Application of the density matrix method to

Application of the density matrix method to multlphoton ionization of molecules A. Boeglin, B. Fain,8) and S. H. Lin Department 0/Chemistry. Arizona State University. Tempe. Arizona 85287 (Received 3 October 1985; accepted 15 January 1986) In this paper we shall apply the density matrix method to study the one-photon ionization and two-photon ionization of molecules. Fano's results for one-photon autoionization will be reproduced by this formalism. It will be shown that the existence of autoionization states will interfere with the determination of ionization thresholds of molecules by using the photoionization method if the autoionization state is located near the ionization threshold. It will also be shown that the density matrix method can treat two-color photoionization spectroscopy and the excited state lifetime measurement by multiphoton ionization method.

I. INTRODUCTION Recent investigations have demonstrated that two-color resonance-enhanced photoionization spectroscopy is a very sensitive method to probe the higher excited states as well as the low-lying ionic states of polyatomic molecules. l -6 In these experiments light at one frequency WI prepares the molecular system into a specific excited electronic state and light at a second frequency W2 ionizes the excited-state species. By tuning WI' ions are produced only when WI comes into resonance with an excited level. An important extension in two-color photoionization studies is to hold WI fixed to a vibronic level in a given excited electronic state, and then tunew2 through the ionization continuum to yield photoionization efficiency curves for the excited states.4-8 In the latter experiments, the intermediate excited state can have (especially for the case of Rydberg states) greatly enhanced Franck-Condon factors with the lower ionic states compared to the ground state. This leads to much more intense and sharper initial thresholds compared to one-photon ionization techniques. This approach, using two-color laser photoionization spectroscopy with two tunable lasers, can overcome many of the problems inherent with conventional techniques of ionization spectroscopy.9-13 Another type of experiment that involves multiphoton ionization (MPl) of molecules is the determination of the excited state lifetime of molecules. This is accomplished by varying the delay betwe.en the arrival of WI and W2' Thus by using the MPI t~hntque, the measurements ofionization potentials, vibratIOnal frequencies of ions, Franck-Condon factors between the parent molecule and its ion and energy levels, and lifetimes of the highly excited states of the neutral molecules have been accomplished for a great number of molecules. ~n previo?s papers,I4-15 we have developed a density matnx formahsm to treat one-photon ionization and multiphoton ionization of molecules, and we have also compared this method lS with the Green's function method developed by Lamprapoulos et al. 16• 17 It is well known that the density matrix method (Le., the master equation method) can take

into account the effect of pure dephasing properly. This effect is important for the case of isolated polyatomic molecules and for the case of molecules in dense media. A main purpose of this paper is to show that the master equation method for MPI of molecules can be used to treat the abovementioned experiments. Another purpose of this paper is to show that Fano's results for one-photon autoionization can easily be obtained from the density matrix method and to demonstrate how the existence of an autoionization state near the ionization threshold can interfere the determination of the ionization energy. It has been experimentally observed that for H 2,18 benzene, 19 naphthalene,3 and diazabicyclooctaneS the ionization threshold determined by using the photoionization technique is slightly different from that by using other experimental techniques. In the usual treatment of autoionization as given by Fano,20 the discrete state is first coupled to the continuum by configuration interaction. The resulting new continuum states contain the discrete state, and the transition is then calculated through the appropriate transition matrix element (depending on experiments) between the initial bound state and the new prediagonalized continuum state. For multiphoton autoionization when the field is strong, this approach is insufficient because in this case the field is no longer a weak probe that simply causes a transition proportional to its intensity. It is part of the interacting system, and since the coupling can be comparable to or even larger than the configuration interaction, the two must be treated on an equal footing. The present paper is organized as follows. In Sec. II, we briefly describe the density matrix formalism applied to treat multiphoton ionization of molecules. In Secs. III and IV, we ~r~en~ the results of one-photon ionization and two-photon IOnizatIOn of molecules, respectively. In Sec. V, we discuss some numerical results. II. GENERAL THEORY

It has been shown that the Liouville equation for a system subjected to a radiation field can be expressed as21 ,22 !)A .!!e.. :" p, at = -iLn-rp=iL r t A

alPennanent address: Department of Chemistry, Tel-Aviv University Tel-Aviv, Israel. ' 4838

J. Chern. Phys. 84 (9),1 May 1986

A

A

:''''

A

(2.1)

A

whereL represents the Liouville operator of the system (in-

0021-9606/86/094838-16$02.10

@ 1986 American Institute of Physics

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Boeglin, Fain, and Lin: Multiphoton ionization

cluding theA interaction between the system and the radiation field) and r denotes the damping operator due to the coupling between the system and the heat bath. The density matrix of the system is characterized by the indices n,m, ... , of the discrete spectrum and by the indices a, p, ... , of the continuous spectrum. The latter may describe the ionization states of the system. We are interested in the time evolution of the density matrix elements Pm,,; thus it is necessary to eliminate from the equation of motion given by Eq. (2.1) the matrix elements like Pma' PaP' etc. that involve the continuum states. A Notice that the Hamiltonian of the system H can be written as A

A

A

(2.2) H=Ho+ V, where Ho represents the zeroth order Hamiltonian which A

provides the basis set of the system and the interaction Hamiltonian consists of two parts, A

A

A

(2.3)

V=D+ U, A

A

Ubeing the perturbation for inducing autoionization and D denoting the interaction Hamiltonian between the system and the radiation field. When the semiclassical theory of A A A radiation is used, D will be a function of time, Le., D = D(t) A A and H = H(t). The matrix elements involving the continuum states like Pma' PaP' etc., can be eliminated by using the projection operator method23 ,24; the resulting master equations (ME) are given by'S ap . ,," + ~ (V"m Pm" -Pnm Vmn ) + r:;'nm Pmm

at

Lm

TI

Lm

t dT K:;'nm'(t,T)Pmm' (t -

+ L

mm,Jo

----at +

(.

WJ mn

i

+ -Ii

+

:~. (t,T) = ~~ {Vrna (t) Vam' (t -

f-

X exp [ - i

+ V"'a (t -

T) Va" (t)

i~ Tmam (td dt,] 8mm, }.

(2,6a)

As is clear from the structure of this expression, the memory kernel decays in time scale T, T~Tc' where Tc is characterized by the ionization states lim· such that Tc (m.) -I. This frequency is in the order of magnitude IoIli. where 10 is the ionization potential. Thus, Tc is usually much shorter than relaxation times ofthe molecule. This condition justifies the use of the Markoff approximation, In this approximation, we may extend the integrations in Eqs. (2.4) and (2.5) to infinity and neglectdn Pm" (t - T), In the case of photo ionization of molecules there is usually no close connection between the size of the molecule and its density of states because in this case, the ionization states are involved. For the case in which the Markoff approximation applies, Eqs. (2.4) and (2,5) reduce to

*

apnn -a;+ -,;i~(V ~ 10m Pm10 -

V ) P10m m"

+ L r:::,m Pmm + L m

R :;'1Om' (t) Pmm' = 0

(2.9)

mm'

T) = 0

(2.4)

apmn - + (.Imm10 + rm") mn Pmn at

i

+ -Ii L( Vmm, Pm'n m'

mn Pmn

Pmm' Vm·1o )

+ L R :~'(t)Pm'n' = 0,

~ (Vmm, Pm'n - Pmm' Vm·n )

(2,10)

m'n'

m

t dT K:~n'(t,T)Pm'n' (t mn')O

Tman (t,) dtl] 81010,

X exp[i

rmn)

+~

T)

and

and

apmn

K

4839

where

T) = 0,

(2.5)

(2.11 )

A

where K(t,T) represents the so-called memory kernel A

AA

[

.(t

A

A

K(t,T) =PL t exp -i)o (l-P)Lt(t,)dt,

and

]

X eXP[if-T(l-P)Lt(t2)dt2](l-P)Lt(t-T)

(2.6) A

and P denotes the projection operator

p~;r = 8

MM

·8NN ·

(1 - ~8aM)(1- ~8PN) .

Notice that r:: represents the dephasing constant, denotes the relaxation rate constant for m_n, and defined by

r"n nn

= -

~'rnn mm·

~

m

From Eq. (2.6), we have

(2.7)

r:;'nm

r:: is (2.8)

(2.12) Notice that rate constants R :~' contain the interaction matrix elements between the discrete states and continuum states and thus include the autoionization and direct photoionization rate constants. For the detailed derivation of Eqs. (2.4), (2.5), (2.9), and (2,10), the previous paper's should be consulted. A main feature of this theoretical technique is the use of the projection operator defined by Eq. (2.7) to eliminate the continuum states for the case H= H(t). In this paper, we shall apply these general results to treat the one-photon ionization and two-color photoionization of molecules.

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4840

Boeglin, Fain, and Lin: Multiphoton ionization

where

Yga ( -w) = Yga (t)e-ito)=Dga ( -w) -iIiJga ( -w) =Dga( -w)

1 ~Dgc( '" +~p n

a --.----- --~

Uca -w)-Wac

c

g-----~-

'" Dgc ( - w) Uca c5 (wac ) . (3.9) - -1Ti ~ Ii c Notice that as usual the rotating wave approximation (RWA) is valid provided that Iw - Wag IWag , R ::, R::, IYag I, Ilrag. This means that besides the resonance condition, the R WA imposes a limitation on the radiation field strength. To compare with Fano's result, we can rewrite Eq. (3.9) as

FIG. 1. Schematic representation of one-photon ionization.

Yga ( - w)

A

o

-w)I~)

= (gID(

-w)I~)(I-~),

(3.10)

1 ~-c '" Uca I ) ±-p

(3.11 )

III. ONE·PHOTON IONIZATION

In this case the ME's are given by (see Fig. I)

a~~ + ~ (Vga Pag -

Pga Vag)

aPaa at + -,;i (Vag Pga +R:~Pag

V)

Pag ga

where

+ R :: Pgg + R ;: Pag

+R:;Pga -WagPaa =0,

(3.2)

+ _1_ Pag + Ragag Pag -0 - ,

r::).

aplIl 2 - a --Im[Yga(t)pag(t)] +R::pgg -PaaWag =0, t Ii (3.4) where

= Vga

- iliR;:

= Vga

- iliJga

g Pag + (i - '"i Vag + R ba) bg Pba

+ (iCO bg +RZ: + apba i -at + '"i Vba (Paa -

1"~)Pbg =0,

1

+ (lhag ) + (r bI2)

(GaI2)

and

Pbb)

E3 = (r I2) b

+ R t + 1"~ )Pba = O.

~ (lhbg ) [(COl>g -COl -co2)

2 +.!.PL IUbeI ]. If c CObe

(4.14)

Again, we introduce the RWA to rewrite the ME's as follows:

(4.21 )

Notice that the yield per unit time for two-color photoionization is given by

Y( 1")

= ga Paa + Pbb + 4 1m [

(4.15)

1-;i) Pba(C02) ]

apaa a1" +2Im[1/IPag(co I )] -21m [1/2( + (1/ag +ga)Paa -1/baPbb =0, apbb a1"

+ 2 1m[1/2( 1 +;i) Pba (CO2) ] + (1 +

1/ba) Pbb = 0,

+ i1/2(1 -

+ ~ + ~ga>l,1/I,1/2'

(J;b = (Jab

(J;g = (Jag + ! ga >1,1/1.1/2 ,

(4.23)

(J ~ = (Jl>g + !> 1,1/1,1/2 . In this case, we find

i[ 1/IPba (co2) -1/2(1 -

;) Pbg (COl

+ CO2) = 0,

(4.22)

case

(ga (J) 1 . +"'2 + ag ( + lEI) Pag(CO I )

+ i1/l( Pgg -Paa)

;:2 Pba (CO2) ].

Analytical solution ofEqs. (4.15) - (4.20) are. in general, not possible. In the following we shall present the analytical results of a couple of important cases. First. we consider the

(4.16)

(4.17)

apag (COl) a1"

[(COba - CO2)

2 2 +.!.PL (lUbel _IDca (C0 2)21 ) ] , If c CObe CO2 -COca

( 4.13)

~ PbgVga + (R:Paa +RZ!Pbb)

+ (iCOba

E2 =

4843

Pbg(CO I + CO2)

=

(J' (l l>g

~)Pag (COl)]

.:

+ IE3

(4.24)

(4.18)

(4.25)

+i1/2(1-;)Pag(CO I ) -i1/1Pba(C02) =0,

(4.19)

(4.26)

and

apba (CO 2) a1"

+

(1- 2+ga- + (Jba )

(4.27) (1

+ iE2) Pba (CO 2) -

i1/1 Pl>g

X(COl + CO2) + i1/2[ (1 - ;)Paa - (1 + ;)Pbb] = 0, (4.20) where

(4.28)

Wag

1/ag

=r-'

(4.29)

b

(Jab = _1_, ga rb1"ab El = (G I2) a

= Ga , (Jl>g = _1_,

rb

~ (l/1"ag)

and

rb1"l>g Y(1") =gaPaa +Pbb

41/~

+ (J;b(1 +~)

[(CO ag -COl) X [:2(Pbb -Paa) -

:2 (Paa +Pbb)] .

(4.30)

In other words, in this case one needs to solve Eqs. (4.27)(4.29) in order to determine yeT). Again the introduction J. Chern. Phys.• Vol. 84. No. 9.1 May 1986 Downloaded 31 Aug 2011 to 140.113.224.113. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions

Boeglin, Fain, and Lin: Multiphoton ionization

4844 o o

~----------------------------~--------------------------~

o co

o

o ID >- •

..... 0

FIG. 3. I vs E for 1'/3 = 0, 1'/4 = 0, 1'/1 = 1.0, 1'/2 = 0.2, q = S.D. Curve 1: l' = 1.0, curve 2: l' = 2.0, curve 3: l' = 3.0, curve 4: l' = 4.0.

T=1.0

~r--_~

o o o

+-----.-----~-----r----_+----~----~----~r_~~

0- 20 • 00

-15.00

-10.00

-5.00

0.00

DETUNING

5.00

10.00

15.00

20.00

of the steady state approximation simplifies the original ME's into the forms of the traditional rate equations. For the weak-weak field case, we have Pbb l1ag, l1ba' we have Paa1

+

2.,.,2 '/2

( 1 + ~)O' 2 ab

0), l1ba

811~] -,---2-:"":""'--Oab(q -1 +2€2q) XPaa = O. (4.37) Notice that ga in Eq. (4.37) is much bigger than the other two terms in the square bracket of Eq. (4.37). Thus, Eq. (4.37) reduces to 211i

(1 +Ei)O~g

Paa

Pgg

+ [ ga + l1ag -

= ga 0'ag (1 + c.) 1

(4.38)

Pgg .

Substituting Eqs. (4.36) and (4.38) into Eq. (4.30) yields q2) 811~] (1 Y(r)= [ ga+ Paa' (q2 _ 1 2€aq) 0 ~b (q2 - 1 2€2q)

+ +

(4.33a)

The asymmetric effect in this case is exactly the same as that in the one-photon ionization case given by Eq. (3.41). On the other hand, if 11ba - •

. -.

,

.... 0

, ,, ,, ,

,,

,,

,,

,

,

,

,,

,, ,,

,,

,,

,

,

,

.'

......... -"

0-20 •00

-15.00

-10.00

- .00

0.00

5.00

15.00

10.00

20.00

OETUNING (-EPSILON 11

above the ionization threshold. The effeet of dephasing processes is also shown in Figs. 15 and 16. Comparing Figs. 15 and 16 with Figs. 10 and 11, we can see that the interference is smaller for the two-photon ionization case. Next, we shall show that the density matrix method discusses in this paper can also be applied to treat the excitedstate lifetime measurement by the MPI technique (see Sec. I). We shall let td denote the time durationA of the (i)llaser. In A other words, for Ol/'Tag , 'Tab' the steady state approximation can be used (see Sees. III and IV) and the rate equations are given by (5.3 ) and.

apbb, b' a'T- + a l Paa + I PM = 0 , where

(5.1)

A

(5.5)

and for t> td , DI (t) = 0 and D 2(t) #0, i.e., A

A

.

A

.

(5.4)

D 2(t) =D2 «(i)2)e-· t"'+D 2( -(i)2)elt. " .

(5.2)

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Boeglin. Fain, and lin: Multiphoton ionization

4850

Tal.O

0

co

T-2.0 T=3.0 Tall. 0

0

.,:r 0

)-

=:.

... 0

"

., ....... ,

. :

I

---"'-- ... ................ .... -...

'-,

'"

"' ..

'.

........

'

,

--, .......... ---- ........

..... -............. .

....................... .

".

ci •••••••••••••••••••••••••••• :::•••••

gr------------------------ci_ 20 •00

-IS.OO

-10.00

- .00

0.00

S.OO

OETUNING (-EPSILON 21

10.00

I .00

20.00

FIG. 13. (a) y(T) VSE2• (b) I{T) VSE2• = 1.0, T2 = 2.0, T) = 3.0, To = 4.0. 1]1 = 112 = 1.0, 1]ag = 0.1, 11"" = 1.0, go = 0.2. q = S.O, {Jag = {Jbg = {J"" = 1.0, E, =0. TI

T=l .0 T-2.0

~ ~

T=3.0 0

T·q.O

Q

~

......,. I

, II

,

..,

'\

,, ,

, """'''''

0

--

-I/)

"

o·

"

II

-- -- .............

...........................

°0 -'0

-w.

)-

.......................

0 \II

ci

g

......... _- ...........

__

~~--~ -IS.OO ----~--------~----~----~-----r~--~~--~ -10.00 -.00 0.00 S.OO 10.00 I .00 20.00

0. 20 •00

OETUNING (-EPSILON 21

Equations (5.3) and (5.4) can easily be solved (see Sec. (5.6)

III):

(5.9) (5.7)

and

and

Paa (I) A; + b ; ,t'11 A 2 - b I ,t'l e + e 2 , (5.10) Paa (0) (A i-A i) (Iii -..t i) where p lUI (0) represents the concentration of Pa