J. Phys. Chem. 1991,95,6149-6155
6149
Absorption Spectrum of the Solvated Electron. 1. Theory Halina Abramczyk Institute of Applied Radiation Chemistry, Technical University, 93-590 Lodz, Wroblewskiego 15, Poland (Received: May 17, 1990)
Optical absorption band-shape theory of the solvated electron in condensed media is presented. The solvated electron motion occurs in the Born-oppenheimer energy potential well and is coupled to the intramolecular vibrational modes of the solvent. Both the energy potential well and the anharmonic coupling are randomly modulated with time. The time evolution of the anharmonic coupling constant is governed by the resonance energy exchange between the vibrational mode and the bath. The fluctuations of the energy potential arising from the direct coupling to the bath obey the stochastic Liouville equation.
One of the most important and frequently investigated physical properties of solvated electrons in liquids and glasses is the optical absorption spectrum, which probes the short time dynamics of the solvated electron. Optical absorption spectra are exceptionally broad, structureless and with an asymmetric high-energy tail. Different theoretical'"' and quantum simulations of the optical ~ p e c t r u m ' ~try * ~to~ explain J~ such characteristic features. The optical absorption maximum position as a function of temperature, polarity, and pressure are relatively well understood2629 in the framework of the semicontinuum models. Most of the existing theories are notably unsuccessful in treating the optical absorption band shape. Both the semicontinuum" and the continuum models16 can only account for less than half of the experimental line width. The theoretical bandwidths predicted from the other are still two small and symmetrical. Although some of the theoretical or simulation calculations claim successful fits10*25J5936 and predict the transition energies corresponding to absorption maxima with reasonable choices of parameters, the broadness and asymmetry of the spectral profile could not be generated without introducing additional parameters such as a distribution of cavity sizez4*2s.35 or finite lifetime of the excited ~tate.3~3'The most general and potentially powerful model for solvated electron spectra has been proposed by Banarjee and S i m o r ~ s The . ~ ~ model includes anharmonic coupling with vibrational degrees of freedom of the solvent molecules. However, neglecting the time dependence of the coupling strength (the operator A, which creates the "dressed" electron, does not depend on time) and the fluctuations of the potential well makes the model unable to introduce in a natural way inhomogeneous band broadening. Generally, one can expect that three factors contribute to the solvated electron band shape: coupling to the vibrational degrees of freedom, inhomogeneous broadening due to molecular motions of the solvent molecules forming the trap, and nonlocalized transitions due to electron tunneling. It would be valuable to produce a unified model that takes into account all these factors. The present approach is based on the linear response theory formalism.42 Strong coupling limit theory with the canonical transformation for the diagonalization of the vibrational Hamiltonian has been used for the description of coupling between the solvated electron and intramolecular vibrational coordinates of the solvent molecules. The anharmonic coupling constant is regarded as a stochastic variable undergoing intramolecular resonant energy exchange with the bath (translational, rotational motions and the low-frequency intermolecular solvent modes). The time evolution of the anharmonic coupling constant is treated by aid of the reduced density operator formalism43with the Liouville equation solved by the method of Louissel and Walker." The
1. Introduction
The properties of the solvated electron have been studied by a variety of experimentall-' and theoretical methods such as a b initio molecular orbital calculations,- computer modelling techniquese" and theoretical
( I ) Feng, D.F.; Kevan, L. Chem. Reo. 1980.80, 1. (2) Pmccdings of the Fifth and Sixth International Conferenca on Excess Electrons and Metal-Ammonia Solutions. J . Phys. Chem. 1980,84, 1065; 1984,88,3699. (3) Lund, A. Res. Chem. Intermed. 1989, 11, 37. (4) Newton, M. D. J . Phys. Chem. 1975, 79, 2795. (5) Rao, B. K.; Kestner, N. R. J . Chem. Phys. 1984,80, 1587. (6) Clark, T.; Illing, G. J . Am. Chem. Soc. 1987, 109, 1013. (7) Murphy, S. V.; Hibbert, D. B. Rudiut. Phys. Chem. 1986, 28, 319. (8) Tichikawa, H.; Ogaaawara, M.; Lingren, M.; Lund, A. J . Phys. Chem. 1988,92, 1712. (9) Chandler, D. J . Phys. Chem. 1984,88, 3400. (10) Rmky, P. J.; Schnitker, J. J . Phys. Chem. 1988, 92, 4277. (1 1) Sprik, M.; Klein, M. L.; Chandler, D. J . Chem. Phys. 1985,83,3042. (12) Sprik, M.; Impey, R. W.; Klein, M. L. J. Chem.Phys. 1985,83,5802. (13) Wallquist, A,; Martyna, G.; Berne, B. J. J . Phys. Chem. 1988, 92, 1721. (14) Jonah, C. D.; Romero, C.; Rahman, A. Chem. Phys. Lett. 1986,123, 209. (15) Jortner, J. Mol. Phys. 1962, 5, 257. (16) Copcland, D.; Kestner, N. R.; Jortner, J. J . Chem. Phys. 1970,53, 1189. (17) Fueki, K.; Feng, D. L.; Kevan, L.; Christofferson, R. J. Phys. Chem. 1971, 75, 2297. (18) Logan, J.; Kestner, N. R. J . Phys. Chem. 1972, 76, 2738. (19) Kelley, C. S. J . Chem. Phys. 1974, 60, 4547. (20) Carmichael, 1.; Webster, B. J . Chem. Soc., Furuduy Truns. 2 1974, 70, 1570. (21) Tachiya, M.; Mozumder, A. J . Chem. Phys. 1974,60,3037. (22) Baird, J. K. J . Phys. Chem. 3975, 79, 2862. (23) Kestner, N . R. Cun. J . Chem. 1977,35, 1937. (24) Bartczak, W. M.; Kroh, J. Chem. Phys. 1979, 44, 251. (25) Bartczak, W. M.; Hilczer, M.; Kroh, J. J . Phys. Chem. 1987, 91, 3834. (26) Fueki, K.; Feng, D. F.; Kevan, L. J . Am. Chem. Soc. 1973,95, 1398. (27) Fueki, K.; Feng, D. F.; Kevan, L. J . Chem. Phys. 1972, 56, 5351. (28) Feng, D. F.; Fueki, K.; Kevan, L. J . Chem. Phys. 1972, 57, 1253. (29) Fueki, K.; Feng, D. F.; Kevan, L. J . Phys. Chem. 1974, 78, 393. (30) Kestner, N. R.; Jortner, J.; Gaathon, A. Chem. Phys. Left. 1973, 19, 328. (31) Gaathon, A.; Jortner, J. Elecfrons in Fluids; Springer-Verlag: New York, 1973; p 429. (32) Katner, N. R. In Electron-Solvent und Anion Solvent Interactions; Kevan, L.. Webster, B., Eds.; Elsevier: Amsterdam, 1976, Chapter 1. (33) Tachiya, M.; Tabata, Y.; h h i m a , K. J . Phys. Chrm. 1973, 77, 263. (34) Webster. B. C.: Carmichael, 1. J . Chem. Phys. 1978,63,4086. (35) Kajiwara, K.; Funabaahi, K.; Naleway, C. Phys. Reo. A 1972,6, 808. (36) Huang, J. T.; Ellison, F. 0. Chem. Phys. Leu. 1974, 28, 189. (37) Tsubomura, T.; Sunakawa. S.Bull. Chem. Soc. Jpn. 1%7,40,2468. (38) Banerjee, A.; Simons, J. J . Chem. Phys. 1978,68,415.
0022-3654/91/2095-6149$02.50/0
(39) Bush, R. L.; Funabashi, K. J . Chem. Soc., Furuduy Truns. 2 1977, 73, 274. (40) Tachiya, M. J . Chem. Phys. 1974,60, 2275. (41) Tachiya, M.; Mozumder, A. J . Chem. Phys. 1974,61, 3890. (42) Kubo, R.J . Phys. Soc. Jpn. 1962, 17, 1100. (43) Blum, K. Density Mufrix Theory Applicufions;Plenum Prcas: New York, 1981. (44) Louisell, W. H.; Walker, L. R. Phys. Reo. 1965, 137, 204. (0
1991 American Chemical Society
Abramczyk
6150 The Journal of Physical Chemistry, Vol. 95, No. 16, 199'1
direct coupling of the bath is assumed to be weak and is treated by perturbation theory using cumulant expansion and the Markov-Gauss approximation for the stochastic processes governing motions of the solvent. 2. Model 2.1. General Description. We consider the one-electron
(solvated electron) approximation. This approximation involves not explicitly treating all of the electrons belonging to the N solvent molecules in order to deal with a one-electron Schrodinger equation. In the present approach, we shall study the solvated electron spectrum in liquid or frozen matrices assuming that the electron motion occurs in a single- or double-well potential (if tunneling exists) created by the solvent molecules. This motion is accompanied by the anharmonic coupling to the intramolecular vibrational modes of the solvent. Both the potential well and the strength of the coupling to the vibrational degrees of freedom are randomly modulated with time. The origin of the fluctuations is molecular motions of the medium in which the electron is dissolved. This coupling regarded as a stochastic process allows one to calculate the absorption band for the trapped electron in terms of a superposition of Franck-Condon lines. 2.2. Band Profile Theory. 2.2.1. Hamiltonian. In the linear response scheme, the absorption band shape T(w) for a solvated electron can be expressed as the Fourier transform of the correlation f~nction'~ of the electronic dipole moment operator M
-qI I I
I(w) = (2?r)-'w[l - exp(-hw/kT)]S+-_ dr e'&(M(r) M(0)) (1)
To calculate I(w) we have to know the time evolution of the dipole moment operator M ( t ) appearing in eq 1 . The symbol ( ) means averaging over electronic and vibrational degrees of freedom and bath. Thus, let us first discuss the Hamiltonian governing the time dependence of M(r). We consider here a solvated electron in the presence of a large number of photons, which make the system jump between two electronic states. The electron moves in a double-well potential (Figure 1 ) . The absorption and emission of photons is considered instantaneous with respect to tunneling. In this case, the initial (ground) and final (first excited) electronic states may be split by tunnelling through the barrier. This would lead to an appearance of additional bands, whose origins would be displaced from the others by the frequencies of tunneling in the ground and excited states. If the solvated electron is trapped in a single potential well, we observe only one transition. Assuming that the solvated electron is coupled to the intramolecularvibrational modes of the solvent molecule, one can write the Hamiltonian of the system in the form H(r,q) = h(r,q) + P / 2 M
(2)
W , q ) = p 2 / 2 m + U(r,q)
(3)
and
'
-2d+ F l p e 1. Double-well potential V(r,q)of the ground 10) and first excited 11 ) elcctronic states of the solvated electron and characteristicparameters V, and AV. The equilibrium value of the electron coordinate r for an electron in 11) is displaced from that in 10) by 2ao.
where V(r,q) is the potential energy of the solvated electron (Figure 1) and U'(q) is the vibrational potential energy operator of the q mode. Assuming that the q mode is harmonic, one can write the total Hamiltonian (eqs 2 and 3 ) in the form
+
Hm(r,q)= Em(q) P 2 / 2 M
*(r,q) = dJ(r*q)x(q) (4) where $(r,q) is the electron eigenfunction of h(r,q) and x(q) is the vibrational eigenfunction. The potential energy operator U(r,q) consists of two contributions U(r,q) = V(r,q) + U'(q) (5)
+ f/ZMwoo2q2
(6)
The basis for the solvated electron is defined as the tunneling stationary basis for the Hamiltonian (eq 3)
h(r,q) = CEmlm)(ml m
(7)
where Em are the electron eigenenergies in the double well of the electron in its ground and first excited states at fixed (equilibrium) vibrational displacements. Using the creation-annihilation operators a, at (with [ a d ] 1) for the vibrational mode q
P = i ( M h ~ ~ / 2 ) ' 1-~a)( a ~ q
where m is the electron mass, M is the reduced mass of the vibrational oscillator, r denotes the coordinate of the solvated electron, and q is the vibrational normal coordinate of the mode coupled to the electron. We assume that only one vibrational mode is modified in the electronic transition. In fact, the electron motion can also be coupled to all of the other normal modes of the solvent. Such couplings, as long as they are linear, introduce no essential complication into the Hamiltonian. Using the Bom4ppenheimer approximation, we can write the eigenfunctions *(r,q) of the Hamiltonian H as a simple product
'I'
7.0
= ( h / 2 ~ w ~ ) ' / 2 (+aat )
(8)
we obtain for the ground and first excited states of the solvated electron
where &q) is the value of Eo(q) for the equilibrium displacement q and a ,,the dimensionless anharmonic coupling constant, reflects the strength of the coupling between the electron and the q mode
Here, the index m at 00" goes over all the electronic transitions from the ground to the first excited electronic state.
Absorption Spectrum of the Solvated Electron
The Journal of Physical Chemistry, Vol. 95, No.16, 1991 6151
The anharmonic coupling between the electron and vibrational motions can be treated in a general method, called the "strongcoupling theory". In this method, the coupling term is treated in an exact, nonperturbative way, on which a canonical transformation is performed as is common in polaron theory. The transformed Hamiltonian describes the dynamic effects of vibrational motion on the electron as it moves between its BornOppenheimer eigenstates. The usual jargon is that the "bare" electron is "dressed" in a cloud of solvent vibrations. Performing the phase transformation on Ho and f l Ho = At(Y2ao)&4(X.o)
H;" = 4Y2.o)H;"A('/z.o)
(11)
with the operator
= exp[j/2ao(at - 4 1
where Hamiltonian H is given by (15). Because H is acting only on the subspace corresponding to the first excited state, the equation is simplified to the form (let us take for example the r matrix element for the electronic transition O+ I-)
-
In the same way, we can express the time evolution of the other elements rm. Thus, the dipole correlation function (eq 18) may be written as G(t) = Tr( (epib(O) At(t)A(0)e2[ro+l-2exp(-i(w{+'- 2a02woo)t) ro+1-2 exp(-i(w{+'- - 2a02wm)r)
+
+
(12)
exp(-hwb/kT)[rrl-2 exp(-i(wb" - 2 a0 oo)t) + ro-I+2exp(-i(wt-I+ - 2a02woo)t)]]l)(23)
Ro = hw(-JOda f l = hwr&ta - ao(a+ + a)hw@J- ao2hw@J+ hog (13)
where wb is the tunneling frequency in the electronic ground state and
A(Y2UO)
and changing the origin of the energy level, we obtain
The vibrational basis corresponding to the Hamiltonian (eq 13) is denoted as la). It is convenient to pass to a new representation, in which the Hamiltonian governing the q mode is diagonal for the solvated electron in its first excited state. Using the phase transformation, which is in this case the displacement operator of the normal coordinates ut and o A(ao)= exp[ao(at - a ) ] we can write finally
(15)
with the vibrational basis la). 22.2. Dipole Correlation Function Calculation. As we can see from eq 1, in order to calculate the absorption band shape, we have to know the time evolution of the dipole correlation function ( M ( t )M ( 0 ) ) . The dipole correlation function can be expressed as
M ( t )M ( 0 ) )
(16) where, following the adiabatic approximation, we may write the total density operator of the system p(0) at time t = 0 as a product of the solvated electron and vibrational density operators P ( 0 ) = PC1(0)P#b(O)
(17) The index 0 at pZb(0) denotes, that the solvated electron is in its ground state. The symbol ( ) means averaging over the bath. First, we average over electronic degrees of freedom. Using a representation of the eigenstates of the electron in the adiabatic basis (eq 7), we may write G(t) = Tr (~"~(0) E bfm(0) M m A t ) Mttn(0))) n.m,n'
(18)
where the sum over m,n, and n' is restricted to O+, 0-,I+, and 1- states (Figure 1). The density operator pcl(0) governing the electronic motion corresponds to an equilibrium po+o+ =
POT
exp(-Eo+/kT)/(Tr
= exp(-Er/kT)/(Tr
pel)
pel)
(19)
In the strong coupling limit theory, the dipole moment for the solvated electron becomes Mnm(0)
A ( ~ or n) m ( 0 )
M!m(t)
At(t) rmn(t)
(20)
where m and n are restricted to n = 0+,0- and m = 1+, I-, whereas A(a0) and At(r) represent the displacement operators at t = 0 (eq 14) and at time t , respectively. The time evolution of rmn(t)is governed by the Heisenberg equation
= exp(-Eo+/kT)/(Tr pel)
pZb(o)
(14)
f i = hU@J(UtU- 2ao2) + hot
G(t) = Tr (pcl(0) ptb(O)
(24) 22.3. Time Evolutioa of tbe Displacemeat Operator A (t). Now, we shall perform the averaging over the vibrational states of mode q, which is coupled to the solvated electron. The density operator ptb(0) governing the vibrational mode corresponds to Boltzman equilibrium when the solvated electron is in its ground state e
= el exp(-ha+a/kT)
(25)
where the trace Tr pZb is equal to cl-I el = 1 - exp(-hoW/kT)
(26)
If we take a representation of the displacement operator A in terms of the basis of the Hamiltonian (eq 13) of the q mode A = CAab la) (81
(27)
a8
we obtain from (23)
W )= (e2cCel exp(-hwoor/kT)(alAt(t)l~)(?IA(O)la)
X
ol
{ro+l-2exp(-i(w{"- - 2ao2woo>r) + ro+1-2exp(-i(wg*I- 2ao2ww)t) exp(-hob/kT)[rr1-2 exp(-i(wr'- 2ao2woo)l)+ ro-p2exp(-i(&+ - 2a02wao)t)]l) ( 2 8 )
+
The matrix elements appearing in (28) are Franck-Condon factors. According to Koidet5 we have for y 1 a
( M 0 ) l a ) = exp(-ao2/2) x Y
(c~!~!)~/*C(-l)~c~~ ~k)!k!(y + - [ ( yk-0
+ k)!]-' (29)
In order to find the time-dependent matrix elements (alAt(t)lv), we must know the time evolution of the total density operator p(t). Let us assume that the vibrational mode q is relaxing to a bath (Le., the vibrational modes other than the q mode coupled with the solvated electron, reorientational, or translational degrees of freedom), when the solvated electron has jumped abruptly to its first excited state. Thus, the total density operator p(0) may be written as
= pYb(o) Pbatb(0)
(30) where pYb(O) is the vibrational density operator at t = 0 when the electron is in its first excited state. The Liouville equation governing the time dependence of the density operator p(r) can be solved by using the method of Louise11 and Walker." Recently, the same method has been used by Boulil et aisMfor a problem of the H-bonded complex stretching mode coupled with the lowp(0)
(45) Koidc, S.2.Naturforsch. 1960, ISA, 123. (46) Boulil, B.;Hcnri-Rousseau, 0. Chcm. Phys. 1988, 126, 263.
6152 The Journal of Physical Chemistry, Vol. 95, No. 16, 1991
frequency mode relaxing to a bath. By use of their method, it can be shown that A(r) is expressed as A ( t ) = exp[a*(t)at - a(t)a] (31) where a(t) = a. exp(-iw,,,,t) exp(-Ft/2) (32) Here I' is the damping parameter given by
r = 2 r ~ 1 ~ ~ 1- 2a,,,,) b(~~
(33)
J
When there is no coupling to the bath (the coupling constants Kj = 0), the expression for a(f) takes the form a ( t ) = a. exp(-iw,,,,t) Using (31), we calculate (nlAt(t)lii.)
(34)
Abramczyk placement with the q mode as we have assumed in the Hamiltonian (eq 9). Additionally, the electron is coupled not only to the q mode but to all of the other vibrational modes of the solvent molecule, which modify the potential well. Thus, the electronic transition between the ground and first excited Bom-Oppenheimer potentials can be regarded as a transition in the same potential fluctuating with time. The influence of these factors on the band shape of the solvated electron is formally included in our model by averaging over molecular motions in eq 36. In this section, we will concentrate on the averaging of the dipole correlation function G(t) in expression 36. Using the static double well (Figure l), that is, a double well in the case of Born-Oppenheimer electron-nucleus separation and working in the basis given by (7), one can write the following Hamiltonian for the isolated system
5=
(alAt(t)ly) = exp[-a2(t) /2] (a!y!)l/2 x U
C(-1Yla(r)lYa(t)l'-"[(cu- j ) ! j ! ( y- a j-0
+ j)!]-I
(35)
Substituting (35) and (29) into (28), we obtain
with
B7u = f, exp{-i[wg - 2ao2wo0+ (y - a)~,,,,]t) (38) FyupJ
= expI-[(r
- a) + 2P + 2jlrt/2)
Ckjlyu3 (yo2(k+j+/+ya)
We can write the same expression for the electron in its first excited state. In order to calculate the time evolution of the energy matrix elements, one has to solve a Liouville equation with stochastic matrix elements. We have used the method proposed recently by Lami and Villani4' with aP e Ap at
f4
= r,,-1+2exp(-hwb/kr)
COB = wQI-
wd = writ
/ P+ +(O\
f, = ro-1-2exp(-hw(,/kT) ob
= w0+]-
t2(y) =
(45)
wf =
exp(-yho,,,,/kr)
and the stochastic vector
As we can see from eqs 36-40, the optical band of the solvated electron potentially may consist of four lines shifted by the tunneling frequencies of the ground and/or first excited states. Each electron transition band consists of many vibrational lines peaked at w g - 2a02w00+ (y - a)wW and differing by their half-width
ryupl = [(Y - a ) + 2p + 2 j i r
(44)
where
where
fi = ro+1-2 f2= r0tp2
(42)
,where Eo+ and Eo-are the energies of the O+ and 0- levels when the electron is in its ground state. Let us simulate the effect of the environment by assuming that the matrix elements of the Hamiltonian undergo fluctuations
(39) (40)
(7 ;o)
(41)
Each vibrational line consists of a number of Lorentzians differing by their intensities, which depend on products of the form Almui, t2(Y), and C&J/y9' 23.4. Tunneling in the Presence of Fluctuations. So far, we have taken into ac&unt the influence of molecular motions in the neighborhood of the trapped electron, which enter indirectly through the coupling of the electron with the vibrational mode q relaxing to the bath. However, the solvent acts also directly through creation of a fluctuating electric field, which perturbs the energy levels of the ground and excited states of the electrons tunneling from one trap formed by a solvent to another. This conception has been used in many statistical models.~JSJs37 There is, however, another factor, that may be much more important, resulting in fluctuations with time of the electron potential well. During an electronic transition, the positions and the velocities of all atomic nuclei forming the trap remain constant (FranckCondon principle), but the electronic charge distribution is altered. Because of the change in the electronic structure, the transition occurs between the energy surfaces characterized by different equilibrium geometries. Generally, the difference between the energy surfaces is much more complicated than only linear dis-
(46)
The matrix A for the Hamiltonian (eq 43) is given by the expression
If the stochastic vector i ( t ) is a Markov process then, it has been shown4*that the joint process (p.7) is also Markovian and its probability density P(p,i,t) (Le., the probability density that at time r the density matrix will be that specified by the vector p for the realization of h indicated by 7) satisfies the equation
(47) Lami, A.; Villani, G. Chem. Phys. 1987, 115, 391. (48) van Kampen, N. G. Stochartlc Processes in Physics and Chemistry; North Holland: Amsterdam, 1981.
The Journal of Physical Chemistry, Vol. 95, No. 16, 1991 6153
Absorption Spectrum of the Solvated Electron where W is the operator that appears in the master equation for the transition probability density of 7, Le. n(i,t/io,O)= Wn(7,t/i0,,o)
with a probability
(49)
Let us return to the special situation considered in our model. The frequency values of electronic transitions wC appearing in (38) fluctuate with time according to the fluctuations of the energy levels governed by the Hamiltonian (eq 43). Thus, one may write LO#(?)
= ( w r ) + Ad"(?)
(50)
where AU'"(~) w r ( f ) - ( w r )
1
1
-3
/
where a is the tunneling frequency jump caused by the fluctuations and fa is the probability per unit time of making a jump. As is well knownSo(see Appendix A), the tunneling frequency correlation function can be expressed as
Returning to (36), we perform the averaging over the environment (term Tu in eq 38)
(Aom'(tl) Awml(tz))= 2aZ exp[-fa(tl - t 2 ) ]
(B;J = f, exp(-i[-2a02ww + (y - a)ww]tl(exp(-io$t)) (51)
where fa can be characterized by the correlation time
It is convenient to express the electronic transitions or in the following way w#(t)
(wh)
+ (w"') + Aw,$(t)+ Awmr(t)
(@It)
=0
(52)
where ( w2')
Aolf(t) = 0
= -( a\ )
( d')= -(
Aw2'( t ) = -Au\
~ 6 )
Ad'( t )
(1)
-Awb( t )
b4')= -((ob)+ ( w \ ) ) Aw4'(t) = -(Awb(t) + Aw\(t))
(53)
where ut,and o! are the tunneling frequencies of the ground and first excited electronic states. The average of the exponential may be expressed as the exponential of an average (central limit theorem, cumulant exwill p a n ~ i o n ~ ~Substituting ). (52) into (51), the average (Tu) take the form
(58)
(59) We can expect that the correlation time, T ~which , characterizes the reconstruction time of the trap, is closely related to the reorientational correlation time 7 , . The reorientational correlation time q can be determined from dielectricrelaxation measurements or IR band shape analysis. Returning to the correlation function (Aok(tl) AwA(t2)),we have shown in Appendix B that it is given by (A4dtl) A&&) = bZ exp[-h(rl - t2)l (60) Substituting (58) and (60) into (54), we obtain 7a
( Bya)
= l/fa
+
= f, exp{-i[ (A@:) + (w") - 2a02ww (y a)ow]t) exp(-2~~[7,2(exp(-t/~~) - 1) rat])exp{-b2[7&exp(-1/q,) - 1) + ~ ~ t (61) ] ]
+
Finally, is we substitute (61) into (36), the dipole correlation function takes the form 4
m
G(t) = ezeel
eZ(7)
I
- f a a m /
c
+
CA/*aj(B~a)Ckj/y~-fap,)
mil a=Oj=OL-O/-Op-O
y=O
(BYa) =
f, exp{-i[-2a02ww
x'
+ (y - a)woolt)exp{-i((wb)+ ( o m ' ) ) t -
d t ~&I2 dtz [(Awb(tJ A~b(t2))+ (Awmr(tl)Awmf(h))l1
(54) Two different processes govern the time dependence of the correlation functions (Aob(r,) Awb(r2)) and (Awmf(tl)Aomf(t2)). The tunneling correlation functions ( Awmf(tl)Awm(t2)) are due to the fluctuationsof the energy levels of the ground or first excited electron states, whereas in the frequency correlation function Awb(tl) Aub(r2))both electronic states are involved. The fluctuations in this case are due to the change of the potential well, which is generally not the same for the ground and first excited electron states. Because electronic transitions are assumed to be instantaneous with respect to tunneling there are no "cross" correlations (Awb(t,) Awm'(t2))in (54). Now, let us concentrate on the tunneling correlation functions. The energy level fluctuations are created by the solvent molecular motions, and we can model this situatio! by assuming that only the diagonal matrix elements of ho and hl (eq 43) are stochastic. According to eq 43, we can write Aw6(t)
~ b ( t -) (
~ 6 ) h - ' [ ~ ( t-)~ o + ( t ) ]
where Almajlre, e l , c2, f? CkllTahave the same meaning as in eqs 36-40, whereas (qa) 1s given by (61). Substituting the time correlation function G(r) (eq 62) into (l), we can calculate the band profile of the solvated electron. 3. Conclusions The basic result of this paper for the absorption spectrum of the solvated electron in condensed media is contained in eq 62. Numerical calculations concerning the solvated electron band shape and comparison with experiment will be presented in the following paper.s1 Here, we will derive only qualitatively some conclusions from our model. Potentially, the band of the solvated electron consists of four bands corresponding to the tunneling transitions O+ 1+, 0--,I-, '0 1-, and 0- 1+. Due to symmetry restrictions, some of them may be negligible or weaker than the others. Each of the electronic lines is a superposition of Lorentzian bands peaked according to the following rule oya= (ob) + (w") -2aoZww + (7 - L Y ) O ~ ~ (63) +
-
-
Each band peaked at wya has its own half-width
From eq 55, we see that the stochastic vector 7 (eq 46) fluctuates among four situations
(64) = [(Y - a) + 2q + 2i1r due to the coupling "electron-vibrational mode bath". Generally however, the bandwidth may be modified by fluctuations of the energy levels of the trap caused by the molecular motions of the environment and resulting from the transition between the energy surfaces characterized by different equilibrium geometries. These effects are given by (61).
(49) Rothtichild, W. G. Dynamics of Molecular Liquids; Wiley: New York, 1985; p 386.
(50) Mukamel, M. Ado. Chem. Phys. 1987, 70, 165. (51) Abramczyk, H.; Kroh, J. J . Phys. Chrm., following p p e r in thio issue.
Ao\(t) = o{(t)- (wf)= h-I[c,-(t)- cl+(t)]
ryaql
(55)
Abramczy k
6154 The Journal of Physical Chemistry, Vol. 95, NO. 16, 1991 GJt) = exp(-2a2[7,2(exp(-r/r,)
- 1)+ ~ , t ] l
Gi(r) = eXp(-b2[Tb2(eXp(-f/Tb) - 1) + Tbt])
(65) (66)
Two limiting situations can then be written down, namely, that the correlation times T. and ?b decay much slower or much faster than the amplitude coherence, Le. ( A w ~ ) , ~ />> ~ ~1, ( Aw2)b1/2Tb >> 1 Slow limit
if we know the initial distribution of the frequencies Po and the conditional transition probabilities I I ( j / k ) at t = r 1 - r2, since ( Awmf(tl)Awmt(t2))= ( Awmr(t)Aw"(0)). Thus, one may write ( Awmr(f
Awmt(t 2 )) =
x I I ( j / k ) Awmr(j) Aomr(k)Po(Awm'(k)) ( A l ) j,k
In our case, the amplitudes are given by the change a in tunneling phase angular speed and the change b in the electronic transition phase angular speed
where I I ( j / k ) is governed by the master equation (eq 49) with W given by (57). Po(Awmf(k))is equal to since each of the four realizations of the stochastic vector (eq 56) have the same probability at t = 0. Solving the master equation by matrix diagonalization
(Aw2),,'/2 r: 21/20 (AJ),1/2 =b In the slow limit, eqs 65 and 66 simplify to G"(t) = exp(-2a2r2/2) Gb(t) = exp(-b2t2/2)
we have
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