∼53% at 2000 K, indicat ing t hat t he H- abst ract ion react ion channels cont ribut e in a m aj or way at com bust ion t em perat ur es. I t should be repeat ed her e t hat t he • OH product in t he O + C6 H 6 r eact ion was obser ved in t he

162

Chapt er VI I : The O + C6 H 6 React ion

crossed m olecular beam kcal/ m ol, K.

13,15,22

16

exper im ent

at

a high collisional ener gy of 16.5

but could not be det ect ed ear lier in t herm al ex perim ent s at T < 1500

I n cont rast , t he fract ion of t he O- addit ion flux alm ost linearly decreases

wit h incr easing t em perat ures, fr om ∼92% at 1000 K t o ∼47% at 2000 K. These result s can explain t he ov er - pr edict ion of t he phenoxy radical y ield in k inet ic m odeling st udies of C6 H 6 / O2 flam es because t he H- abst ract ion pat hw ay was not t aken int o account . 14 Th e low e st - lyin g sin gle t sur f ace . As m ent ioned abov e, alm ost all exper im ent al st udies12,18–21 show singlet phenol t o be one of m ost im port ant product s for t he t it le react ion. I n addit ion, benzene ox ide was exper im ent ally det ect ed ear ly 18 and very r ecent ly cyclohexadienone, but adieny l- k et ene, and benzene ox ide hav e been observ ed in an argon m at rix st udy. 23 Moreov er , t he product CO w as found by som e groups. 19,21 All t hese product s m ust be form ed on a singlet PES, r eached via a spin- forbidden m echanism st art ing from t he init ial t r iplet



C6 H 6 O• adduct .

Hence, t he t it le r eact ion m ust also inv olv e t he ( lowest - lying) singlet surface following an I SC pr ocess. Our RRKM calculat ions of t he decom posit ion rat e of t he chem ically act ivat ed t riplet • C6 H 6 O• T1 v ia t he t wo channels T1 •

c- C6 H 5 O +



TS1

O + C6 H 6 and T1

TS2

H ( see Figur e VI I .1) show t hat t he sum m ed < k ( E) > averaged ov er

t he nascent energy dist ribut ion incr eases sharply w it h t em perat ur e, i.e. ~ 10 10 s –1 at 300 K, t hr ough ~ 3.5× 10 11 s–1 at 1000 K, and t o ~ 3× 10 12 s–1 at 2000 K. So, t he lifet im e of T1 is v er y short , especially at higher t em perat ur es; i.e. ~ 100 ps at 300 K, ~ 3 ps at 1000 K, and down t o ~ 0.35 ps at 2000 K. Therefor e, t r iplet - t osinglet int ersyst em crossing of



C6 H 6 O• is only ex pect ed t o occur at low t o

m oderat e t em perat ur es ( T < 1000 K) since t he I SC rat e can be est im at ed t o be ca. 10 10 s–1 . 56,57 I n addit ion, a ( near - ) cr ossing region of t he t r iplet and singlet surfaces is expect ed t o occur in t he v icinit y of t he equilibr ium geom et r y of t r iplet •

C6 H 6 O• T1 , since a sm all t r iplet - singlet energy gap of 2.7 kcal/ m ol was obt ained

at t he CASPT2/ / CASSCF lev el and spin- orbit coupling is sm all for first - row elem ent s. Thus, at low t em perat ures w her e t he decom posit ion lifet im e of T1 is ~ 100 ps, t he chem ically act iv at ed, v ibrat ing T1 adduct can access t he cr ossing region m any t im es, r esult ing in an incr eased t r ansit ion probabilit y and I SC rat e. Because t her e ar e 33 degr ees of freedom for C6 H 6 O species, it is pr ohibit iv e t o charact er ize all possible crossing seam s bet w een t he t wo elect r onic st at es, and

1

3

A’

A’, of ox ybenzene. One could r est r ict t he search for a crossing region by

det erm ining w hich m odes would pr om ot e t he non- radiat ive 3 A’

Chapt er VI I : The O + C6 H 6 React ion

1

A’ t r ansit ion in

163

oxybenzene. Such dy nam ic calculat ions should be ver y helpful, but beyond t he scope of t his paper.

Figure VI I .6: Crossing seam s bet w een t r iplet and singlet sur faces: ( A) leading t o singlet phenol; ( B) y ielding singlet benzene oxide.

We charact erized t w o singlet • C6 H 6 O• species wit h differ ent elect ronic st at es, 1 A’ and 1 A" , denot ed hereaft er as S1 and S1 e x , ly ing 2.7 and 5.1 k cal/ m ol higher in energy, r espect iv ely , t han t r iplet



C6 H 6 O• T1 , com put ed at t he CASPT2/ CASSCF

level. We t hus find t hat singlet S1 lies slight ly higher t han it s t riplet count erpart T1 , w her eas singlet phenol ( S2 ) and singlet benzene ox ide ( S6 ) lie m uch low er t hat t heir t r iplet count er part s T2 and T6 . Therefor e, it can be expect ed t hat t her e are int ersect ions bet w een t he t riplet and singlet pot ent ial energy cur ves along t hese t w o react ion pat hways, nam ely : T1 S2 ; and T1

TS8

TS1 0

T6 int ersect ing w it h S1

T2 int ersect ing wit h S1 S6 . We t herefor e used t riplet

UB3LYP opt im ized geom et r ies along t wo I RC r out es, T1 TS8

TS1 0

T2 and T1

T6 , t o com put e energies of t he t r iplet and singlet wav efunct ions at t he

CASPT2( 8,8) / cc- pVDZ lev el, also allow ing us t o exam ine t he vert ical energy difference bet w een t he t riplet and singlet surfaces. The r esult s are present ed in Figures VI I .6A and VI I .6B for phenol and benzene ox ide, r espect iv ely. These

164

Chapt er VI I : The O + C6 H 6 React ion

figur es show t hat t he crossing regions are v er y close t o t he harm onic vibrat ional regions of t r iplet • C6 H 6 O• T1 . As show n in Figur e VI I .6, t wo crossing point s were locat ed, each about 5 kcal/ m ol abov e T1 , one int ersect ing t he singlet sur face dir ect ly connect ing t o phenol, t he ot her t o benzene ox ide. The int ersect ion leading t o phenol appears m uch m or e pr one t o result in act ual I SC, as t he t r iplet and singlet sur faces are m uch closer t o one anot her ov er a larger RC range t han for t he int ersect ion yielding benzene ox ide. I t should be m ent ioned t hat in crossed m olecular beam ex per im ent t he I SC process was found t o speed up when t he collision energy incr eased from 2.5 t o 6.5 k cal/ m ol, 12 suggest ing t hat crossing is facilit at ed by act ivat ion of t he t r iplet adduct T1 .

Figure VI I .7: Part of t he low est - ly ing singlet PES st art ing at singlet phenol, as com put ed at t he CBS- QB3 lev el of t heor y . The arr ow indicat es I SC crossing from t he init ial t r iplet t o singlet surfaces.

Chapt er VI I : The O + C6 H 6 React ion

165

Figure VI I .8: Part of t he low est - ly ing singlet PES st art ing at singlet benzene ox ide, as com put ed at t he CBS- QB3 level of t heory . The ar row indicat es I SC crossing from t he init ial t r iplet t o singlet sur faces. Dashed lines present a connect ion fr om singlet benzene ox ide t o singlet 2,4- cyclohexadienone ( S3 ) in Figure VI I .7.

I n any case, once form ed from t he init ial t r iplet adduct T1 by I SC, t he chem ically act ivat ed singlet oxybenzene will rapidly rearr ange eit her t o singlet phenol by 1,2- H m igrat ion or t o singlet benzene ox ide by ring- closure. The var ious react ion pat hways st art ing at singlet phenol or benzene ox ide ar e schem at ically present ed in React ion Schem e VI I .2, and t he m ost im port ant channels ar e depict ed in Figures VI I .7 and VI I .8. I t should be not ed t hat , in t he course of t his w ork , a v ery recent

art icle of Xu and Lin 61 addr essed ab init io–based k inet ics for t he

unim olecular r eact ion c- C6 H 5 OH

CO + c- C5 H 6 . Various react ion rout es on t he

singlet PES st art ing at singlet phenol w ere t heoret ically charact er ized at t he G2M level and discussed in det ail by t hese aut hors. 61 Our CBS- QB3 values are in good agreem ent w it h t he G2M r esult s, generally w it hin 2- 3 kcal/ m ol ( see Table VI I .1) , sim ilar t o t he energy differences found in our pr ev ious st udies. 52,53,62 Ther efor e, it

166

Chapt er VI I : The O + C6 H 6 React ion

is unnecessary t o repeat t he discussion of t he singlet PES in gr eat det ail here. I nst ead, we sum m arize t he energet ically m ost favorable react ion r out es, nam ely : S2

S3

S1 0

S9

CO + c- C5 H 6 and S6 / S7

S3

S1 0

S9

CO + c-

C5 H 6 . So, under low pressure condit ions, t he product cyclo- pent adiene t oget her wit h CO is t heoret ically predict ed t o be m aj or, whereas ot her product s such as H 2 O + c- C6 H 4 or c- C6 H 5 O• +



H all appear t o be m inor. How ev er, at m oderat e t o

high pressur es t he chem ically act ivat ed singlet phenol and/ or singlet benzene ox ide/ ox epin w ill be t herm ally st abilized rapidly by deact ivat ing collisions wit h t he bat h gas. As a result , under t hese condit ions singlet phenol and/ or singlet benzene ox ide/ oxepin are t he m aj or product s form ed on t he singlet PES.

Figure VI I .9: Product dist ribut ion on t he t r iplet PES as a funct ion of t em perat ur e and pr essur e: ( A) for H • + c- C6 H 5 O• ; ( B) for O + C6 H 6 ; and ( C) for collisional st abilizat ion of t riplet ox y benzene.

Chapt er VI I : The O + C6 H 6 React ion

167

V I I .3 .2 . Qu a nt ifica t ion of t he pr oduct dist ribu t ion r e sult ing fr om Oa ddit ion The ov erall cont r ibut ions of t he independent

O- addit ion and H- abst ract ion

channels, as a funct ion of t em perat ur e, have alr eady been discussed ( see e.g. Figure VI I .5) . I n t his sect ion, we w ill first discuss t he product dist ribut ions result ing fr om O- addit ion, as predict ed for t he separat e t r iplet and singlet surfaces, consider ed as adiabat ic. On t h e t r iple t PES. As already m ent ioned abov e, t here are t w o lowest - ly ing

t riplet surfaces ( i.e. 3 A DQG 3 A VHH)LJXUHVI I .1) . For t he indiv idual 3 A VXUIDFH re- dissociat ion back t o t he init ial react ant s is t he predom inant fat e at high t em perat ur es ( T• . , whereas int er nal conversion ( I C) of T1 e x t o T1 prev ails at

low

t em perat ures ( T”. .

At

t em perat ur es in- bet ween,

re-

dissociat ion and I C bot h cont r ibut e and com pet e w it h each ot her, such t hat t he

product dist ribut ion on t his 3 A  VXUIDFH st rongly depends on t he I C rat e, w hich can be obt ained only by very dem anding dy nam ic calculat ions t hat ar e how ev er beyond t he scope of t his paper. Ther efore, in a first approx im at ion, w e assum ed t hat T1 e x eit her com plet ely r e- dissociat es back t o t he r eact ant s or goes ent irely

t owards T1 , which subsequent ly proceeds along various react ion pat hways on t he 3

A VXUIDFH

We w ill now discuss t he pr oduct dist ribut ion on t he individual 3 A  VXUIDFH Not e t hat t he I SC fract ion of T1 t ow ards singlet phenol or benzene ox ide is not consider ed here. There are only t hr ee channels of any im port ance occurr ing on

t he 3 A Vur face: r e- dissociat ion back int o t he r eact ant s, product ion of c- C6 H 5 O• + •

H, and collisional st abilizat ion of t he init ial adduct T1 . The y ields of t hese t hr ee

product channels, as a funct ion of t em perat ur e and pr essur e ( T = 300- 2000 K, P = 10 –1 - 10 5 Tor r ) , hav e been obt ained by solv ing t he w eak - collision m ast er equat ion, and ar e pr esent ed in Figure VI I .9. As can be seen, t he fract ion of r edissociat ion back t o t he init ial r eact ant s incr eases wit h t em perat ur e, but is alm ost pressur e- independent . Howev er, t he re- dissociat ion rout e r em ains m inor ev en at T = 2000 K w it h a y ield < 10% . The m ost im port ant pr oduct is c- C6 H 5 O• +



H,

which act ually m ak es up close t o 100% of all t rue pr oduct s ov er w ide T and P

ranges. I t is only at low T ( ” 500 K) and very high P ( •00 at m ) t hat collisional

st abilizat ion of t he init ial oxybenzene adduct T1 becom es im port ant also as a product rout e, increasing m onot onously w it h increasing pr essure. Act ually, under all r elevant condit ions of hydr ocarbon com bust ion, phenox y radical + H is t he quasi- unique O- addit ion product form ed on t he t riplet surface, w hereas t he y ield of st abilized T1 is negligible, < 1% ev en at T = 1500 K and P = 10 5 Torr . This result is a consequence of t he v ery short lifet im e of t he nascent chem ically

168

Chapt er VI I : The O + C6 H 6 React ion

act ivat ed T1 adduct ( < 1 ps at T= 1500 K) , giv en also t hat it r equir es m any collisions t o br ing it s energy below t he low est - ly ing decom posit ion t r ansit ion st ruct ur e, TS2 . Any t herm ally st abilized t r iplet oxybenzene T1 is pr edict ed t o t hen undergo t he I SC st ep t o t he singlet surface, follow ed by eit her r ing- closur e t o lead t o singlet benzene oxide/ oxepin or a 1,2- H shift t o y ield singlet phenol. I t is wort h st ressing here t hat t he nascent , v ibrat ionally excit ed t r iplet oxybenzene can also undergo a sam e I SC pr ocess leading t o singlet phenol or benzene ox ide, in com pet it ion w it h it s chem ically act ivat ed fragm ent at ion on t he t riplet surface, leading t o phenoxy + H. On t h e sin gle t PES. Singlet phenol and benzene ox ide w hen form ed from t r iplet oxybenzene by I SC w ill undergo subsequent isom er izat ion/ decom posit ion st eps or collisional st abilizat ion. The pr oduct dist r ibut ion on t he singlet sur face is of course dependent on t he rat io of t he init ial singlet phenol and benzene ox ide for m at ion, i. e. t he r at io of t he I SC r at es fr om t r iplet oxybenzene t o singlet phenol and benzene ox ide, r espect iv ely . Again, I SC r at e calculat ions are beyond our capabilit ies. Therefore, calculat ions of t he product dist r ibut ion on t he singlet surface w er e carr ied out for t wo ext r em e cases: t he first for t he I SC of T1 yielding only singlet phenol and t he second for t his process giving only singlet benzene ox ide. Not e howev er t hat t he ent ire singlet PES is included in t he m ast er equat ion analyses, i.e. bot h regions show n in Figur es VI I .7 and VI I .8 are included in t he k inet ic r eact ion schem e at all t im es. A num ber of ot her im port ant point s should also be m ent ioned her e first . ( i) Since t he c- C6 H 5 OH

c- C6 H 5 O• +



H

channel is barrier - less, we charact er ized t he kinet ic bot t leneck st ruct ur e using var iat ional t ransit ion st at e t heory . The v ariat ional TS t hus locat ed has a O- H bond lengt h of 2.0 Å at an int er nal energy of 6.5 kcal/ m ol abov e t he react ant s; t he rov ibrat ional param et er s of t his TS will be used in subsequent pr oduct dist ribut ion analyses. ( ii) The benzene oxide ( S6 )

ox epin ( S7 ) int er nal r earr angem ent

appears t o occur very fast , quick ly achiev ing a m icro- canonical equilibr ium before decom posit ion can t ak e place, such t hat t heir effect iv e densit y of st at es is t ak en as

t he

sum

of

t he

t wo

indiv idual

densit ies.

( iii)

Ring- opening

of

2,4-

cyclohexadienone ( S3 ) leads t o but adieny l- k et ene ( S8 , denot ed as BDK) ( see Figure VI I .7) , which by rapid int er nal rot at ion around t he C- C ax is giv es r ise t o var ious r ot am ers. We charact er ized in t ot al 8 different rot am ers, four of Z ( cis) configurat ion and 4 of E ( t rans) configurat ion, according t o t he or ient at ion of t he m olecule about t he cent ral C= C bond ( see Table VI I .1) , in agr eem ent w it h prev ious findings. 23 Howev er , only Z- configur at ions can readily be generat ed by t he int ernal r ot at ion ar ound t he C- C bond st ar t ing at S8 a in a cis- for m , w her eas it r equires som e 60 kcal/ m ol t o break t he cent r al C= C π- bond and so form an E-

Chapt er VI I : The O + C6 H 6 React ion

169

configurat ion. At high t em perat ur es, S8 m ay cont ain sufficient init ial energy for t his ( see Figur e VI I .7) , but ot her pr ocesses are energet ically m uch m ore favorable and m uch fast er. On t he ot her hand, rat es of int ernal r ot at ion about CC of BDK- Z ar e m uch fast er t han t hose of r ing- closur e or decay st eps. Thus, it is ent irely j ust ified t o t r eat t he densit y of st at es for t he BDK S8 int erm ediat e as t he sum of t he indiv idual densit ies of t he 4 Z configurat ions. Not e t hat each BDK st ruct ur e w it hout sym m et ry ( C1 ) has a m ir ror configurat ion, t he densit y of st at es of w hich was also included. Yields of v arious product channels com put ed as a funct ion of T = 300- 2000 K and P = 10 –4 - 10 5 Torr ar e present ed in Figure VI I .10 for 100% singlet phenol init ially for m ed by I SC from t r iplet oxybenzene; and in Figur e VI I .11 for 100% init ial singlet benzene oxide. An im port ant observat ion is t hat t he pr oduct dist ribut ions predict ed for t hese t wo ext r em e cases do not differ m uch, except of course for st abilized phenol and benzene ox ide at t he highest pressur es. Figur es VI I .10 and VI I .11

also show

t hat

t he product s c- C5 H 6 +

CO,

phenol,

and benzene

ox ide/ ox epin are m aj or , at least in som e condit ions, w hile t he pr oduct s phenox y radical + H, c- C6 H 4 + H 2 O, t he ket one S3 and BDK S8 are always m inor. These result s ar e in agr eem ent wit h ear lier exper im ent al obser v at ions for key product s such as phenol and CO. 19- 21 As can be seen, t he fract ions of t he various product s depend in a com plex way on T and P. I n general, an increase of t em perat ur e enhances t he y ields of c- C5 H 6 + CO, phenox y radical + H, and c- C6 H 4 + H 2 O. On t he ot her hand, incr easing t he pressur e will r educe t he y ields of c- C5 H 6 + CO, phenoxy radical + H, and c- C6 H 4 + H 2 O, but incr ease t hose of phenol and benzene ox ide/ oxepin. These t em perat ur e and pressure effect s are ent ir ely in keeping w it h t he com pet it ion bet ween isom erizat ion/ decom posit ion r eact ions on one hand, and collisional st abilizat ion on t he ot her .

170

Chapt er VI I : The O + C6 H 6 React ion

Figure VI I .10: Pr oduct dist ribut ion on t he singlet PES as a funct ion of t em perat ur e and pressure, assum ing t hat singlet phenol is t he only init ial singlet int erm ediat e result ing from I SC of t r iplet ox ybenzene.

Chapt er VI I : The O + C6 H 6 React ion

171

Figure VI I .11: Pr oduct dist ribut ion on t he singlet PES as a funct ion of t em perat ur e and pr essur e, assum ing t hat singlet benzene ox ide is t he only init ial singlet int erm ediat e result ing fr om I SC of t r iplet ox ybenzene.

172

Chapt er VI I : The O + C6 H 6 React ion

Com pa r ison of pr e dict ions w it h e x pe r im e n t . Let us com pare our predict ed m aj or product s t o t he exper im ent al obser vat ions. First , follow ing O- addit ion, t he rout e t o phenox y radical + H is pr edict ed t o be t he dom inant if not t he sole product pat hway on t he t riplet PES under all hy drocarbon com bust ion condit ions. The phenox y pr oduct was det ect ed not only in ear lier collision- fr ee exper im ent , 12 but also in r ecent t herm al, m ult i- collision ex per im ent s. 13 Second, phenol was est ablished as an im port ant pr oduct of t he t it le r eact ion in several ex per im ent al st udies, 12,18–21 and r ecognized t o result solely from spin- forbidden I SC from t he t riplet t o singlet sur faces. Our present st udy show s t hat phenol is preferably

generat ed under low T ( ”. condit ions wher e I SC t o t he singlet surface can m or e easily out r un t he decom posit ion of t he t riplet adduct T1 t o phenox y + H ( see furt her ) . While form at ion of phenol in m ult i- collision condit ions is easily explained by I SC and collisional st abilizat ion by bat h gas m olecules/ at om s, t he observ at ion of phenol in single- collision cr ossed m olecular beam experim ent s 12 is not underst ood com plet ely. Lee et al. 12 rat ionalized t heir det ect ion of im port ant am ount s of phenol at a flight t im e of 200 µs aft er t he r eact iv e O + C6 H 6 encount er as being due t o a long lifet im e of t he hot singlet c- C6 H 5 OH int erm ediat e for m ed upon I SC. To t est t his assum pt ion, w e inv est igat ed t he t im e- evolut ion of t he abundance of t he v arious int erm ediat e and product species in t he O + C6 H 6 react ion, assum ing t hat of t he 6.5 k cal/ m ol collision energy in t he m ent ioned exper im ent , only t he 4.1 kcal/ m ol required t o surm ount t he addit ion bar r ier cont r ibut es t o t he int er nal energy of t he init ial adduct and ensuing singlet int erm ediat es, i. e. eit her phenol or benzene ox ide. I n t his RRKM analysis, t he m ot ion of t he hydr oxy l- H about t he O- C ax is in phenol was t reat ed as a free int ernal r ot at ion, quit e j ust ifiable at t he 105.6 k cal/ m ol int er nal energy of t he hot phenol at issue. The result s, plot t ed in Figur e VI I .12A and VI I .12B, indicat e t hat aft er 200 µs m uch of t he init ial singlet phenol/ benzene ox ide should st ill surv iv e, while m ost of t he r em ainder should have decayed t o CO + c- C5 H 6 . Given t he sharp abundance changes predict ed ar ound 100 µs ( see Figur e VI I .12) , and consider ing t he est im at ed uncert aint ies of a fact or of 2 or 3 regarding our t heor et ical RRKM lifet im es of t he singlet int erm ediat es, t he present result s are consist ent w it h t he Lee et al. findings and rat ionalizat ions, 12 even t hough t heir observ at ion t hat

CO is only a m inor product

aft er 200 µs suggest s an

ov er est im at ion on our part of t he hot phenol/ benzene ox ide decom posit ion rat e. At t he sam e t im e, our analysis offers an qualit at iv e explanat ion why in t he crossed- beam exper im ent of Sloane, 21 at only 0.6 kcal/ m ol collision energy and hence longer flight t im es, CO and C5 H 6 were observ ed as m aj or product s, at least on a par w it h singlet [ C6 H 6 O] com pounds. I t m ay be not ed t hat only t he

Chapt er VI I : The O + C6 H 6 React ion

173

fragm ent at ion pr oduct s CO + c- C5 H 6 and in a m inor way H + phenox y result ing from t he hot singlet [ C6 H 6 O] int erm ediat es should be consider ed as t rue singlet surface “ end” - product s in single- collision condit ions.

Figure VI I .12: Tim e- evolut ion of abundances of species on t he lowest - ly ing singlet surface under single- collision condit ions at t ot al ener gy assum ed equal t o t he low est - ly ing ent rance TS, i.e. 4.1 k cal/ m ol abov e t he r eact ant s ( see t ext ) . Case A st art ing at singlet phenol and case B st art ing at singlet benzene ox ide, respect ively , as init ial act ivat ed singlet int erm ediat e.

Third, in t herm al condit ions, t he pr oduct benzene ox ide/ ox epin is predict ed t o be produced on t he singlet PES at low T and m oderat e P; it was indeed det ect ed ear lier as a t ransient int erm ediat e 18 in t he O + C6 H 6 react ion and also observ ed in a recent argon m at r ix st udy at 12 K. 23 Fourt h, H- abst ract ion pat hways on t r iplet surfaces leading t o pheny l plus hydr oxy l radicals are t heoret ically charact er ized and evaluat ed t o cont r ibut e significant ly at high t em perat ures; e.g. at T = 2000 K t heir cont r ibut ion is pr edict ed t o be com parable t o t hat of O- addit ion. The • OH radical was indeed det ect ed in cr ossed m olecular beam exper im ent s at a high collision energy of 16.5 kcal/ m ol. 16 Finally, t he product ion of CO, w hich should be for m ed t oget her wit h c- C5 H 6 , has been a subj ect of som e debat e. I n our present analysis, w e resolv e t his cont roversy by r at ionalizing t he product ion of CO as a m aj or end- pr oduct in t he part icular condit ions of crossed m olecular beam exper im ent s at longer flight t im es as report ed by Sloane, 21 m ak ing clear at t he sam e t im e w hy in sim ilar but short flight - t im e exper im ent s such as t hat of Lee et

174

Chapt er VI I : The O + C6 H 6 React ion

al. 12 , CO form at ion is st ill less im port ant . I t should be not ed t hat t he t wo ot her

st udies w ho r eport ed a negligible CO pr oduct ion w it h y ield ”  of Nicovich et

al. 22 as well as of Baj aj and Font ij n, 13 w ere bot h conduct ed under t her m al, m ult icollision condit ions, at T = 298- 950 K / P = 100 Tor r and at T = 405 K / P = 3- 12 Torr, r espect iv ely. Again, our present inv est igat ion clar ifies why CO + c- C5 H 6 product ion in m ult i- collision condit ions differs great ly fr om t hat in single- collision exper im ent s. Figures VI I .10 and VI I .11 show t hat t he predict ed CO y ield from t he singlet surface is < 10% in T and P ranges of bot h t he abov e- m ent ioned t herm al exper im ent s. 13,22 I f we r oughly assum e t hat 50% of t he react iv e flux goes t hr ough I SC, t he overall CO y ield is est im at ed t o be < 5% , in agreem ent w it h t he exper im ent al findings. 13,22 A quant it at iv e predict ion of t he ov erall product branching rat ios spanning all pot ent ial energy sur faces is at t his t im e difficult due t o t he need for accurat e dynam ic calculat ion on t he rat es of t he I C and I SC pr ocesses inv olv ed. For t hese syst em s such calculat ions are ext r em ely dem anding and beyond our cur rent com put at ional r esources. An alt er nat iv e appr oach, used successfully in ear lier wor k, 52,53 where t he rat io of I SC cr ossing versus on- surface unim olecular react ions is calibrat ed against experim ent al product m easur em ent s, is not possible her e due t o t he curr ent lack of sufficient ly com plet e ex per im ent al product dist r ibut ion dat a.

V I I .3 .3 . Ove r a ll t he r m a l ra t e coe fficie n t The ov erall t em perat ur e- dependent rat e coefficient k ( T) overall for t he O( 3 P) + C6 H 6 react ion can be com put ed according t o:

k (T )overall = k (T )O −add + k (T ) H −abs

( VI I .14)

wher e t he k( T) in t he r ight - hand- side ( RHS) ar e t he rat e coefficient derived from m ult i- st at e t ransit ion st at e t heor y: re TS 1 re TS 1ex k (T )O − add = (1 − γ TS 1 ) × k (T )O − add + (1 − γ TS 1ex ) × k (T ) O − add

( VI I .15)

wit h 1ex k (T )TS O − add =

≠ ≠ kbT QTS 1ex exp( − ETS 1ex / RT ) × h QOQC6 H 6

( VI I .16)

k (T )OTS−1add and k (T ) H −abs alr eady defined in eqs ( VI I .13) and ( VI I .12) above, respect ively ; and γre is t he fract ion of r e- dissociat ion of t he init ial adduct s back t o t he init ial react ant s, as a funct ion of t em perat ure and pressur e ( for exam ple, see Figure

VI I .9B) . At low t em perat ur es ( T ”  . , r e- dissociat ion is m inor if not Chapt er VI I : The O + C6 H 6 React ion

175

negligible and t he value of γre is close t o 0. Abov e 1000 K, r e- dissociat ion becom es non- negligible, but it s cont r ibut ion is t he result of a com plicat ed com pet it ion bet ween r e- dissociat ion, furt her isom er isat ion, and I C/ I SC processes ( I C for T1 e x ) . Accurat e quant ificat ion of γr e t her efor e again r equires dynam ic calculat ions t hat are bey ond our capabilit ies. We t herefore lim it ed ourselves t o

t he calculat ion of k( T) overall for T ” 00 K; t em perat ur es com parison.

ex per im ent al The

dat a

ar e

w ell

rot at ional sym m et r ies for

furt herm or e, for t hese lower est ablished C6 H 6 ,

and

av ailable

t he O- addit ion,

and

for H-

abst ract ion t ransit ion st at es ar e 12, 1, and 2, r espect iv ely, such t hat t he react ion pat h degeneracy is 12 for each O- addit ion channel, but 6 for each H- abst ract ion channel as already discussed ear lier. The elect ronic part it ion funct ion of t he O at om

explicit ly

includes t he t hree low est - lying elect ronic st at es ( 3 P2 wit h

elect r onic degeneracy g= 5,

3

P1 w it h g= 3, and

3

P0 wit h g= 1) , w it h r elat ive

energies of 0.0000, 0.4525, and 0.6490 kcal/ m ol, respect ively. 63 Also, t he elect r onic degener acy of 3 for t he t ransit ion st ruct ur es, having a t r iplet elect ronic st at e, is duly t ak en int o account .

Figure VI I .13: Ov erall t herm al rat e coefficient k( T) for t he O + C6 H 6 r eact ion as a funct ion of t em perat ur e in t he T = 300- 800 K r ange, com put ed using Mult i- St at e TST t heory . The m ost recent exper im ent al dat a ar e present ed for t he purpose of com parison.

176

Chapt er VI I : The O + C6 H 6 React ion

The rat e pr edict ions ar e plot t ed in Figur e VI I .13 and can be well- repr esent ed by a m odified Ar rhenius equat ion k ( T) ov erall = 3.7 × 10 −16 × T1.66 × exp ( −1830 K/ T) cm 3 m olecule −1 s −1 ; recent exper im ent al dat a are also show n for com parison. Our k( T) result s are in near - perfect agr eem ent w it h t he experim ent al dat a of Nicovich et al. 22 and Av ram enk o et al. 24 ov er t he ent ire range 300 t o 800 K, w hile t hey are slight ly below t he m easur em ent s by ot her aut hors at low t em perat ures. At room t em perat ur e, our pr edict ed rat e coefficient is 1.0 × 10 –14 cm 3 m olecule −1 s−1 , in good agreem ent w it h t he 1.2 × 10 –14 cm 3 m olecule −1 s −1 m easurem ent by Nicovich, 22 but ca. 40% below t he 1.7 × 10 –14 cm 3 m olecule −1 s−1 r ecom m ended in t he

lit erat ure. 30

How ev er,

at

higher

t em perat ur es,

t he

agr eem ent

w it h

exper im ent al dat a im pr ov es considerably. I t should be not ed t hat our CBS- QB3 com put ed bar r ier height s prove t o be r eliable and wit hin “ chem ical accuracy” , giv en t hat 0.5 kcal/ m ol differ ence alt ers t he com put ed k ( T) values at r oom t em perat ur e by a fact or of 2.3.

V I I .4 . Con clu ding r e m a r k s I n t he pr esent t heor et ical st udy , t he lowest - lying t riplet and singlet pot ent ial energy surfaces for t he O( 3 P) + C6 H 6 react ion w er e charact er ized, uniform ly using t he high lev el quant um chem ical CBS- QB3 m et hod. RRKM- Mast er Equat ion calculat ions t o evaluat e prim ary product dist ribut ion for each of t hese sur faces separat ely and t o qualit at iv ely predict t he ov erall m aj or product s, were carr ied out using t he ex act st ochast ic sim ulat ion m et hod. I n addit ion, ov er all t herm al rat e coefficient s were det erm ined in t he 300- 800 K range using m ult i- st at e t ransit ion st at e t heory. A num ber of im port ant result s em erge from t his st udy and can be sum m ar ized as follows: ( i) The O( 3 P) + C6 H 6 r eact ion is confirm ed t o occur m ainly, but not exclusively , via an elect rophilic O- addit ion m echanism as t he first react ion st ep. The predict ed m aj or pr oduct s from t his addit ion r eact ion ar e c- C6 H 5 O• + H • t oget her wit h phenol and/ or benzene ox ide/ ox epin. c- C6 H 5 O• + H • ar e t he m ost im port ant product s in flam e condit ions and nearly exclusiv ely form ed on t he low est - t r iplet PES, w her eas phenol and/ or benzene ox ide/ oxepin ar e m ainly produced fr om t he lowest - ly ing singlet sur face follow ing an I SC pr ocess; t hese r esult s confirm t he av ailable exper im ent al obser vat ions. CO + c- C5 H 6 ar e predict ed t o be form ed in flam es, but wit h yields t hat can be im port ant only around at m ospher ic pressur e or below . I n single- collision crossed m olecular beam exper im ent s, 21 CO + c- C5 H 6 should be m aj or end- pr oduct s at long enough flight t im es, result ing fr om t he decom posit ion

Chapt er VI I : The O + C6 H 6 React ion

177

of chem ically act ivat ed singlet phenol/ benzene ox ide form ed aft er I SC. Ket ones should be produced as m inor pr oduct s under m oderat e T and P condit ions. The O + C6 H 6

c- C5 H 5 • +

relevant

com bust ion

conclusions.



CHO channel is predict ed t o be unim port ant under all condit ions,

in

cont rast

wit h

prev ious

t heor et ical

14

( ii) Separat e fr om O- addit ion, H- abst ract ion by O pr oceeds on t w o elect r onic surfaces, 3 B1 and 3 B2 , and r esult s in OH( X2 Π) + c- C6 H 5 • pr oduct s, predict ed t o be m aj or at high t em per at ur es. The cont r ibut ion of H- abst ract ion t o t he overall product form at ion is est im at ed t o be ca. 50% at 2000 K. Furt her ex per im ent al st udies are await ed t o v alidat e t hese predict ions. ( iii) The ent rance bar r ier height s and react ion ent halpies com put ed at t he CBSQB3 level of t heory ar e in good agr eem ent wit h available ex per im ent al dat a, wit hin 0.5 kcal/ m ol. ( iv ) The lack of accurat e dynam ic calculat ions for I SC and I C rat es and/ or of available exper im ent al product branching rat ios prohibit s us from quant it at iv ely predict ing t he ov erall pr im ary pr oduct dist r ibut ion for t he t it le r eact ion. Howev er , t he pr esent

st udy

elucidat es t he det ailed

react ion m echanism

and

sem i-

quant it at iv ely predict s all pr oduct s, as obser ved in w idely differ ent react ion condit ions. ( v ) The Mult i- St at e TST com put ed ov erall rat e coefficient for t his com plex m ult ichannel r eact ion, ov er t he range 300- 800 K: k ( T) = 3.7 × 10 −16 × T1.66 × exp( −1830 K/ T) cm 3 m olecule −1 s−1 , is in good agreem ent w it h t he ex per im ent al dat a in t he lit erat ure, w it hin a fact or of < 2.

178

Chapt er VI I : The O + C6 H 6 React ion

Re fe r e nce s ( 1) Glassm an I . Com bust ion, 2 nd ed. ; Academ ic press: Flor ida, 1987. ( 2) Gardiner W. C., Jr. Com bust ion Chem ist ry ; Spr inger - Ver lag: New York , 1984. ( 3) Miller, J. A.; Klippenst ein, S. J. J. Phys. Chem . A 2 0 0 3 , 107, 7783. ( 4) Richt er, H.; Howard, J. B. Phys. Chem . Chem . Phys. 2 0 0 2 , 4, 2038. ( 5) Brezinsky, K. Prog. Energy Com bust . Sci. 1 9 8 6 , 12, 1. ( 6) Saw yer, R. F. Tw ent y- Fourt h Sym p. ( I nt .) Com bust . 1 9 9 2 , 1423. ( 7) Bit t ner, J. D.; Howard, J. B. Eigt eent h Sym p. ( I nt .) Com bust . 1 9 8 1 , 1105. ( 8) Chai, Y. ; Pfeffer le, L. D. Fuel 1 9 9 8 , 77, 313. ( 9) Dav is, S. G.; Wang, H.; Br ezinsk y , K.; Law, C. K. Tw ent y- Six Sym p. ( I nt .) Com bust . 1 9 9 6 , 1025. ( 10) Tan, Y.; Frank , P. Twent y - Six Sym p. ( I nt .) Com bust . 1 9 9 6 , 677. ( 11) Fr om NI ST w eb page: ht t p: / / srdat a.nist .gov / cccbdb/ ( 12) Sibener, S. J.; Buss, R. J.; Casav ecchia, P.; Hirooka, T.; Lee, Y. T. J. Chem . Phys. 1 9 8 0 , 72, 4341. ( 13) Baj aj , P. N. ; Font ij n, A. Com bust . Flam e 1 9 9 6 , 105, 239. ( 14) Hodgson, D.; Zhang, H. Y.; Nim los, M. R.; McKinnon, J. T. J. Phys. Chem . A 2 0 0 1 , 105, 4316 and r efer ences t her ein. ( 15) Ko, T.; Adusei, G. Y.; Font ij n, A. J. Phys. Chem . 1 9 9 1 , 95, 8745. ( 16) Bar ry, N. J. ; Flet cher, I . W.; Whit ehead, J. C. J. Phy s. Chem . 1 9 8 6 , 90, 4911. ( 17) Barck holt z, C.; Bar ckholt z, T. A.; Hadad, C. M. J. Phys. Chem . A 2 0 0 1 , 105, 140. ( 18) Mani, I . ; Sauer, Jr. M. C. Advan. Chem . Ser. 1 9 6 8 , 82, 142. ( 19) Boocock, G.; Cv et anov ic, R. J. Can. J. Chem . 1 9 6 1 , 39, 2436. ( 20) Bonanno, R. A.; Kim , P.; Lee, J. H.; Tim m ons, R. B. J. Chem . Phys. 1 9 7 2 , 57, 1377 and r eferences t herein. ( 21) Sloane, T. M. J. Chem . Phys. 1 9 7 7 , 67, 2267. ( 22) Nicov ich, J. M.; Gum p, C. A.; Rav ishankar a, A. R. J. Phys. Chem . 1 9 8 2 , 86, 1684. ( 23) Park er , J. K.; Dav is, S. R. J. Am . Chem . Soc. 1 9 9 9 , 121, 4271. ( 24) Av ram enko, L. I .; Kolesnik ova, R. V.; Sav inov a, G. I . Bull. Acad. Sci. USSR Div. Chem . Sci. ( Engl. Transl.) 1 9 6 5 , 24- 29. ( 25) Colussi, A. J.; Singlet on, D. L.; I rw in, R. S.; Cv et anov ic, R. J. J. Phys. Chem . 1 9 7 5 , 79, 1900. ( 26) At k inson, R.; Pit t s, J. N. Chem . Phy s. Let t . 1 9 7 9 , 63, 485. ( 27) Cv et anov ic, R. J. J. Phys. Chem . Ref. Dat a 1 9 8 7 , 16, 261. ( 28) Leidr eit er, H. I .; Wagner, H. G. Z. Phys. Chem . N. F. 1 9 8 9 , 165, 1. ( 29) Tappe, M.; Schliephak e, V.; Wagner , H. G. Z. Phys. Chem . N. F. 1 9 8 9 , 162, 129. ( 30) Baulch, D. L.; Cobos, C. J.; Cox, R. A.; Frank, P.; Haym an, G.; Just , Th.; Ker r, J. A.; Mur rells, T.; Pilling, M. J.; Tr oe, J.; Walker, R. W.; War nat z, J. J. Phys. Chem . Ref. Dat a 2 0 0 5 , 34, 854. ( 31) Nguyen, M. T.; Kry achko, E. S.; Vanquickenbor ne, L. G. in “ The Chem ist ry of Phenols” , Edit or: Rappoport Z., Wiley 2 0 0 3 , Chichest er, U. K., Chapt er 1. ( 32) Alzuet a, M. U. ; Glarborg, P.; Dam - Johansen, K. I nt . J. Chem . Kinet . 2 0 0 0 , 32, 498. ( 33) Cost a, I . D.; Fournet , R.; Billaud, F. ; Bat t in- Lecclerc, F. I nt . J. Chem . Kinet . 2 0 0 3 , 35, 503. ( 34) Lindst edt , R. P.; Skev is, G. Com bust . Flam e 1 9 9 4 , 99, 551. ( 35) Tan, Y.; Frank , P. Twent y - Sixt h Sym p. ( I nt .) Com bust . 1 9 9 6 , 677. ( 36) Becke A. D. J. Chem . Phys. 1 9 9 3 , 98, 5648. ( 37) St ev ens P. J.; Devlin F. J. ; Chablowsk i C. F.; Fr isch M. J. J. Phy s. Chem . 1 9 9 4 , 98, 11623. ( 38) Gonzalez C. ; Schlegel H. B. J. Chem . Phys. 1 9 8 9 , 90, 2154. ( 39) Gonzalez C. ; Schlegel H. B. J. Phys. Chem . 1 9 9 0 , 94, 5523.

Chapt er VI I : The O + C6 H 6 React ion

179

( 40) Mont gom ery J. A. Jr.; Fr isch M. J.; Ocht erski J. W.; Pet ersson G. A. J. Chem . Phys. 1 9 9 9 , 110, 2822. ( 41) Werner H. J.; Knowles P. J. J. Chem . Phys. 1 9 8 5 , 82, 5053. ( 42) Know les P. J.; Wer ner H. J. Chem . Phy s. Let t . 1 9 8 5 , 115, 259. ( 43) Celani P.; Wer ner, H. J., J. Chem . Phys. 2 0 0 0 , 112, 5546. ( 44) Frisch M. J.; Tr uck s G. W. ; Schlegel H. B. et al. Gaussian 03, Gaussian, I nc., Pit t sburgh, PA, ( 2 0 0 4 ) . ( 45) DALTON, a m olecular elect r onic st ruct ure program , wr it t en by Helgaker T.; Jensen H. J. Aa. ; Joergensen P.; Olsen J.; Ruud K.; Aagr en H.; Auer A. A. et al., Release 1.2 ( 2 0 0 1 ) . ( 46) MOLPRO is a package of ab init io pr ogram s wr it t en by Wer ner H.- J.; Know les P. J.; Schüt z M.; Lindh R.; Celani P. ; Korona T.; Rauhut G.; Manby F. R. ; Am os R. D.; Ber nhardsson A.; Ber ning A.; Cooper D. L.,; Deegan M. J. O.; Dobby n A. J. ; Eckert F; et al. ( 2 0 0 2 ) . ( 47) Gillespie D. T. J. Com put . Phy s. 1 9 7 6 , 22, 403. ( 48) Gillespie D. T. J. Phys. Chem . 1 9 7 7 , 81, 2340. ( 49) Gillespie D. T. J. Com put . Phy s. 1 9 7 8 , 28, 395. ( 50) Ver eeck en L.; Huy ber echt s G.; Peet ers J. J. Chem . Phy s. 1 9 9 7 , 106, 6564. ( 51) Hippler H.; Troe J; Wendelken H. J. J. Chem . Phys. 1 9 8 3 , 78, 6709. ( 52) Nguy en, T. L.; Ver eeck en, L.; Peet ers, J. J. Phys. Chem . A 2 0 0 6 , 110, 6696. ( 53) Nguyen, T. L.; Vereeck en, L.; Hou, X. J. ; Nguy en, M. T. ; Peet ers, J. J. Phys. Chem . A 2 0 0 5 , 109, 7489. ( 54) Nguy en, T. L. ; Peet ers, J. ; Vereecken, L. J. Phys. Chem . A 2 0 0 6 , 110, 12166. ( 55) Malick, D. K.; Pet ersson, G. A.; Mont gom ery, Jr., J. A. J. Chem . Phy s. 1 9 9 8 , 108, 5704. ( 56) Klessinger M. ; Michl J. Ex cit ed St at es and Phot ochem ist ry of Or ganic Molecules; VCH: New York , 1995. ( 57) Haas Y.; Klessinger M.; Zilberg S. Chem . Phys. 2 0 0 0 , 259, 121 and references t herein. ( 58) Liu, R.; Morokum a, K.; Mebel, A. M.; Lin, M. C. J. Phys. Chem . 1 9 9 6 , 100, 9314 and refer ences t her ein. ( 59) Eckart C., Phys. Rev. 1 9 3 0 , 35, 1303. ( 60) Johnst on H. S.; Heick len, J., J. Phys. Chem . 1 9 6 6 , 66, 532. ( 61) Xu, Z. F.; Lin, M. C. J. Phys. Chem . A 2 0 0 6 , 110, 1672. ( 62) Nguyen, T. L.; Dils, B. ; Car l, S. A.; Vereecken, L.; Peet ers, J. J. Phys. Chem . A 2 0 0 5 , 109, 9786. ( 63) NI ST w eb page: ht t p: / / phy sics.nist .gov/ PhysRefDat a/ Handbook / per iodict able.ht m . ( 64) Nguy en, T. L.; Le, T. N.; Mebel, A. M. J. Phys. Or g. Chem . 2 0 0 1 , 14, 131 and references t herein. ( 65) Went hold, P. G.; Squir es, R. R.; Lineberger, W. C. J. Am . Chem . Soc. 1 9 9 8 , 120, 5279. ( 66) Luo, Y. R. Handbook of Bond Dissociat ion Ener gies in Or ganic Com pounds; CRC Pr ess : Boca Rat on, FL, 2003.

180

Chapt er VI I : The O + C6 H 6 React ion

Ge n e r a l Con clusion s

In

t his

t hesis,

com binat ion

we

w it h

describe t he

high- lev el

st at e- of- t he- art

quant um

chem ical

st at ist ical

k inet ic

calculat ions analysis

in

in a

com prehensiv e st udy of t he oxidat ion r eact ions of t riplet ox ygen at om s wit h several

sm all

unsat urat ed

hydrocarbons

including

acet ylene,

et hene

( t et rafluoroet hene) , allene, and benzene, which ar e crucial int er m ediat es in hydrocarbon

com bust ion

chem ist ry

and

flam es

in

general.

The

react ion

m echanism , t he pr oduct dist r ibut ion r esult ing from sev eral elect ronic st at e pot ent ial energy sur faces, and t he overall t herm al rat e coefficient s for t hese react ions wer e syst em at ically pr esent ed for t he first t im e. Our com put ed result s are in good agr eem ent wit h t he available exper im ent al dat a, and show t hat t he elect r ophilic O- addit ion m echanism s ont o a car bon at om of t he double or t r iple CC bond in t hese hydr ocar bons ar e usually t he dom inant channels up t o com bust ion t em perat ur es. One of t he m aj or product s t ypically pr edict ed for t he addit ion rout e is t he form at ion of an alk yl- ox y radical or a vinox y radical plus a hydrogen at om , m ainly produced from

t he t r iplet surface v ia a spin- conserv ing ex change-

m echanism of t he t r iplet O at om against an H at om of t he hy drocarbon. Ot her predict ed m aj or pr oduct s for t he addit ion pat hw ays depend fundam ent ally on t he nat ur e of init ial unsat urat ed hydr ocarbons, and are sum m arized in det ail in t he conclusions of each indiv idual chapt er. H- abst ract ion m echanism s can play im port ant r oles at v ery high com bust ion t em perat ur es, but ar e in general negligible at low er t em perat ur es. The t heoret ical m odel chem ist ry ( i.e. CBS- QB3, CBS- APNO, G2M, and G3) used ext ensively in t his wor k prov ides “ chem ical accuracy” for relat ive energies of st at ionary point s on t he pot ent ial energy surfaces of int erest , i.e. a 1- 2 kcal/ m ol dev iat ion com par ed t o exper im ent al dat a av ailable. However, such chem ical accuracy is st ill insufficient for pr ecise calculat ions of t herm al rat e coefficient s at present , giv en t hat t heor et ical k ( T) v alues ar e quit e sensit iv e t o t he accuracy of barr ier height s, especially at low t em perat ur es: at r oom t em perat ure a differ ence in bar r ier height by 0.5 kcal/ m ol alt ers t he com put ed k ( T) v alues by a fact or of 2.3. Ro- v ibrat ional par am et ers com put ed at t he DFT- B3LYP, QCI SD or CCSD( T) m et hods appear t o be reasonable for st at ist ical k inet ic analy sis in t his sy st em , prov ided t hat hinder ed int er nal r ot ors are t r eat ed proper ly. Furt her m or e, w hen t he wav e- funct ion of st at ionary point s possesses a m ult i- r efer ence charact er, applicat ion of an appropr iat e m ult i- configur at ion CASPT2 or MRCI

m et hod

account ing for dynam ic- elect r on correlat ions is necessary .

181

Based on t he high- lev el quant um chem ical dat a t hus calculat ed, t he product dist ribut ion calculat ions by solut ion of a weak - collision m ast er equat ion using t he Gillespie’s exact st ochast ic sim ulat ion m et hod ( ESM) , w hich was im plem ent ed ear lier

in

our

laborat ory ,

y ielded

result s

in

fair ly

good

agr eem ent

w it h

exper im ent al observat ions. I n addit ion t o regular unim olecular r eact ions and collisional ener gy t ransfer processes, som e of t he r eact ions of t riplet ox ygen at om s wit h unsat urat ed hydrocarbons also inv olve com plicat ed int er nal conv ersion ( I C) and int ersyst em crossing ( I SC) processes, whose rat es can be obt ained only by accurat e dy nam ic calculat ions; such calculat ions are well out of t he scope of t his wor k. Thus, quant it at iv e pr edict ions of t he overall pr oduct dist ribut ions for t he react ions of O at om s wit h allene and benzene by com bining t he k inet ics ov er t he indiv idual pot ent ial

energy

surfaces wit h

det ailed

int er - surface

dy nam ics cannot

be

achiev ed. For ot her r eact ions, w e w ere able t o use available exper im ent al dat a t o deduce t he r elat ive im port ance of pr oduct form at ion over t he indiv idual elect ronic surfaces, allow ing us t o der iv e com plet e over all product dist r ibut ion predict ions ev en for product hit hert o unobser ved in ex perim ent al work. This st udy illust rat es again t hat high- lev el quant um chem ical calculat ions followed by

accurat e

st at ist ical

kinet ic

analysis

ar e

powerful

t ools

for

a

bet t er

underst anding and elucidat ion of det ailed r eact ion m echanism s, as well as for t he quant it at iv e predict ion of t he pr oduct br anching rat ios. Fut ure work along t he lines of t his t hesis includes t he t heor et ical inv est igat ion of ot her react ions of t riplet oxygen at om s wit h unsat urat ed species of im port ance in com bust ion processes, such as propy ne, n- propene, n- but ene, iso- but ene, diacet y lene, each of w hich has a specific role in com bust ion chem ist ry.

182

THE MORE ACCURATE THE CALCULATIONS BECOME, THE MORE THE CONCEPTS TEND TO VANISH INTO THIN AIR. R.S. MULLIKEN, J.C.P. 43, S2 (1965)

THE MORE PROGRESS PHYSICAL SCIENCES MAKE, THE MORE THEY TEND TO ENTER THE DOMAIN OF MATHEMATICS, WHICH IS A KIND OF CENTRE TO WHICH THEY ALL CONVERGE. WE MAY EVEN JUDGE THE DEGREE OF PERFECTION TO WHICH A SCIENCE HAS ARRIVED BY THE FACILITY WITH WHICH IT MAY BE SUBMITTED TO CALCULATIONS. ADOLPHE QUETELET, IN INSTRUCTIONS POPULAIRES SUR LE CALCUL DES PROBABILITIES (1928)