J. Phys. Chem. 1982, 86, 2197-2205
that near-maximum benefits will be obtained by leaving such developments in the hands of computer scientists, who are generally disinterested in numerical computation and are lacking, of course,in knowledge of specific chemical applications. For the optimum advance of the field, computational chemists should become more actively involved in areas that are traditionally thought of as computer science, computer engineering, etc.
2197
Acknowledgment. The authors gratefully acknowledge financial support for this research from the National Science Foundation under Grant MCS79-20698. This research was also sponsored in part by Control Data Corporation, Grant 79C13, and by the Defense Advanced Research Projects Agency (DARPA), ARPA Order No. 3597, monitored by the Air Force Avionics Laboratory under Contract F33615-78-C-1551.
Effect of Excitation on Non-Markovian Vibrational Energy Relaxation Blman Bagchl and Davld W. Oxtoby' The James Franck Institute and The Depertment of Chemistry, The University of Chicago, Chicago, Illinois 60637 (Received July 22, 108 1)
In the non-Markovian limit the process of excitation can have a profound influence on subsequent relaxation processes. We study this effect for vibrational energy relaxation in liquids. We carry out exact quantum mechanical simulationsof a three-level system where the ground state is radiatively coupled to one of two closely spaced excited vibrational levels which are strongly coupled to a bath of known statistical properties (either Poisson or Gaussian). We follow the energy exchange between the two excited levels as the light is turned off. Two types of excitation pulses, square and Gaussian, with different time duration (7')of excitation are used. Pronounced oscillations are found in the non-Markovian limit for the Poisson simulation for values of T comparablewith the bath correlation time. These oscillations may be observable in picosecond laser experiments. Oscillations are less pronounced for the Gaussian bath. We use the stochastic Liouville equation to explain the results of the simulation. Exact agreement with simulation results is obtained for the Poisson case, while for a Gaussian bath, good agreement is obtained by including a small number of bath decay modes. We discuss the experimental situations where these non-Markovian effects may be detectable.
I. Introduction The problem of vibrational energy relaxation in liquids has received considerable attention in recent years. Experiments have demonstrated a tremendous range of behavior, with single-level relaxation times ranging from picoseconds to seconds,l depending on the molecule involved and the solvent used. The slower relaxations can certainly be described by a rate-equation model in which rate constants for state-to-state relaxation are introduced and the population of each level is a sum of exponentials. We refer to this rate equation limit as Markovian behavior, where the molecules stay in a particular vibrational state long enough that any memory effects have disappeared. Another way of putting this is to say that there is a time scale separation between slowly relaxing vibrational modes and fast bath degrees of freedom such as rotational and translational motion. When the vibrational time scales reach into the picosecond or subpicosecond regime, however, the separation of time scales no longer holds true and some interesting qualitatively new effects can arise. This non-Markoviah limit is the subject of the present paper. The role of excitation in the preparation of excited states in molecules and its influence on their subsequent decay has been the subject of many recent theoretical investigations.2-8 These investigations reveal some new and interesting features that can appear in the radiationless relaxation of excited molecules in the condensed state: in the non-Markovian limit the excitation can have a profound effect, even a t long times, on the relaxation of the prepared state. In this paper we study these effects in the *Alfred P. Sloan Foundation Fellow and Camille and Henry Dreyfus Foundation Teacher-Scholar. 0022-3654/82/2086-2 197$01.25/0
case of vibrational energy relaxation in liquids. The first systematic investigation of the effects of excitation on the subsequent relaxation was made by Rhodes2-6 who concluded that, for isolated molecules, increasing the time duration T of the exciting radiation may change the fluorescence from an oscillatory decay to an exponential one or it may even lead to a drastic change in the observed lifetime. The dependence of molecular decay on the nature of the exciting light has also been emphasized by Robinson and LanghoffG who were able to include partially the effect of finite bandwidth and finite durations of excitation pulse and arrived at conclusions similar to those of R h ~ d e s . ~But . ~ none of these authors considered the effect of a stochastic driving force on the relaxation of the excited state. It was Middleton and Schieveg who, working in a somewhat different context, showed that the generalized master equation of Zwanzig, owing to i b inherent non-Markovian nature, can lead to an oscillatory decay for the occupation probability of the discrete state in the Friedrich's'O model. Therefore, oscillatory decay can arise from two seemingly unrelated sources. However, Grigolini and Lami' and GrigolinP have stressed that, in the non-Markovian limit, the process of excitation cannot be separated in a clear-cut way from that of relaxation due to the fact that the excited bath states D. W. Oxtoby, Adv. Chem. Phys., 47,487 (1981). W.Rhodes, J. Chem. Phys., 50, 2885 (1969).
W.Rhodes, Chem. Phys. Lett., 2, 179 (1971). W.Rhodes, Chem. Phys., 4, 259 (1974). W.Rhodes, Chem. Phys., 22,95 (1977). G. W.Robinson and C. A. Langhoff, Chem. Phys., 5, 1 (1974). P. Grigolini and A. Lami,Chem. Phys., 30,61 (1978). (8)P. Grigolini, Mol. Phys., 31, 1717 (1976). (9)J. W.Middleton and W. C. Schieve, Physica, 37, 139 (1973). (10)J. L. Pietenpol, Phys. Rev., 162,1301 (1967).
0 1982 American Chemical Society
2198
The Journal of Physical Chemistty, Vol. 86, No. 12, 1982
can get significantly populated during the excitation process and can thus be driven far from equilibrium. These authors7i8have emphasized the difficulty (or even impossibility) of ascertaining whether a quantum beat phenomenon may be traced back to the non-Markovian properties of the bath or to the resonance. It is well-known that in the Markovian limit, where the bath relaxes rapidly on the time scale of the molecular relaxation, the decay of the prepared excited state becomes exponential. Thus we are faced with the interesting possibility that a damped oscillatory nonexponential decay can possibly be used as a measure of the non-Markovian character of the bath. Unfortunately, no unambiguous experimental observations of these non-Markovian effects are yet available; however, it seems likely that, with the advent of picosecond and subpicosecond laser techniques, they may soon become available. Grigolini and ~ o - w o r k e r s ~ have ~ ~ J ~also J ~ studied the effect of the intensity of the exciting laser pulse on the subsequent decay. They concluded that decay following a high-intensity pulse is the same as that following a weak excitation pulse of short duration in that both effectively "decouple" the molecular system and the bath and prepare the system in the radiative state isolated from the bath. In order to understand better the effect of excitation on the subsequent decay and also to supplement the lack of experimental data, we have carried out an exact quantum mechanical simulation for the vibrational energy relaxation in a molecular system strongly coupled to a bath of known statistical properties. We have considered both the Markovian and non-Markovian limits with emphasis on the latter. The preparation of the excited state by interaction between the system and the electromagnetic field of the radiation, and the subsequent irreversible decay which occurs when the radiation is switched off, has been considered. These simulations can be thought of as a direct generalization of the simulations recently reported by Abbott and Oxtobylawho carried out, for the first time, exact quantum mechanical simulations of vibrational relaxation in a molecular system strongly coupled to a bath of known statistical properties. In their simulations, Abbott and Oxtoby13did not consider the role of excitation and the system was initially placed in the excited instantaneous eigenstate of the time-dependent total Hamiltonian (system + bath). In that work, two different stochastic properties of the bath were considered in the first, the bath was modeled as a two-state Poisson process, in which the coupling matrix elements jump at random intervals between two possible values. This roughly mimics the effects of hard repulsive binary collisions in liquids. In the second case, the bath was modeled as a Gaussian random process to imitate the relaxation behavior due to longer-ranged attractive interactions like those that can occur in dipolar liquids. T h e most surprising result of their simulation was that the qualitative, and to some extent even quantitative, details of relaxation were the same for both the Poisson and the Gaussian bath, even though these baths have very different statistical properties. The second interesting result was the appearance of weak oscillations in the Poisson case which were absent for the Gaussian bath. In the simulations discussed in this paper, we have considered in detail several aspects of the excitation process: the duration, intensity, and shape of the pulse. Two types of pulses have been studied extensively: square and (11)P.Grigolini, J. Chem. Phys., 74, 1517 (1981). (12)P. Grigolini, preprint. (13)R. J. Abbott and D. W. Oxtoby, J. Chem. Phys., 72,3972(1980).
Bagchi and Oxtoby
Gaussian. While square pulses are the ones mostly used in the theoretical analysis?l4 Gaussian pulses are certainly more realistic for the experimental situations15J6in the picosecond and subpicosecond range. The simulations are described in section 11. In section 111, we show that the stochastic Liouville equation approach developed by Kubo17J8and applied to problems in electron spin resonance by Freedlg and to relaxation problems by Grigolini,"J2 can be used successfully to explain the results of our simulations. For the Poisson case, the agreement between theory and simulation is excellent. In the Gaussian case, where the stochastic diffusion operator has an infinite number of eigenstates, we get good agreement by including 6-8 of those states. In treating the interaction between the exciting light and the system, we have avoided making the rotating wave approximation.'* This enabled us to treat high-intensity excitation situations accurately. Section IV concludes with a discussion of the results.
11. Simulation In our simulations, we model the vibrational degrees of freedom as a three-level system involving a ground state lg) and two closely spaced excited states 11)and 12). The energies of these levels in the isolated molecule are denoted by tg, el, and e2, respectively; we take the energy difference el - tg to be much greater than e2 - tl so that relaxation from the excited states to the ground state 18) is negligible on the time scale of interest. For simplicity we also assume that there is radiative coupling only between states (8) and 12). The excited states may either be levels in the ground electronic manifold accessible by infrared radiation or a pair of states in an upper electronic manifold accessible by visible light. The Hamiltonian has the form
H
= Ho
+ Hext(t)+ H B + V
(11.1)
where Ho is the Hamiltonian for the vibrational degrees of freedom, Hext(t)is the interaction between the system and the electromagnetic radiation, HBis the Hamiltonian for the bath (rotational and translational) degrees of freedom, and V is the interaction between the system and the bath which couples levels 11) and 12). We assume that the bath can be described as a high-temperature classical stochastic force so that in the interaction representation the vibrational system evolves under the time-dependent Hamiltonian H ( t )
H ( t ) = Ho
+ H"'t(t) + V(t)
(11.2)
In the notation introduced above
Ho = t,Ig)(gl
+ tlll)(ll+ e212)(21
(II.3a)
V12,Vzl, Vzz,and Vll are the time-dependent matrix elements of the stochastic driving force, and are therefore stochastic functions of the time. For F ( t ) we shall make (14)M. Sargent, M. 0. Scully, and W. E. Lamb in "Laser Physics", Addison-Wesley, Reading, MA, 1974. (15)D.J. Bradley and G. H. C. New, Proc. IEEE, 62, 313 (1974). (16)G. R.Fleming, Adu. Chem. Phys., in press. (17)R. Kubo. Adu. Chem. Phvs.. 15. 101 (1969). (18)R. Kubo; J. Phys. SOC.Jin.; 26, (Suppl.), 1 (1969). (19)J. H.Freed, G. V. Bruno, and C. F. Polnaszek, J.Phys. Chem., 75,3385 (1971).
Non-Markovian Vibrational Energy Relaxation the semiclassical approximation F ( t ) = A ( t ) H , cos ut
The Journal of Physical Chemi$try, Vol. 86, No. 72, 1982 2199 1.20
(11.4)
where A(t) gives the temporal shape of the exciting pulse and H,is the well-known Rabi frequency14equal to p E where ~1 is the transition dipole matrix element between levels lg) and (2),and E gives the magnitude of the electric field. w is the frequency of the radiation and is chosen close to the frequency difference between the levels 12) and Ig). For a square pulse A ( t ) is given by A ( t ) = 1 for -T < t < 0 = 0 otherwise (11.5) We monitor the relaxation from level 2 to level 1after the exciting light is turned off at t = 0. For the Gaussian pulse, the situation is a little more complicated because it extends from t = -m to t = m and to do a realistic calculation, we have to use a cutoff at some intensity. We cut the pulse off on both sides symmetrically when it has reached an intensity which is a fraction f of the peak intensity, and we vary f to determine the effect of the cutoff. In order to make an effective comparison between the Gaussian and square pulse, we choose the full-width at half-maximum (fwhm) of the Gaussian pulse to be equal to the time duration T of the corresponding square pulse, and we take the area under the two pulses to be the same. Since our simulations are quite similar to those described by Abbott and Oxtoby,13we shall not repeat the technical details of the simulation. In what follows, we discuss only those features which are new to the simulations presented in this paper. At time t = -T,the system is placed at the ground state lg) of the molecular system. The subsequent time evolution of the total wave function I$(t)) involves the full Hamiltonian and can be written as I$(t))= exp,[iJ:
dt’fW?]lg)
(11.6)
where expoindicates time ordering. Since we are interested in the population of the second level, we must project the total wave function I$@))onto 12). We define the population evolution function P J t ) for the ith level by pi(t) =
(1 (il$(t) ) 12)
(11.7)
where the outer brackets indicate an ensemble average. To monitor the time evolution of the population in the second excited level after the field is turned off at t = 0, we introduce the normalized time correlation function C2(t) through C2(t) = [P2(t)- f/zP1(0)+ P,(O))I/[P,(O)
P2(0)11 = 2[P2(t) - f/ZPl(O)
- 72P1(0) + + P2(0)lI/[P2(0) - Pl(0)I
(11.8)
C2(t)decreases from 1 to 0 as t increases from 0 to
a.
To calculate the ensemble average, we must obtain simulation results for a very large number N of runs and then average over them. For each run k, the random part Wk)(t)of the total Hamiltonian Hk)(t)is obtained through a random number generator which gives the matrix elements Vi,(t). The other time-dependent part Hext(t)and the time-independent part Ho are simply added to W k ) ( t ) to obtain H(k)(t).From the total Hamiltonian Hck)(t),the is found. P2(t)is then obexact wave function l$(k)(t)) tained by projecting it onto (2),squaring, and averaging over the N different runs
-__
z
D c u
T-3.0 7-10.0
0.00
TIME
Figure 1. Simulation results for Poisson bath wffh increase of time duration T of a square pulse excRation.
The total Hamiltonian changes continuously with time and it is necessary to approximate it as piecewise constant. This can easily be achieved by replacing t by nh where h is the length of the time step, and by taking h shorter and shorter until no further change in the results is seen. The simulation of the stochastic part of H ( t ) follows the method described in ref 13 for Poisson and Gaussian baths. The Poisson bath jumps between two states (which we call and -) at an average rate b / 2 and the probability that a jump will occur in a time interval At is given by (b/2) exp[-(b/2)At]. The coupling matrix elements then randomly jump between the two sets of values (Vll+, V22+, , = V21-). V12+= V21+)and (Vll-, V 2 ~V12For the Gaussian bath, the time evolution of Vij(t) can be represented by a discretized ordinary Langevin equation Vijn+l= exp(-bh)Vijn wijn (11.10)
+
+
where h is the time interval between step n and step n + 1, b is the time constant characteristic of the Gaussian process, and wijn is a Gaussian random variable with mean square given by ( ( w ~ ~ ”= ) ~[)l -
-
e~p(-2bh)](V,0)~
(11.11)
where Vi? defines the strength of the fluctuations in Vi,. In the limit h 0, this describes a true Gaussian process. In our simulations we have set the diagonal elements Vll and V2, equal to zero. In the Poisson case, we chose v12* = v21* = & V where V is the coupling strength of the perturbation while in the Gaussian case we took Vl,O = V2,0 = V. Following Abbott and Oxtoby,13 we have worked in dimensionless units in which b, V, and t-’ are measured in units of wo = (e2 - tl)/h, the frequency difference between states 12) and 11) in the isolated molecule. We chose V = 0.5 and b = 2.0 for the Markovian limit and V = 0.5, b = 0.2 for the non-Markovian limit. The significance of these choices is discussed by Abbott and 0xt0by.l~We further chose t1 - tg = 10(t2- el) for most of our simulations. The results are not very sensitive to this choice within a given range. Figures 1and 2 show the results in the non-Markovian limit of our simulations for the Poisson and the Gaussian bath, respectively. The excitation is a weak square pulse. For the Poisson case, we see that the correlation function C2(t)oscillates strongly for pulses of time duration shorter than or comparable with the bath correlation time b-’. The / where Am oscillation has a well-defined period of 2 i ~Am, is the frequency difference between the eigenstates of the full Hamiltonian and is equal to (4p+ m2)1/2. But as the time of excitation is increased, these oscillations become weaker and in the limit of pulse duration time T a,they
-
2200
The Journal of Physical Chemistry, Vol. 86, No. 12, 1982 -7-T-
7-77
TT77 T
-
. . .
_ _ _ _
T
T
0.50-
