Ab initio study of the n- * electronic transition in ...

JOURNAL OF CHEMICAL PHYSICS

VOLUME 111, NUMBER 1

1 JULY 1999

Ab initio study of the n - p * electronic transition in acetone: Symmetry-forbidden vibronic spectra D. W. Liao Institute of Physical Chemistry, Department of Chemistry and State Key Lab of Physical Chemistry on Solid Surfaces, Xiamen University, Xiamen, Fujian 361005, China

A. M. Mebela) Institute of Atomic and Molecular Sciences, Academia Sinica, P.O. Box 23-166, Taipei 10764, Taiwan, ROC and Department of Chemistry, Tamkang University, Tamsui 25137, Taiwan, ROC

M. Hayashi Institute of Atomic and Molecular Sciences, Academia Sinica, P.O. Box 23-166, Taipei 10764, Taiwan, ROC

Y. J. Shiu Department of Chemistry, National Taiwan University, Taipei 106, Taiwan, ROC

Y. T. Chen Institute of Atomic and Molecular Sciences, Academia Sinica, P.O. Box 23-166, Taipei 10764, Taiwan, ROC and Department of Chemistry, National Taiwan University, Taipei 106, Taiwan, ROC

S. H. Lin Institute of Atomic and Molecular Sciences, Academia Sinica, P.O. Box 23-166, Taipei 10764, Taiwan, ROC, Department of Chemistry, National Taiwan University, Taipei 106, Taiwan, ROC, and Department of Chemistry and Biochemistry, Arizona State University, Tempe, Arizona 85287-1604

~Received 15 January 1999; accepted 29 March 1999! Ab initio calculations of geometry and vibrational frequencies of the first singlet excited 1 A 2 ( 1 A 9 ) state of acetone corresponding to the n- p * electronic transition have been carried out at the CASSCF/6-311G** level. The major geometry changes in this state as compared to the ground state involve CO out-of-plane wagging, CO stretch and torsion of the methyl groups, and the molecular symmetry changes from C 2 v to C s . The most pronounced frequency changes in the 1 A 9 state are the decrease of the CO stretch frequency v 3 by almost 500 cm21 and the increase of the CH3 torsion frequency v 12 from 22 to 170 cm21. The optimized geometries and normal modes are used to compute the normal mode displacements which are applied for calculations of Franck–Condon factors. Transition matrix elements over the one-electron electric field operator at various atomic centers calculated at the state-average CASSCF/6-3111G** level are used to compute vibronic couplings between the ground 1 A 1 , 1 A 2 , and Rydberg 1 B 2 (n-3s), 2 1 A 1 (n-3p y ), 2 1 A 2 (n-3p x ), 2 1 B 2 (n-3 p z ), and 1 B 1 (n-3d xy ) electronic states, and the Herzberg–Teller expansion of the electronic wave function is applied to derive the transition dipole moment for 1 A 1 → 1 A 2 as a function of normal coordinates. The results show that the intensity for this transition is mostly borrowed from the allowed 1 A 1 - 1 B 2 (n-3s) transition due to vibronic coupling between 1 A 2 and 1 B 2 through normal modes Q 20 , Q 22 , and Q 23 and, to some extent, from the 1 A 1 - 1 B 1 transition due to Q 19 ~CO in-plane bend! which couples 1 A 2 with 1 B 1 (n-3d xy ). The calculated total oscillator strength for the n- p * transition through the intensity-borrowing mechanism, 3.6231024 , is in close agreement with the experimental value of 4.1431024 . Ninety-four percent of the oscillator strength comes from the perpendicular component (b 1 inducing modes! and 6% from the parallel component (b 2 modes!. Calculated spectral origin, 30 115 cm21 at the MRCI/6-311G** level, underestimates the experimental value by ;300 cm21. Calculated positions of the most intense peaks in the spectra also reasonably agree with the experimental band maximum. The presence of numerous weak vibronic peaks densely covering a broad energy range ~;12 000 cm21! explains the diffuse character of the experimental n- p * band. Most of the bands observed in fluorescence excitation spectra @Baba and Hanazaki, Chem. Phys. Lett. 103, 93 ~1983!; Baba, Hanazaki, and Nagashima, J. Chem. Phys. 82, 3938 ~1985!# can be assigned based on the computed spectrum. © 1999 American Institute of Physics. @S0021-9606~99!30324-X#

a!

Electronic mail: [email protected]

0021-9606/99/111(1)/205/11/$15.00

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© 1999 American Institute of Physics

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I. INTRODUCTION

Acetone, (CH3!2CO, is an important probe molecule in experimental and theoretical photophysical chemistry. Excited electronic states of acetone have been investigated extensively.1–16 For the electronic transition to the first excited singlet n- p * ( 1 A 2 ) state, a very weak diffuse band extending from 250 to 340 nm with a broad maximum at 290 nm was observed in the gas phase by Noyes et al. back in 1934.1 More recent measurements by Walzl et al.2 using the electron impact spectrometry gave the vertical excitation energy of 283 nm for this transition. Worden3 reported the oscillator strength for the n- p * transition to be 4.14 31024 . Baba et al.4,5 were able to resolve the 1 A 1 - 1 A 2 band in the acetone spectra through measurements of fluorescence excitation spectra in a supersonic nozzle beam. They have assigned a number of vibronic peaks in the energy range up to ;1500 cm21 from the spectral origin in terms of active normal modes, such as CH3 torsions ( v 12 and v 24!, CO outof-plane wagging v 23 , etc. They have also located the band origin at 30 435 cm21. The 1 A 1 → 1 A 2 (n- p * ) electronic transition in (CH3!2CO is symmetry-forbidden. The appearance of the corresponding band in the absorption spectra can be explained on the basis of an intensity-borrowing mechanism where the oscillator strength is borrowed from the symmetry-allowed transitions such as 1 A 1 → 1 B 2 , 1 A 1 →2 1 A 1 , or 1 A 1 → 1 B 1 due to vibronic coupling of 1 A 2 with higher excited states 1 B 2 , 2 1 A 1 , and 1 B 1 . Although the qualitative understanding of intensity-borrowing goes back to Herzberg and Teller,17 quantitative calculations of individual peak intensities and the total oscillator strength are still rare. Ab initio calculations of vibronic coupling in the application to symmetryforbidden vibronic spectra have been reported for formaldehyde,18–24 benzene,25–27 and recently for ethylene.28 Since a significant amount of experimental data is available for the n- p * transition in acetone, this band is a good candidate to test the theoretical procedure for ab initio calculations of symmetry-forbidden spectra suggested by us earlier.28 This would also allow us to make a more definite assignment of the spectra and to elucidate the details of the intensity-borrowing mechanism for acetone.

II. THEORETICAL METHODS

Ab initio calculations of geometry and vibrational frequencies of (CH3!2CO in the ground electronic state were performed at the hybrid density functional B3LYP/6-311G** level.29 The geometry and frequencies of the first excited singlet n- p * state were computed using the CASSCF/6-311G** approach30 with the active space including 10 electrons distributed on 11 orbitals (2a 1 14b 2 13b 1 12a 2 in C 2 v symmetry or 5a 8 16a 9 in C s !. For comparison, the ground state geometry was also recalculated at the same CASSCF level, but the result is very similar to that obtained at B3LYP ~see Fig. 1!. The B3LYP frequencies were scaled by 0.961431 and the CASSCF frequencies of the excited state were scaled by 0.93. The scaling factor for CASSCF was chosen to match the calculated CH stretch

FIG. 1. CASSCF~10,11!/6-311G** and B3LYP/6-311G** ~italics! optimized geometries of the ground 1 A 1 ~1! and first singlet excited 1 A 9 ~2! electronic states of acetone ~bond lengths are in Å and bond angles are in degrees!.

frequencies of the ground and excited states in the 2900– 3020 cm21 energy range assuming that they should not change significantly upon the n- p * electronic transition. The vertical and adiabatic excitation energies for the S 1 state were refined at the MRCI~10,11!/6-311G** level32 including Davidson corrections for quadruple excitations. The optimized geometries, normal modes, and vibrational frequencies of the two electronic states were used in calculations of Franck–Condon factors for the case of displaced, distorted, and mixed normal modes.33 In order to properly describe higher excited states of acetone which have a Rydberg character, we carried out CASSCF/6-3111G** calculations with the active space of 6 electrons on 13 orbitals: 3a 1 15b 2 14b 1 11a 2 , i.e., 3 highest occupied orbitals 2b 2 11b 1 , all vacant valence orbitals with addition of the 3s and 3 p Rydberg orbitals. For calculations of the 1 B 1 (n-3d xy ) state we add to the active space one more Rydberg a 2 (3d xy ) orbital. Correspondingly, the transition dipole moments and transition matrix elements over the one-electron electric field operator needed for calculations of vibronic coupling of 1 A 1 and 1 A 2 with the Rydberg states were obtained at the state-average CASSCF~6,13! level or at CASSCF~6,14! for 1 A 1 - 1 B 1 and 1 A 2- 1B 1 . The following ab initio programs were employed in this study: GAUSSIAN 9434—for B3LYP calculations of the ground state; DALTON35 –for CASSCF geometry optimization and frequency calculations of the n- p * excited state; and MOL36 PRO 96 —for MRCI and state-average CASSCF calculations of the energies, transition dipole moments, and matrix elements. The theory of vibronic coupling is described elsewhere.17–19,28,37 Summarizing, the expression for vibronic coupling between two electronic states F a and F b due to the normal mode Q i can be written as follows: VC ~ ab ! i 5 5

^ F 0a u ~ ] V/ ] Q i ! 0 Q i u F 0b & E ~ F 0a ! 2E ~ F 0b ! ( satomsZ s e 2 ( j l sj i W sj ~ ab !

where j5x,y,z,

E ~ F 0a ! 2E ~ F 0b !

Qi ,

~1!

n-p transition in acetone


207

TABLE I. Calculated and experimental excitation energies ~eV! of the vertical excited states of acetone.

State 1 1 2 2 1 1

CASSCF~6,13! /6-3111G**

CASSCF~6,14! /6-3111G**

MRCI~10,11! /6-311G**

4.64 6.24 7.47 7.36 7.36

4.65 6.16 7.37 7.27 7.27 8.33

4.44

A 2 (n y 2 p * ) B 2 (n y 23s) 1 A 1 (n y 23p y ) 1 A 2 (n y 23p x ) 1 B 2 (n y 23p z ) 1 B 1 (n y 23d xy ) 1 1

CASPT2a

Expt.

4.18 6.58 7.26 7.34 7.48 8.20

4.38b 6.35c 7.41d 7.36d 7.45d 8.17a

a

From Ref. 13. From Ref. 2. c From Ref. 41. d From Ref. 42. b

l sj i 5

] s sj , ]Qi

~2!

K U(

electrons

~ s sj 2q aj !

a

r s3 a

W sj ~ ab ! 5 F 0a

U L

F 0b ,

~3!

in which s sj and q aj are the Cartesian coordinates of nucleus s and electron a, respectively. electrons

~ s sj 2q aj !

(a

r s3 a

is the jth component of the electric field operator for nucleus s and W sj (ab) can be computed as the transition matrix element over the one-electron electric field operator at the atomic center s. Using the Herzberg–Teller expansion for the wave function,17 one can write the transition dipole moment for a symmetry-forbidden transition as a function of normal coordinates: Dab ~ Q ! 5

H

0 VC ~ aa 8 ! i ^ F a 8 u Ru F 0b & (i ( a

1

8

0 ^ F 0a u Ru F b 8 & VC ~ b 8 b ! i ( b

8

J

,

~4!

where the a 8 →b and a→b 8 transitions are symmetryallowed. Then, if for a particular inducing normal mode Q i the transition dipole moment is expressed as Dab (Q i ) 5Diab Q i , the oscillator strength of the vibronic transition a n →b n 8 is calculated as 2 FC a n j ,b n 8 . f a n ,b n 8 5 DE u Diab u 2 u I a n i ,b n 8 u 2 i j 3 jÞi

)

~5!

I a n i ,b n 8 5 ^ x a n i ~ Q i ! u Q i u x b n 8 ~ Q 8i ! &

~6!

Here, i

i

are vibronic overlap integrals for the inducing mode Q i and FC a n j ,b n 8 5 u ^ x a n j u x b n 8 & u 2 j

j

are Franck–Condon factors for modes jÞi.

~7!

III. COMPARISON OF GEOMETRY, VIBRATIONAL FREQUENCIES, AND NORMAL MODES OF ACETONE IN S 0 AND S 1 STATES

Optimized geometries of the acetone molecule in the ground ~1! and first excited singlet n- p * ~2! electronic states are shown in Fig. 1. In S 0 ~CH3!2CO has C 2 v symmetry with a planar CCCO fragment. The first excited singlet state corresponding to the n- p * electronic transition is 1 A 2 . At the MRCI~10,11!/6-311G** level, vertical 1 A 1 → 1 A 2 excitation energy is calculated to be 4.44 eV ~Table I!. This result is close to 4.37 eV38 obtained earlier at the CASPT2/6-31G** level and to the experimental value of 4.38 eV.2 Upon geometry optimization, the symmetry of the S 1 state relaxes from C 2 v to C s and the electronic term changes to 1 A 9 . The major geometry changes from 1 A 1 to 1 A 9 are the pyramidalization of the carbonyl C atom and the CO bond stretch. In 2, the OCCC dihedral angle is 132.4° and the CCO angle decreases by ;10° as compared to that in 1. The CO bond in 1 A 9 is elongated to 1.36 Å from 1.21 Å in 1 A 1 due to an increase of p* electrons. There is also a conformational change of the CH3 groups in 2. While in 1 the HCCO dihedral angles are 0° and 6120°, in 2 they took the values of 51°, 269°, and 171°. The other geometric parameters, including the CC and CH bond lengths and the CCC and CCH bond angles are similar in both electronic states. It should be mentioned that the optimized geometry of 1 A 9 calculated here at the CASSCF~10,11!/6-311G** level is close to that obtained by Setokuchi et al.15 at CASSCF~6,5!/6-31G** , except the CO distance which is about 0.04 Å shorter in the present study. The computed adiabatic 1 A 1 → 1 A 9 excitation energy is 3.73 eV ~30 115 cm21 or 332 nm! at the MRCI level including ZPE corrections. This result is fairly close to the experimental value of 30 435 cm21 reported by Baba et al.4,5 Vibrational frequencies of 1 and 2 are shown in Table II. The frequency n 3 corresponding to CO stretching undergoes the largest change in the S 1 state, from 1733 cm21 in 1 to 1226 cm21 in 2, due to elongation of the CO bond. On the contrary, the CH3 torsion is hindered in 1 A 9 ; the corresponding frequencies n 12 ~22 cm21! and n 24 ~133 cm21! increase to 170–196 cm21. The calculated frequencies agree fairly well with the experimental data,6,39,40 except n 12 , 22 vs 77 cm21 in experiment.40 This deviation is apparently due to an un-

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J. Chem. Phys., Vol. 111, No. 1, 1 July 1999 TABLE II. Calculated and experimental ~in parentheses!a vibrational frequencies ~cm21! of ~CH3!2CO in the 1 A 1 and 1 A 9 electronic states. 1

Frequency

n 1 (a 1 →a 8 ) n 2 (a 1 →a 8 ) n 3 (a 1 →a 8 ) n 4 (a 1 →a 8 ) n 5 (a 1 →a 8 ) n 6 (a 1 →a 8 ) n 7 (a 1 →a 8 ) n 8 (a 1 →a 8 ) n 9 (a 2 →a 9 ) n 10(a 2 →a 9 ) n 11(a 2 →a 9 ) n 12(a 2 →a 9 ) n 13(b 2 →a 9 ) n 14(b 2 →a 9 ) n 15(b 2 →a 9 ) n 16(b 2 →a 9 ) n 17(b 2 →a 9 ) n 18(b 2 →a 9 ) n 19(b 2 →a 9 ) n 20(b 1 →a 8 ) n 21(b 1 →a 8 ) n 22(b 1 →a 8 ) n 23(b 1 →a 8 ) n 24(b 1 →a 8 )

Ac

Assignment

3019 2928 1226~1124! 1481 1408 1113 768~757! 320 2992 1483 962 170 3021 2927 1470 1399 1294 952 360 2986 1500 942 395~373! 196

CH3 degenerate stretch CH3 symmetric stretch CO stretch CH3 symmetric deformation CH3 degenerate deformation CH3 rock CC symmetric stretch CCC bend/symmetric CCO bend CH3 degenerate stretch CH3 degenerate deformation CH3 rock CH3 torsion CH3 degenerate stretch CH3 symmetric stretch CH3 degenerate deformation CH3 symmetric deformation CC asymmetric stretch CH3 rock/CC asymmetric stretch CO in-plane bend CH3 deg. stretch/CO out-of-plane bend CH3 degenerate deformation CH3 rock/CO out-of-plane bend CO out-of-plane bend/CH3 torsion CH3 torsion/CO out-of-plane bend

A 1b

3020~3019! 2914~2937! 1733~1731! 1414~1435! 1330~1364! 1041~1066! 750~777! 364~385! 2960~2963! 1410~1426! 848~877! 22~77! 3018~3019! 2907~2937! 1404~1410! 1332~1364! 1183~1216! 848~891! 515~530! 2967~2972! 1431~1454! 1075~1091! 471~484! 133~124.5!

1

a

From Refs. 5, 6, 39, 40. B3LYP/6-311G** frequencies scaled by 0.9614. c CASSCF/6-311G** frequencies scaled by 0.93. b

harmonic character of CH3 torsion. As will be shown in Sec. VI, the experimental S 0 -S 1 fluorescence excitation spectra4,5 can be assigned based on the calculated frequencies for the excited state. Table III shows displacements DQ of normal modes of acetone from S 0 to S 1 . The normal modes of a 2 and b 2 symmetry (Q 9 -Q 19) correspond to a 9 in the 1 A 9 state and are not displaced. On the other hand, normal modes Q 1 -Q 8 (a 1 ) and Q 20-Q 24(b 1 ) belong to the a 8 irreducible representation of the C s symmetry group and can be displaced. The largest displacements are found for the modes involving some contribution from CO out-of-plane bending: DQ 2450.860 bohr amu1/2 for CH3 torsion/CO out-of-plane bend, DQ 2250.449 bohr amu1/2 for CH3 rock/CO out-ofplane bend, DQ 2050.443 bohr amu1/2 for CH3 stretch/CO out-of-plane bend, and DQ 2350.189 bohr amu1/2 for pure CO out-of-plane bend. Other significantly distorted modes include Q 21(0.345 bohr amu1/2) for CH3 degenerate deformation, Q 8 (0.223 bohr amu1/2) for CCO symmetric bend,

Q 3 (0.204 bohr amu1/2) for CO stretch, and Q 4 (0.192 bohr amu1/2) for CH3 symmetric deformation. Comparison of normal coordinates in the ground and excited electronic states indicates that the modes of the a 9 (a 2 1b 2 ) symmetry are not mixed in 1 A 9 . Some mixing or rotation is found for the a 8 (a 1 1b 1 ) modes. For instance, the normal modes Q 3 , Q 6 , Q 22 , and Q 23 are significantly mixed meaning that the CO stretch, CH3 rock and CO out-of-plane bend are strongly coupled in the excited state. The corresponding normalized Duschinsky matrix for the four modes is presented in Table IVa. Also, there is a small mixing of symmetric and degenerate CH3 deformations Q 4 and Q 21 ~Table IVb!. Rotation of the other normal modes is insignificant. IV. FRANCK–CONDON FACTORS FOR 1 A 1 ˜ 1 A 9 VIBRONIC TRANSITIONS

Using the data presented in the previous section, including the normal mode displacement, distortion, and rotation,

TABLE III. Displacements DQ ~bohm amu1/2! of normal modes in the excited 1 A 9 electronic state of ~CH3!2CO. Normal mode Q1 Q5 Q9 Q 13 Q 17 Q 21

DQ

Normal mode

DQ

Normal mode

DQ

Normal mode

DQ

0.0825 0.0545 0.0 0.0 0.0 0.3448

Q2 Q6 Q 10 Q 14 Q 18 Q 22

0.0889 0.0957 0.0 0.0 0.0 0.4493

Q3 Q7 Q 11 Q 15 Q 19 Q 23

0.2037 0.1544 0.0 0.0 0.0 0.1888

Q4 Q8 Q 12 Q 16 Q 20 Q 24

0.1915 0.2232 0.0 0.0 0.4431 0.8598


J. Chem. Phys., Vol. 111, No. 1, 1 July 1999 TABLE IV. Normalized Duschinsky matrices.

Q 83 Q 68 Q 822 8 Q 23

Q 48 Q 821

Q3

~a! Q6

Q 22

Q 23

0.5901 0.2793 0.5156 2.4710

0.3998 0.7633 2.5052 0.2516

0.7117 2.6562 2.3693 0.1111

2.1658 2.0132 2.5340 2.8012

Q4

~b! Q 21

0.9348 2.3180

0.3358 0.9555

we can calculate Franck–Condon factors for the S 0 →S 1 (n- p * ) transitions. Since the electronic transition is one-photon forbidden, the Franck–Condon factors are directly related to the intensities of allowed two-photon excitations. Nevertheless, the Franck–Condon factors are also relevant to calculations of intensities in one-photon absorption or fluorescence spectra presented in Sec. VI. Franck–Condon factors computed for individual normal modes and for groups of mixed modes are collected in Table V. Normal modes Q 1 , Q 2 , Q 7 , and Q 8 have high Franck– Condon factors for the 0-0 transition ~0.91–93! and the factors for 0-1 are in the 0.07–0.09 range. Therefore, the activity of these modes in vibronic spectra should be low. For Q 5 , the 5 10 Franck–Condon factor is even lower, ;0.02. Due to the large displacements DQ, normal modes Q 20-Q 22 and

Q 24 exhibit relatively long series of Franck–Condon factors with nonnegligible values up to 0-3 or 0-4 transitions. For instance, for the high frequency mode Q 20 , the calculated Huang–Rhys factor is 2.4 and the largest Franck–Condon factors are found for the 0-1 ~0.2136!, 0-2 ~0.2591!, and 0-3 ~0.2100! transitions, 2986, 5972, and 8958 cm21 from the origin, respectively. Notice that for this mode the S factor and the Franck–Condon factors are very sensitive to small geometric changes and the Franck–Condon factors for higher vibronic transitions ~0-2, etc.! might be somewhat overestimated. For Q 3 and Q 4 , only 0-1 transitions besides 0-0 can be significant. Q 6 and Q 23 which are mixed with Q 3 and Q 22 have small Franck–Condon factors for nonzero transitions and would not be active in an electronically allowed spectrum. Normal mode Q 12 of a 2 symmetry is not displaced, but the large distortion from 22 to 170 cm21 results in a long series of Franck–Condon factors; even the 1210 0 value is higher than 0.01. Since DQ 1250, only even transitions are allowed. For this mode the calculated frequency significantly deviates from the experimental value of 77 cm21.40 Using the latter for the ground state frequency and 8 for the excited state, we obtain a much the theoretical n 12 shorter series of Franck–Condon factors: 1200 50.9264, 1220 50.0657. In any case, the 1220 transition should play some role in the spectra. Summarizing, the following normal modes would determine the shape of an allowed two-photon vibronic spectrum of acetone corresponding to the n- p * electronic transition: Q 20 (CH3 stretch coupled with CO outof-plane bend, S 1 frequency is 2986 cm21!, Q 21 (CH3 defor-

TABLE V. Calculated Franck–Condon factors for individual normal modes ~groups of mixed modes! in the n- p * electronic transition of ~CH3!2CO. Normal mode Q1 Q2 Q 3 ,Q 6 ,Q 22 ,Q 23

Q 4 ,Q 21

Q5 Q7 Q8 Q 12

Q 20 Q 24

209

Franck–Condon Factors 1 00 50.9181, 1 10 50.0784, 1 20 50.0033 2 00 50.9089 2 10 50.0869, 2 20 50.0041 02050.2994, 2320 50.0144, 2210 50.2305,2210 2310 50.0219, 2210 2320 50.0099, 2210 2330 50.0032, 2220 50.0857,2220 2310 50.0302, 2220 2330 50.0205, 2230 50.0857, 2230 2310 50.0160, 2230 2320 50.0005, 2240 50.0035, 2240 2310 50.0051, 6 10 50.0149, 6 10 2210 50.0084, 6 10 2210 2310 50.0025, 6 10 2220 2310 50.0019, 6 20 50.0016, 3 10 50.0677, 3 10 2310 50.0072, 3 10 2320 50.0029, 3 10 2330 50.0010, 3 10 2210 50.0378, 3 10 2210 2310 50.0185, 3 10 2210 2330 50.0025, 3 10 2220 50.0095, 3 10 2220 2310 50.0122, 3 10 2230 50.0014, 3 10 2230 2310 50.0038, 3 10 2230 2320 50.0015, 3 10 6 10 2210 50.0005, 3 10 6 10 2210 2310 50.0010, 3 20 50.0095, 3 20 2310 50.0030, 3 20 2210 50.0039, 3 20 2210 2310 50.0041, 3 20 2220 2310 50.0017 02050.3873, 2110 50.2724, 2120 50.1044, 2130 50.0288, 2140 50.0064, 2150 50.0012, 4 10 50.0877, 4 10 2110 50.0587, 4 10 2120 50.0215, 4 10 2130 50.0057, 4 10 2140 50.0012, 4 10 2150 50.0002, 4 20 50.0114, 4 20 2110 50.0073, 4 20 2130 50.0006, 4 30 50.0011, 4 30 2110 50.0007, 4 30 2120 50.0002 5 00 50.9833, 5 10 50.0166 7 00 50.9276, 7 10 50.0697, 7 20 50.0026 8 00 50.9300, 8 10 50.0698, 8 30 50.0002 1200 50.6370, 1220 50.1893, 1240 50.0843, 1260 50.0418, 1280 50.0217, 12 14 16 1210 0 50.0116, 120 50.0063, 120 50.0035, 120 50.0019, 18 20 120 50.0011, 120 50.0006 2000 50.0883, 2010 50.2136, 2020 50.2591, 2030 50.2100, 2040 50.1280, 2050 50.0626, 2060 50.0256, 2070 50.0090 2400 50.6034, 2410 50.2373, 2420 50.1032, 2430 50.0370, 2440 50.0129, 2450 50.0042, 2460 50.0013, 2470 50.0004

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J. Chem. Phys., Vol. 111, No. 1, 1 July 1999 TABLE VI. Vibronic couplinga between different electronic states of acetone. 1

A 1 -2 1 A 2

2 1A 1- 1A 2

23.0631023 Q 9 5.0831023 Q 10 8.0931023 Q 11 27.3031023 Q 12

20.1072 Q 9 20.0138 Q 10 5.6131024 Q 11 3.9031024 Q 12

1

A 1- 1B 1

3.5831023 Q 20 26.4331023 Q 21 29.1331023 Q 22 20.0319 Q 23 20.0107 Q 24

a

1

A 1- 1B 2

22.3331023 Q 13 20.0610 Q 14 0.0100 Q 15 0.0572 Q 16 20.1363 Q 17 20.0501 Q 18 0.1026 Q 19

1

B 2- 1A 2

0.2570 Q 20 20.0265 Q 21 0.2536 Q 22 20.2229 Q 23 0.0721 Q 24 1

A 1 -2 1 B 2

23.4331024 Q 13 0.0345 Q 14 0.0132 Q 15 0.0166 Q 16 20.1091 Q 17 29.0131023 Q 18 0.1165 Q 19

2 1B 2- 1A 2 20.1131 Q 20 25.3931023 Q 21 20.0250 Q 22 0.0808 Q 23 0.0215 Q 24 1

B 1- 1A 2

0.0193 Q 13 0.0664 Q 14 20.0159 Q 15 0.0221 Q 16 0.0277 Q 17 20.0147 Q 18 20.0908 Q 19

Experimental energy differences E(F a )2E(F b ) shown in Table I are used for calculations of vibronic couplings.

mation, 1500 cm21!, Q 22 (CH3 rock coupled with CO out-of plane bend, 942 cm21!, Q 24 (CH3 torsion, 196 cm21!, Q 12 ~CH3 torsion, 170325340 cm21!, and, in a less extent, Q 3 ~CO stretch, 1226 cm21! and Q 4 (CH3 symmetric deformation, 1481 cm21!, for which the 0-1 vibronic transition could be seen. In a two-photon allowed spectrum, the intensity of each peak should be proportional to the product of individual Franck–Condon factors.33 For example, for the spectral origin we obtain the total Franck–Condon factor of 2.79 31023 compared with 6.7431023 , 8.1731023 , and 6.63 31023 for 2010 , 2020 , and 2030 , respectively. Since the number of active modes is large, the vibronic peaks with Franck– Condon factors of similar orders of magnitude are numerous and they densely cover the energy range from the origin to ;12 000 cm21. In experiment,1 the absorption spectrum of acetone is a diffuse band extending from 250 to 340 nm ~the width is 10 588 cm21! with a broad maximum at 290 nm ~4000–5000 cm21 from the origin! and the electron-impact energy loss spectrum2 has a similar structure. We shall return to the direct comparison with experiment after consideration of the intensity-borrowing mechanism for this symmetryforbidden transition. V. VIBRONIC COUPLING

The intensity for the n- p * ( 1 A 1 - 1 A 2 ) transition can be borrowed from the n-3s, n-3p, and n-3d Rydberg transitions to the electronic states of 1 B 2 , 1 A 1 , and 1 B 1 symmetries. As seen in Table I, state-average CASSCF~6,13! and CASSCF~6,14!/6-3111G** calculations including seven electronic states reproduce experimental vertical excitation energies13,41,42 of these states fairly well. Therefore, the transition matrix elements over the one-electron electric field operator at the atomic centers were computed at these levels of theory. The vibronic couplings between various states are shown in Table VI. The intensity for the symmetry-forbidden 1 A 1 → 1 A 2 transition can be borrowed from the 1 A 1 → 1 B 2 (n-3s) and 1 A 1 →2 1 B 2 (n-3p z ) transitions due to vibronic coupling be-

tween 1 A 2 and 1 B 2 or 1 A 2 and 2 1 B 2 , respectively. As seen in Table VI, VC( 1 B 12 ,A 2 ) is large and reaches 60.22-0.26Q i due to normal modes Q 20 , Q 22 , and Q 23 of b 1 symmetry. VC(2 1 B 2 , 1 A 2 ) is substantially smaller with the highest value of 0.1131Q 20 . Hence, we did not include into our consideration higher states of 1 B 2 symmetry, 3 1 B 2 (n-3d x 2 2y 2 ) and 4 1 B 2 (n-3d z 2 ). The 1 B 2 and 2 1 B 2 states are also vibronically coupled with the ground 1 A 1 state, although in a less extent. The 1 A 1 - 1 B 2 /2 1 B 2 vibronic couplings generate the intensity-borrowing from the allowed 1 A 2 → 1 B 2 and 1 A 2 →2 1 B 2 transitions. In this case, the inducing modes are of b 2 symmetry. Normal modes Q 17 ~CC asymmetric stretch! and Q 19 ~CO in-plane bend! give rise to the largest vibronic coupling of 1 A 1 with 1 B 2 and 2 1 B 2 . However, the major source of the intensity-borrowing with the b 2 inducing modes is vibronic coupling of 1 A 2 with 1 B 1 (n-3d xy ). The 1 A 1 → 1 B 1 transition has a significant dipole moment ~0.39 a.u.! and the CO in-plane bend Q 19 with VC( 1 B 1 , 1 A 2 ) 520.09Q 19 makes a sizable contribution into the 1 A 1 - 1 A 2 transition dipole moment. Normal modes of a 2 symmetry participate in D( 1 A 1 , 1 A 2 ) through two mechanisms. The intensity is borrowed from the 1 A 1 →2 1 A 1 (n-3p y ) transition because of vibronic coupling of 1 A 2 with 2 1 A 1 and from the 1 A 2 →2 1 A 2 (n-3p x ) transition due to vibronic coupling of 1 A 1 with 2 1 A 2 . In both cases, VC(2 1 A 1 , 1 A 2 ) and VC( 1 A 1 ,2 1 A 2 ) are rather small. The largest contribution to VC(2 1 A 1 , 1 A 2 ) comes from Q 9 (20.1072Q 9 ). Again, in this consideration we neglect the vibronic couplings with higher 3 1 A 1 (n-3d yz ) and 3 1 A 2 (n-3d xz ) states assuming that they should be smaller than VC(2 1 A 1 , 1 A 2 ) and VC( 1 A 1 ,2 1 A 2 ). Overall, the transition dipole moment D( 1 A 1 , 1 A 2 ) is calculated as follows: ~1! for normal modes of b 1 symmetry (Q 20-Q 24): D~ 1 A 1 , 1 A 2 ! 5VC~ 1 A 1 , 1 B 1 ! D~ 1 B 1 , 1 A 2 ! 1D~ 1 A 1 , 1 B 2 ! VC~ 1 B 2 , 1 A 2 ! 1D~ 1 A 1 ,2 1 B 2 ! VC~ 2 1 B 2 , 1 A 2 ! ;

~8!



211

TABLE VII. Calculated transition dipole moments ~a.u.! for symmetry-allowed transitions ~a! and for the 1 A 1 - 1 A 2 transition as a function of normal coordinates ~b!. ~a! Allowed transition 1

A 1 -2 1 A 1 1 A 1- 1B 2 1 B 1 -2 1 A 2 1 A 1- 1B 1 2 1B 2- 1A 2

Dipolea moment

Allowed transition

Dipolea moment

0.1163k 0.5478i 23.1054i 0.3935j 20.1248j

2 1A 2- 1A 2 2 1A 1- 1B 2 1 A 1 -2 1 B 2 2 1A 1- 1B 1 1 B 2 -2 1 A 2

20.0872k 3.1590i 0.0634i 2.7327j 23.4649j

1

Function of Q i 22

21.22310 Q 9 k 6.8231024 Q 12k 26.9731023 Q 15i 29.4631023 Q 18i 21.6231022 Q 21j 3.8531022 Q 24j a

Mean-square

~b! A 1 - 1 A 2 dipole moment function of Q i

24

2.48310 9.4931024 2.9931024 6.7031024 3.3331023 2.5931022

23

22.04310 Q 10k 7.4031023 Q 13i 1.2131022 Q 16i 24.0531022 Q 19i 1.3531021 Q 22j

Allowed transition 2 1B 2- 1B 2 1 B 1- 1A 2 2 1 A 1 -2 1 B 2 1 B 2- 1A 2 2 1 B 2 -2 1 A 2

Mean-square 25

8.72310 1.4831024 5.4631024 4.6231023 3.2031022

Function of Q i 24

26.40310 Q 11 k 23.6231022 Q 14i 1.1531022 Q 17i 1.3431021 Q 20j 21.2431021 Q 23j

Dipolea moment 23.3526k 0.2185i 20.3411i 0.0953j 20.6609j

Mean-square 4.5331025 7.5031024 5.8531024 1.9131022 4.4331022

i, j, k are unit vectors in the x, y, and z directions, respectively.

VI. VIBRONIC SPECTRUM FOR THE n - p * TRANSITION

~2! for normal modes of b 2 symmetry (Q 13-Q 19): D~ 1 A 1 , 1 A 2 ! 5D~ 1 A 1 , 1 B 1 ! VC~ 1 B 1 , 1 A 2 ! 1VC~ 1 A 1 , 1 B 2 ! D~ 1 B 2 , 1 A 2 ! 1VC~ 1 A 1 ,2 1 B 2 ! D~ 2 1 B 2 , 1 A 2 ! ; ~3! for normal modes of a 2 symmetry (Q 9 -Q 12): D~ 1 A 1 , 1 A 2 ! 5VC~ 1 A 1 ,2 1 A 2 ! D~ 2 1 A 2 , 1 A 2 !

~9!

1D~ 1 A 1 ,2 1 A 1 ! VC~ 2 1 A 1 , 1 A 2 ! .

~10!

Using the above equations with vibronic couplings from Table VI and transition moments for the allowed transitions from Table VIIa we obtain D( 1 A 1 , 1 A 2 ) shown in Table VIIb. Thus in the 1 A 1 → 1 A 2 transition dipole moment the largest contributions come from the modes of b 1 symmetry, mostly due to the 1 A 2 → 1 B 2 vibronic coupling. Contribution from the b 2 modes through the 1 A 2 → 1 B 1 vibronic coupling are less significant and those from the a 2 modes are negligibly small. It should be noted that vibronic couplings of the ground state with 1 B 2 , 2 1 B 2 , 2 1 A 2 , and 1 B 1 do not play and important role in the calculation of D( 1 A 1 , 1 A 2 ). Table VIIb also shows the mean-square electronic transition moments due to the vibronic coupling induced by each normal mode computed as28,37

u D~ 1 A 1 , 1 A 2 ! Q i u A^ Q 2i & 0 5 u D~ 1 A 1 , 1 A 2 ! Q i u

A

\ . 2vi

~11!

The largest values are obtained for the following b 1 modes: the CO out-of-plane bend Q 23(0.044 a.u.), CH3 rock Q 22(0.032 a.u.), and CH3 torsion Q 24(0.026 a.u.). Among normal modes of b 2 and a 2 symmetry, the largest meansquare transition moment is found for the CO in-plane bend Q 19(4.631023 a.u.).

Now we are in position to consider the vibronic spectrum for the n- p * transition of acetone. First, we compute the vibronic overlap integrals I 0n 8 for 02n 8 transitions of inducing normal modes. Table VIII shows the u Di u 2 u I 0n 8 u 2 values as well as the oscillator strengths f @ Q i (02n 8 ) # related to each 02n 8 transitions computed as 2 f @ Q i ~ 02n 8 !# 5 DE Q i ~ 02n 8 ! u Di u 2 u I 0n 8 ~ Q i ! u 2 . 3

~12!

DE Q i (02n 8 ) here is the difference between the energy of acetone in the ground electronic state and zero vibrational state for the mode Q i and the energy of the excited state in vibrational state n 8 . Multiplied by Franck–Condon factors for other normal modes j ( jÞi) @see Eq. ~5!#, these oscillator strengths give approximate intensities of individual vibronic peaks in the symmetry-forbidden spectra. Summing up the oscillator strengths in Table VIII, we obtain a lower estimate for the total oscillator strength of the n- p * transition in (CH3!2CO as 3.6231024 . This result is in excellent agreement with the experimental value of 4.14 31024 . 3 Earlier38 we found that in the acetone-water complexes where the n- p * transition becomes allowed due to the break of C 2 v symmetry, its oscillator strength increases by 7.131025 . Thus the present calculation confirms the conclusion38 that the effect of the vibronic coupling on the n- p * spectral intensity in acetone is stronger than that of the coordination of water molecules. Ninety-four percent of the total oscillator strength are originated from the b 1 inducing modes. This corresponds to the perpendicular transition polarized in y direction. The b 1 normal modes involving the out-of-plane CO wagging have also been thought to be active in the intensity-borrowing mechanism in formaldehyde and other alkyl-ketones.18–24 Baba and Hanazaki4 who measured

212

Liao et al.


TABLE VIII. Vibronic overlap integrals and oscillator strengths corresponding to particular vibronic transitions of the promoting normal modes. n8

Q9 u D i u 2 u I 0n 8 u 2 26

1 3 5

3.033 10 1.31310210 4.74310215

f @ Q i (02n 8 ) # 27

3.053 10 1.56310211 6.49310216

n8 1 3 5

Q 10 u D i u 2 u I 0n 8 u 2 27

1.783 10 1.70310210 1.35310213

Q 12 n8

u D i u 2 u I 0n 8 u 2 27

1 3 5

3.293 10 2.973 1027 2.183 1027

n8

u D u u I 0n 8 u

2

26

1 3 5

2.083 10 1.653 1029 1.08310212

f @ Q i (02n 8 ) # 28

n8

u D i u 2 u I 0n 8 u 2 26

3.033 10 2.733 1028 2.053 1028

1 3 5

1.093 10 4.04310213 1.25310219

u D i u 2 u I 0n 8 u 2 26

1 3 5

6.323 10 3.163 1028 1.32310210

f @ Q i (02n 8 ) #

n8

u D u u I 0n 8 u

27

2.003 10 1.73310210 1.23310213

1 3 5

n8

u D u u I 0n 8 u

0 1 2 3

7.883 1025 6.593 1025 7.243 1026 1.023 1025

2

2

26

6.603 10 5.963 1029 4.48310212

1 3 5

28

2.893 10 1.72310210 8.52310213

f @ Q i (02n 8 ) # 2.733 1029 1.72310211 9.04310214

Q 14 f @ Q i (02n 8 ) # 27

n8

u D i u 2 u I 0n 8 u 2 25

1.103 10 4.81310214 1.71310220

1 3 5

2.723 10 4.79310210 7.04310215

f @ Q i (02n 8 ) #

n8

u D u u I 0n 8 u 2

f @ Q i (02n 8 ) # 2.733 1026 5.66310211 9.57310216

Q 17

27

6.323 10 6.21310210 5.06310213

i 2

26

f @ Q i (02n 8 ) # 6.393 1027 9.92310211 1.28310214

1 3 5

6.723 10 9.72310210 1.17310213

f @ Q i (02n 8 ) #

n8

u D u u I 0n 8 u 2

f @ Q i (02n 8 ) #

3.653 1027 3.963 1028 3.003 1027 2.503 1027

0 1 2 3

3.473 1024 1.103 1025 2.933 1024 2.663 1024

3.173 1025 1.043 1026 2.853 1025 2.663 1025

Q 19 f @ Q i (02n 8 ) # 27

n8

u D i u 2 u I 0n 8 u 2 24

5.963 10 3.173 1029 1.40310211

1 3 5

1.833 10 8.603 1026 3.383 1027

f @ Q i (02n 8 ) #

n8

u D u u I 0n 8 u

7.213 1026 6.633 1026 7.943 1027 1.213 1026

0 1 2 3

3.983 1026 4.123 1027 2.983 1026 2.373 1026

Q 20 i 2

1.713 10 1.79310211 1.54310214

Q 11 u D i u 2 u I 0n 8 u 2

Q 16 i 2

Q 18 n8

28

n8

Q 13

Q 15 i 2

f @ Q i (02n 8 ) #

f @ Q i (02n 8 ) # 1.693 1025 8.153 1027 3.273 1028

Q 21 i 2

Q 23

2

Q 22 i 2

Q 24

n8

u D i u 2 u I 0n 8 u 2

f @ Q i (02n 8 ) #

n8

u D i u 2 u I 0n 8 u 2

f @ Q i (02n 8 ) #

0 1 2 3

1.073 1024 1.603 1023 2.563 1024 5.703 1027

9.743 1026 1.483 1024 2.413 1025 5.423 1028

0 1 2 3

2.353 1024 1.033 1024 1.253 1024 1.073 1024

2.153 1025 9.493 1026 1.163 1025 1.003 1025

fluorescence excitation spectra of the 1 A 2 state of acetone, mentioned an appreciable contribution from the parallel ~xpolarized! transition. Our calculations show that the parallel component contribution amounts to ;6% of the total oscillator strength and is mostly due to the b 2 normal mode Q 19 , in-plane CO bend. Next, we compute the intensities of individual vibronic peaks. Table IX demonstrates the results for vibronic transitions where only inducing modes are excited. For each transition, we separated the intensity into two parts: due to selfpromotion and due to promotion by other active modes. The self-promoted intensities are calculated as follows: f self@ Q i ~ 02n 8 !# 2 5 DE Q i ~ 02n 8 ! u Di u 2 u I 0n 8 ~ Q i ! u 2 FC Q j ~ 0-0 ! . 3 jÞi

)

~13!

Summation of f self@ Q i (0-0) # for the displaced promoting modes Q 20-Q 24 of b 1 symmetry gives the intensity of 5.45

31027 for the spectral origin. The intensity due to promotion by other modes is related to the intensity of the origin by the corresponding Franck–Condon factors: f others@ Q i ~ 02n 8 !# 5

FC Q i ~ 02n 8 ! FC Q i ~ 0-0 !

f self@ Q j ~ 0-0 !# . ( jÞi ~14!

Therefore, the larger the Franck–Condon factor for Q i (0 2n 8 ), the larger the intensity due to the promotion by other modes. Let us consider, for example the 2010 , 2020 , and 2310 transitions. The other modes promotion intensity of 2010 is lower than that of 2020 because of the lower Franck–Condon factor ~Table V!. However, the self-promotion intensity of 2010 is about an order of magnitude higher than that of 2020 . As a result, the overall intensity of the 2010 peak, 9.78 31027 , is slightly higher than that for 2020 . For 2310 , the other modes promotion intensity is very low owing to the



213

TABLE IX. Intensities of vibronic peaks related to the promoting normal modes. Intensity Mode

Excitation

Other mode promotion

Self-promotion

0-0 0-1 0-2 0-3

3.18310 7.6931027 9.3331027 7.5631027

2.27310 2.0931027 2.5031028 3.8231028

5.4531027 9.7831027 9.5831027 7.9431027

Q 21

0-0 0-1 0-2 0-3

5.4331027 3.8231027 1.4631027 4.0431028

2.103 1029 2.28310210 1.733 1029 1.443 1029

5.4531027 3.8231027 1.483 1027 4.183 1028

Q 22

0-0 0-1 0-2 0-3

3.5731027 2.7531027 1.0231027 2.4431028

1.883 6.193 1.703 1.583

1027 1029 1027 1027

5.453 1027 2.813 1027 2.723 1027 1.823 1027

Q 23

0-0 0-1 0-2 0-3

5.1631027 5.50310210 2.4831028 7.98310211

2.863 1028 4.343 1027 7.073 1028 1.59310210

5.453 1027 4.353 1027 9.553 1028 2.39310210

0-0 0-1 0-2 0-3

4.4631027 1.7531027 7.6231028 2.7331028

9.923 4.383 5.353 4.623

1028 1028 1028 1028

5.453 1027 2.193 1027 1.303 1027 7.353 1028

0-1 0-1 0-1 0-1 0-1 0-1 0-1 0-1 0-1 0-1 0-1 0-3

0 0 0 0 0 0 0 0 0 0 0 0

8.50310210 4.76310211 7.61310212 1.33310210 3.06310210 7.603 1029 5.57310210 1.763 1029 1.783 1029 1.663 1029 4.713 1028 2.273 1029

8.50310210 4.76310211 7.61310212 1.33310210 3.06310210 7.6031029 5.57310210 1.763 1029 1.783 1029 1.663 1029 4.713 1028 2.273 1029

Q9 Q 10 Q 11 Q 12 Q 13 Q 14 Q 15 Q 16 Q 17 Q 18 Q 19

27

Total

Q 20

Q 24

27

FIG. 2. Calculated vibronic spectrum of acetone corresponding to the n- p * electronic transition ~zero on the energy scale corresponds to the spectral origin!. Only most intense peaks are shown.

the low energy bands. This could be due to unharmonic behavior of the low frequency modes corresponding to CH3 torsion. According to our results, the most distinct vibronic peaks are the following: origin ~0 cm21, intensity 5.4531027 ), 2310 ~ 395, 4.3531027 ! ,

2210 ~ 942, 2.8131027 ! ,

2210 2310 ~ 1337, 3.5831027 ! , 2220 ~ 1884, 2.7231027 ! ,

2110 2310 ~ 1895, 3.0631027 ! ,

2010 ~ 2986, 9.7831027 ! ,

2010 2310 ~ 3381, 1.0531026 ! ,

2010 2210 2310 ~ 4323, 8.4131027 ! , 2010 2220 ~ 4870, 5.6131027 ! ,

2010 2110 ~ 4486, 6.8531027 ! ,

2020 ~ 5972, 9.5831027 ! ,

2020 2310 ~ 6367, 1.2731026 ! , 2020 2210 2310 ~ 7309, 1.0031026 ! , 2020 2110 ~ 7472, 6.7031027 ! ,

very small Franck–Condon factor for this transition. But the self-promotion is significant and the total intensity amounts to 4.3431027 . Normal modes of b 2 and a 2 symmetry are not displaced in the n- p * transition and they cannot contribute to the intensity of the origin. They can have nonzero self-promoted intensities for the odd transitions 0-1, 0-3, etc. Of those, only intensity of the 1910 peak is considerable, 4.7131028 . On the other hand, the distorted mode Q 12 has nonzero Franck– Condon factors for a series of even transitions and such transitions are promoted by the other modes. Self-promotion for the odd transition 1210 is much smaller than the intensity of the 1220 peak promoted by the Q 20-Q 24 active modes. The data from Table IX are used to compute the intensities of other significant vibronic peaks. The calculated vibronic spectrum of acetone is presented in Fig. 2. In Table X we suggest an assignment of various bands resolved by Baba et al. in the fluorescence excitation spectra.4,5 As seen in this table, most of experimental bands can be assigned using our theoretical frequencies for the 1 A 9 electronic state. There are some deviations of theory from experiment, especially, for

2110 ~ 1500, 3.8331027 ! ,

2030 ~ 8958, 7.9431027 ! ,

2030 2310 ~ 9353, 1.0331026 ! , 2030 2210 2310 ~ 10295 ,8.1531027 ! , 2030 2110 ~ 10458, 5.5631027 ! , etc. The normal modes active in the spectra are Q 20-Q 24 of b 1 symmetry. Interestingly, modes Q 20 , Q 21 , Q 22 , and Q 24 are active mostly due to the large Franck–Condon factors, but mode Q 23 ~pure CO out-of-plane bend! is active because of its promoting ability. In these calculations the peaks with highest intensities are related to the 2020 excitation and located at about 6000 cm21 from the spectral origin. In different experiments, however, the maximum intensity of the diffuse band corresponding to the n- p * transition is found at 290 nm ~;4000 cm21 from the origin! or at 283 nm ~;5000 cm21 from the origin!.1,2 The discrepancy could be due to an overestimation of displacement DQ 20 and the corresponding Huang–Rhys factor for this high frequency mode. If DQ 20 is somewhat smaller, the Franck–Condon factor for 2010 would be larger than that for 2020 . In this case, the highest peaks would be located at 2986 cm21(2010 ), 3381 cm21(2010 2310 ), and

214

Liao et al.


TABLE X. Positions, Franck–Condon factors, and intensities of vibronic peaks for the symmetry-forbidded n- p * electronic transition in ~CH3!2CO in comparison with positions of experimental bands observed in the fluorescence excitation spectra. Peak exptb

Energya theor

0

1.783 1029

170

1210

0

1.33310210

1163

7 10 2310

2.2231027

3.263 1028

196

2410 8 10 1220

27

1182

2320 2420

2.3031025

1.813 1028

320 340 360

395

544 578 624 645 681 697 747

779 791 832 863 886 902 933

510 536 588 592

1034 1073 1099

680 706 732 768 787 790 856 876

a

1230 1220 2410 2430 2310 2410

1240 1230 2410 1220 2420 1220 2310 7 10 2310 2420 2320 8 10 1220 2410 1240 2410

902

1230 2420

928

1220 2430 2210 1810 7 10 2410 2320 2410 1260 1240 2420 7 10 1220 6 10 1220 2320 2210 2410 7 10 2420

952 986 1020 1072 1108 1130 1138

1180

2.09310

24

8.28310 0

2.193 10

4.76310

26

2.97310

1216

1260 2410

7.2231025

1.443 1028

27

1243

1226

3 10

6.3031024

1.233 1027

1263

1262

8 10 2210 1240 2310 2410 1220 2210 1230 2320 7 10 1220 2410 6 10 2410 1220 2320 2410 2210 2420 2210 2310 7 10 2310 2410 1280 2320 2430 1610 5 10 1260 2420 1260 2310 3 10 2410 7 10 1240 6 10 1220 8 10 2210 2410 1240 2310 2420 1240 2320 1220 2210 2410 4 10 2110 7 10 1220 2310 6 10 2420 6 10 2310 1220 2320 2420 2210 2430 2210 2310 2410 3 10 8 10 1280 2410 3 10 1220

1.6131025

2.153 1028

27

2.263 1028

24

6.36310

8.533 1028

0

5.72310212

1.623 10

5.21310 24

1211

4.093 10

4.713 10

0

28

28

1271

211 27

1.303 10

1282 1294

27

1300 1304

4.35310

0

1.19310 24

3.27310

24

1.71310

26

1.17310

210

1349

28

1326 1334

6.513 10

28

1337

7.353 10

27

1359

1.713 10

24

3.69310 0

28

1.33310

8.8331027 2.0931024 5.0831027 1.3431024 2.4531025 1.4531024

1399

7.213 10 4.68310

24

211 28

3.863 10

1408 1429

1.293 1027

1412 1415

4.103 1028 7.443 1028

1378

1422 1456

9.553 1028

1448 1453

4.893 1029

1458

2.903 1028

1467 1470

1113 1124

24

1386

964 982

1.10310

1360

942 955

1910 1210 2410 2420 2310

23

1309

735 757

Intensity

1710

392

506

FC factor

1160

366

473 487

Energya Assignment theor

5.453 1027

172

345 373

Peak exptb

2.7931023

0

341

Intensity

origin

0

333

FC factor

Assignment

1160

0 25

5.08310

23

2.14310 0

2.04310211

1478

28

1481

2.183 10

27

2.81310

1.663 10 25

8.27310

25

5.29310

24

1.83310

25

6.30310

25

6.22310

24

1.39310

25

3.98310

24

8.42310

25

3.58310

1505

1500

29

1503

28

1505

1.653 10

28

1508

5.603 10

28

3.583 10

28

1.723 10

28

1.223 10

28

2.713 10

28

2.843 10

27

1.143 10

1522

1522 1530 1533 1546 1556 1566

1.55310

25

4.893 1029

25

1.343 1028

25

1.663 1028

24

7.623 1028

24

3.583 1027

28

1.283 1028

25

1.863 1028

26

8.22310

7.793 1029

0

1.763 1029

2.46310 6.72310 1.57310 3.67310 2.04310 8.79310 9.49310

25

9.203 1029

25

8.533 1029

27

2.853 1028

24

4.953 1028

25

5.423 1029

25

8.063 1029

25

8.563 1029

28

9.853 1029

25

1.77310

1.263 1028

2.5031024

3.383 1028

6.3131024

4.093 1028

1.9631023

3.833 1027

6.6331028

6.423 1029

2.3731025

6.473 1029

27

2.163 1028

26

5.383 1029

24

4.813 1028

25

1.413 1027

25

9.2531029

25

7.463 1029

24

3.663 1028

4.70310 3.12310 1.94310 2.49310 2.77310 4.12310 6.32310 6.72310

2.23310 6.84310 1.32310 8.01310 4.73310 3.75310 1.87310

29

9.773 10

In cm21, from the spectral origin. From Ref. 5.

b

4323 cm21(2010 2210 2310 ) from the origin, i.e., in the region of the observed absorption maximum. The present results explain the broad diffuse character of the n- p * band in acetone and the difficulties in its resolution. Numerous vibronic peaks with weak intensities densely cover a broad energy region. VIII. CONCLUSIONS

In this study, we carried out ab initio calculations of geometry and vibrational frequencies of the first singlet ex-

cited state of acetone corresponding to the n- p * electronic transition. The major geometry changes in this state as compared to the ground state involve CO out-of-plane wagging, CO stretch, and torsion of the methyl groups. The molecular symmetry changes from C 2 v to C s . The vibrational frequenrelated to the CO bend decrease, while the frequencies for CH3 deformation, rocking, and torsional motions slightly increase as compared to those in the ground state. The most pronounced frequency changes in the 1 A 9 state are the decrease of the CO stretch frequency n 3 by almost 500 cm21



and the increase of the CH3 torsion frequency n 12 from 22 to 170 cm21. The optimized geometries and normal modes are used to compute the normal mode displacements which are applied for calculations of Franck–Condon factors. Normal modes involving CO out-of-plane bend exhibit the largest DQ. Transition matrix elements over the one-electron electric field operator at various atomic centers are used to compute vibronic couplings between the ground 1 A 1 , 1 A 2 (n- p * ), and Rydberg 1 B 2 (n-3s), 2 1 A 1 (n-3 p y ), 2 1 A 2 (n-3p x ), 2 1 B 2 (n-3p z ), and 1 B 1 (n-3d xy ) electronic states. Then, the Herzberg–Teller expansion of the electronic wave function is applied to derive the transition dipole moment for 1 A 1 → 1 A 2 as a function of normal coordinates. The results show that the intensity for this transition is mostly borrowed from the allowed 1 A 1 - 1 B 2 (n-3s) transition due to vibronic coupling between 1 A 2 and 1 B 2 through normal modes Q 20 , Q 22 , and Q 23 . Among modes of b 2 symmetry coupling 1 A 2 with 1 B 1 (n-3d xy ), Q 19 ~CO in-plane bend! is the most relevant for the intensity-borrowing from 1 A 1 → 1 B 1 . The calculated total oscillator strength for the n- p * transition through the intensity-borrowing mechanism is 3.6231024 , in close agreement with the experimental value of 4.14 31024 . Ninety-four percent of the oscillator strength comes from the perpendicular component (b 1 promoting modes! and 6% from the parallel component (b 2 modes!. The 1 A 1 → 1 A 2 transition dipole moments and Franck– Condon factors are applied to compute the intensities of various peaks in vibronic spectra. Calculated spectral origin, 30 115 cm21 at the MRCI level, underestimates the experimental value by about 300 cm21. The presence of numerous weak vibronic peaks densely covering a broad energy range ~;12 000 cm21! explains the experimental observations of the diffuse broad band for the n- p * transition in (CH3!2CO. Calculated positions of the most intense peaks in the spectra also reasonably agree with the experimental band maximum. Most of the bands observed in fluorescence excitation spectra can be assigned based on the computed spectrum. Normal modes Q 20-Q 24 of b 1 symmetry should be most active in the spectra. ACKNOWLEDGMENTS

This work was supported in part by the Chinese Petroleum Corporation and by the National Science Council of ROC. 1

W. A. Noyes, Jr., A. B. F. Dunacan, and W. M. Manning, J. Chem. Phys. 2, 717 ~1934!. K. N. Walzl, C. F. Koerting, and A. Kuppermann, J. Chem. Phys. 87, 3796 ~1987!. 3 E. E. Worden, Jr., Spectrochim. Acta 22, 21 ~1966!. 4 M. Baba and I. Hanazaki, Chem. Phys. Lett. 103, 93 ~1983!. 5 M. Baba, I. Hanazaki, and U. Nagashima, J. Chem. Phys. 82, 3938 ~1985!. 6 H. Zuckerman, B. Schmittz, and Y. Haas, J. Phys. Chem. 93, 4083 ~1989!. 7 G. A. Gaines, D. J. Donaldson, S. J. Strickler, and V. Vaida, J. Phys. Chem. 92, 2762 ~1988!. 8 D. J. Donaldson, G. A. Gaines, and V. Vaida, J. Phys. Chem. 92, 2766 ~1988!. 9 S. A. Buzza, E. M. Snyder, D. A. Card, D. E. Folmer, and A. W. Castleman, Jr., J. Chem. Phys. 105, 7425 ~1996!. 2

10

215

B. Hess, P. J. Bruna, R. J. Buenker, and S. D. Peyerimhoff, Chem. Phys. 18, 267 ~1976!. 11 V. Galasso, J. Chem. Phys. 92, 2495 ~1990!. 12 X. Xing, R. McDiarmid, J. G. Philis, and L. Goodman, J. Chem. Phys. 99, 7565 ~1993!. 13 M. Merchan, B. O. Roos, R. McDiardmid, and X. Xing, J. Chem. Phys. 104, 1791 ~1996!. 14 S. R. Gwaltney and R. J. Bartlett, Chem. Phys. Lett. 241, 26 ~1995!. 15 O. Setokuchi, M. Sato, S. Matuzava, Y. Simizu, J. Mol. Struct.: THEOCHEM 365, 29 ~1996!. 16 G. Herzberg, Molecular Spectra and Molecular Structure. III. Electronic Spectra and Electronic Structure of Polyatomic Molecules ~Van Nostrand Reinhold, New York, 1966!, p. 658. 17 G. Herzberg and E. Teller, Z. Phys. Chem. Abt. B 21, 410 ~1933!. 18 S. H. Lin, Proc. R. Soc. London, Ser. A 352, 57 ~1976!. 19 ~a! J. A. Pople and J. W. Sidman, J. Chem. Phys. 27, 1270 ~1955!; ~b! J. N. Murrel and J. A. Pople, Proc. R. Soc. London, Ser. A 69, 245 ~1956!. 20 J. M. F. van Dijk, M. J. H. Kemper, J. H. H. Kerp, and H. M. Buck, J. Chem. Phys. 69, 2453, 2462 ~1978!. 21 M. J. H. Kemper, J. M. F. van Dijk, and H. M. Buck, J. Chem. Phys. 70, 2854 ~1979!. 22 F. Pauzat, B. Levy, and Ph. Millie, Mol. Phys. 39, 375 ~1980!. 23 M. J. H. Kemper, L. Lemmens, and H. M. Buck, Chem. Phys. 57, 245 ~1981!. 24 T. Nakajima and S. Kato, J. Chem. Phys. 105, 5927 ~1996!. 25 ˚ gren, S. Knuts, B. F. Minaev, and P. Jørgensen, Chem. Phys. Y. Luo, H. A Lett. 209, 513 ~1993!. 26 ˚ gren, and O. Vahtras, Chem. Phys. 175, 245 B. F. Minaev, S. Knuts, H. A ~1993!. 27 ˚ gren, and B. F. Minaev, J. Mol. Struct.: THEOCHEM 311, S. Knuts, H. A 185 ~1994!. 28 ~a! A. M. Mebel, M. Hayashi, and S. H. Lin, Chem. Phys. Lett. 274, 281 ~1997!; ~b! M. Hayashi, A. M. Mebel, K. K. Liang, and S. H. Lin, J. Chem. Phys. 108, 2044 ~1998!. 29 ~a! A. D. Becke, J. Chem. Phys. 98, 5648 ~1993!; ~b! C. Lee, W. Yang, and R. G. Parr, Phys. Rev. B 37, 785 ~1988!. 30 ~a! H.-J. Werner and P. J. Knowles, J. Chem. Phys. 82, 5053 ~1985!; ~b! P. J. Knowles and H.-J. Werner, Chem. Phys. Lett. 115, 259 ~1985!. 31 A. P. Scott and L. Radom, J. Phys. Chem. 100, 16502 ~1996!. 32 ~a! H.-J. Werner and P. J. Knowles, J. Chem. Phys. 89, 5803 ~1988!; ~b! P. J. Knowles and H.-J. Werner, Chem. Phys. Lett. 145, 514 ~1988!. 33 ~a! A. M. Mebel, Y.-T. Chen, and S. H. Lin, Chem. Phys. Lett. 258, 53 ~1996!; ~b! A. M. Mebel, Y.-T. Chen, and S. H. Lin, J. Chem. Phys. 105, 9007 ~1996!. 34 M. J. Frisch, G. W. Trucks, H. B. Schlegel, P. M. W. Gill, B. G. Johnson, M. A. Robb, J. R. Cheeseman, T. Keith, G. A. Petersson, J. A. Montgomery, K. Raghavachari, M. A. Al-Laham, V. G. Zakrzewski, J. V. Ortiz, J. B. Foresman, J. Cioslowski, B. B. Stefanov, A. Nanayakkara, M. Challacombe, C. Y. Peng, P. Y. Ayala, W. Chen, M. W. Wong, J. L. Andres, E. S. Replogle, R. Gomperts, R. L. Martin, D. J. Fox, J. S. Binkley, D. J. Defrees, J. Baker, J. P. Stewart, M. HeadGordon, C. Gonzalez, and J. A. Pople, GAUSSIAN 94, Revision E.2 ~Gaussian, Inc., Pittsburgh, PA, 1995!. 35 MOLPRO is a package of ab initio programs written by H.-J. Werner and P. J. Knowles, with contributions from J. Almlo¨f, R. D. Amos, M. J. O. Deegan, S. T. Elbert, C. Hampel, W. Meyer, K. Peterson, R. Pitzer, A. J. Stone, P. R. Taylor, and R. Lindh. 36 ˚ gren, T. Helgaker, H. J. Aa. Jensen, P. Jørgensen, J. Olsen, K. Ruud, H. A T. Andersen, K. L. Bak, V. Bakken, O. Christiansen, P. Dahle, E. K. Dalskov, T. Enevoldsen, H. Heiberg, H. Hettema, D. Jonsson, S. Kirpekar, R. Kobayashi, H. Koch, K. V. Mikkelsen, P. Norman, M. J. Packer, T. Saue, P. R. Taylor, and O. Vahtras, Dalton, an ab initio electronic structure program, Release 1.0 ~1997!. 37 A. M. Mebel, M. Hayashi, and S. H. Lin, in Trends in Physical Chemistry, Vol. 6 ~Research Trends, India, 1997!, p. 315. 38 D.-W. Liao, A. M. Mebel, Y.-T. Chen, and S. H. Lin, J. Phys. Chem. 101, 9925 ~1997!. 39 T. Shimanouchi, Tables of Molecular Vibrational Frequencies ~U.S. National Bureau of Standards, Washington, 1972!, NSRDS-NBS 39. 40 J. G. Philis, J. M. Berman, and L. Goodman, Chem. Phys. Lett. 167, 16 ~1990!. 41 J. G. Philis and L. Goodman, J. Chem. Phys. 98, 3795 ~1993!. 42 R. McDiarmid and A. Sabljic, J. Chem. Phys. 89, 6086 ~1988!.