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Quantum effects in molecular reaction dynamics in solids: photodissociation of HI in solid Xe
Volume 173. number 4
CHEMICAL PHYSICS LETTERS
12 October 1990
Quantum effects in molecular reaction dynamics in solids: photodissociation of HI in solid Xe R.B. Gerber ‘sband R. Alimi a ’ Department of Physical Chemistry and The Fritz Haber Research Center for Molecular Dynamics, The Hebrew University, Jerusalem 9 I904, Israel ’ Department of Chemistry, Universityo/California. Irvine. CA 9.2717, USA
Received 2 1 May 1994 in final form 24 July 1990
The photodissociation of a molecular HI impurity in a host Xe crystal is studied theoretically with focus on the role of quantum effects. The method used treats the H atom by timedependent wavepackets, the heavy atoms classically. Comparison is made with fully classical molecular dynamics simulations. Important differences are found between the quantum and the classical behavior, due to tunneling and zero-point energy effects for the H atom. The results demonstrate the desirability of using mixed quantum/classical molecular dynamics methods in simulating chemical processes in condensed matter at low temperatures.
The experimental study of molecular reactions in low-temperature rare-gas solids has been a topic of considerable activity in recent years [l-7]. This is part of the motivation for the theoretical interest in these systems, which are appealing also because they offer a framework for studying chemical dynamics in solvents of relatively simple structure and of known interaction potentials. These advantages are important in search for first-principles understanding of molecular reaction dynamics in condensed phases. There have been several theoretical studies of simple photochemical reactions of molecular impurities in rare-gas solids [g-l 11. The method employed in these studies were many-particle classical trajectory (molecular dynamics) simulations [ 121. That indeed has also been the main theoretical tool so far for gaining insight into the microscopic details of chemical processes in liquids [ 131. This raises the important question as to what is the role of quantum effects in such processes. Such effects are expected to be specially important at low temperatures, and should in general be most dramatic for processes in which very light atoms are involved. The present paper explores the nature and role of quantum effects in photochemical reaction in a low temperature solid in which a hydrogen atom is involved. The specific process studied is photodissociation of an HI im-
purity in a Xe host crystal. This system was chosen because it has already been the topic of experimental work [ 61, and because it was extensively studied theoretically by classical molecular dynamics simulations [8]. The process considered here is the photodissociation of HI caused by electronically exciting this molecule in its site within the host Xe crystal from the ground state ‘E+ to the purely repulsive excited state ‘II. Description of the process requires knowledge of the interaction potentials for both the ground and the excited electronic state. We assume that the potential function of the system, both before and after electronic excitation, can be modelled as a sum of pairwise interactions between the atoms of the system. The pair potentials for Xe-Xe, Xe-I, Xe-H and H-I were obtained by fitting gas-phase spectroscopic and scattering data, and are described in ref. [ 81. For the H-I pair interaction, the gas-phase ‘C+ and ‘II potential curves were used respectively for the ground and for the excited state. The crudest approximation in the potential functions used is the assumption that the electronic ground state interaction between each Xe atom and the I-IImolecule can be written as a sum of atom-atom potentials. We tested the consequences of this assumption, which simplifies the present calculations, in the framework of classical
0009-2614/90/$03.50 0 1990 - Elsevier Science Publishers B.V. (North-Holland)
393
Volume 173,number 4
CHEMICALPHYSICS LETTERS
MD simulations. It was found not to affect the main, semiquantitative aspects of the dynamics. In view of the large mass separation between H and the heavier atoms in this system, it is expected that the main quantum effects will be associated with the H atom. We estimate that already around 5 K (the temperature of the simulations) the behavior of Xe and I in processes of the type considered here is nearly classical. We therefore adopt a method that describes the H atom by time-dependent wavepackets, and the Xe and I atoms by classical trajectories - mixed quantal/classical molecular dynamics [ 14-l 71. This approach was used in several recent studies of processes somewhat related to the present topic [ 14- 17] , The coupling between the quantum and classical degrees of freedom in this approach is introduced by means of the mixed quantum/classical version of the time-dependent self-consistent field (TDSCF) approximation [ 14,161. Strong support for the validity of this approximation is that the method was tested very recently against exact quantum mechanical calculations for photodissociation of HI in the collinear cluster XeHI, and found to be of excellent accuracy [ 181. The scheme is as follows. Let V(r, R) be the potential function of the system, r the H atom position vector, R a collective label for the heavy particle coordinates. If R( t) is the trajectory function of the heavy particles, then VQ(r, t) = V(r, R(t) ) is used as the time-dependent potential for the H atom motion. The time evolution of the H atom wavefunction is determined from iA
Wr, t)
~=[T,+V,(r,f)lyl(r,t), at
(1)
where T, is the H atom kinetic energy operator. As the potential for the motions of the classical particles the method employs the mean field: V,(R,t)=(~(r,1)IY(r,R)ldu(r,t)).
(2)
Eq. ( I ) for t,~(r, I), and Newton’s equations of motion for R(t), with the potential of eq. (2), must be solved self-consistently, In solving eq. (1) we employed the following algorithm, which proved very efficient. v(r, t) was expanded in a basis set, that consists of two types of functions g, (r, t ) and &(r): vtr, t)=
1 a;(t)g,(r,
f) + C
bj(O$ji(r)
.
(3)
The gi( r, t) are basis functions that move with time. 394
12October 1990
Specifically, each g,(r, t) was modelled as a Gaussian wavepacket, the center of which moves according to a classical trajectory ri( t) for the H atom. The classicaltrajectories were obtained from classicalMD simulations of the system. The gi( r, t) thus span the wavefunction of the H in the classically allowed region. The 4,(r) are time-independent basis states, chosen to efficiently span the classically forbidden regions of w( r, t). They are practically most important in representing the events in which the H atom tunnels from its initial site to some final site, following photodissociation. The @j(r) were thus taken as localized functions along the tunneling paths. The Schrijdinger equation ( 1) is solved numerically for the unknown coefficients ai( bj( t) of both the moving and the “static” basis states. The difficulties from the nonorthogonality of the basis states did not prove a major obstacle in this work. Thepower ofthe algorithm is due to: (I) the ef$ciency with which the moving part of the basis set spans the classically allowed part of configuration space (an advantage stemmingfrom the fact that the gi(V,t) move with the classical trajectoriesfor the same system); (2) the fact that the tunneling paths for this system occur along narrow “tubes” in conjig-urationspace, and can be easily covered with a few basis states #,{qjin the classically forbidden region. Details of the method, and aspects of its convergence, etc., will be published elsewhere [ 191. The use of moving basis sets has been proposed by Heller [ 201 and developed by several authors [21-241. It should be noted that the present, still restricted version of the method cannot allow for wavepacket spreading. Each Gaussian represents now a peak of the wavepacket, and with the basis used here, there is no way for each peak to “broaden” or to get “narrower”. At long times this could affect the validity of eq. (2). This is not a problem in the present case because of the short time scale. This weakness can, however, be easily repaired by extending the basis to cope with the problem more generally. Around each center (trajectory) one should put several Gaussians of different widths, each with an independent coefficient (that varies with time). This should work in principle since Gaussians having the same center but different widths are linearly independent. In the system studied, the HI center-of-mass occupies before photolysis a substantial site in the fee
CHEMICAL PHYSICS LETTERS
Volume 173, number 4
Xe lattice. A slab of 108 Xe atoms (with the HI in it) was taken to represent the solid, periodic boundary conditions imposed at the ends. With such a slab, the classical MD simulations of the same system were converged with respect to size [ 8 1. Before photodissociation, the system was propagated on the ground state potential surface, the classical particles equilibrated to a temperature of 5 K, and the H vibrations described by their zero-point motions. Photolysis was assumed to occur by an extremely fast pulse: The wavefunction of the H atom was then used as an initial state for propagation on the excited potential surface, likewise the coordinates of the classical particles were integrated on the new potential from the initial values at the instant of the pulse (by the Franck-Condon principle). The calculations reported here are for photodissociation (photon) energy of 4.5 eV (corresponding to translational energy of 1.5 eV for H produced from HI). The results can be summarized as follows: For this excess energy, at very low temperature ( w 2 K), with extensive initial site sampling, we found that classically there is essentially zero probability for the separation of the photoproducts, and for exit of the H atom from the reagent case. At the same excess energy and temperature, the quantum result is very different. Fig. 1 shows P(t), the probability of finding H outside the reagent case. The build-up of probability with time is appreciable, and translated into macroscopic terms predicts a measurable yield of H and I separation. The mixed quantum/classical MD simulations thus predict a purely quantum, tunneling photoseparation process in these conditions. Fig. 2
lime (psec) Fig. 2. Kinetic energy of H atom after photolysis. The solid and dashed lines are the quantum and classical results respectively.
shows the kinetic energy of the H atoms that remain in the cage, comparing classical MD with the mixed quantum/classical MD scheme. The initial rapid fluctuations in the kinetic energy are due to the collisions of the H with the surrounding walls. Initially, the classical and quantum results are semiquantitatively in accord. For t> 0.25 ps, a large difference is seen, the energy of the quantum calculations reaching a limit larger by m0.15 eV than the classical result. This difference was found to be dw to the zeropoint motion of the H atom in its site, after it underwent energy relaxation by collisions with the Xe atoms. The “violation” of the zero-point motion by the classical calculation leads to a difference that is of major importance for a system of this type. This should imply e.g. very large differences between the classical and the quantum calculations for the vibrational spectrum of the H atom. Finally, we consider fig. 3, which shows the time dependence of I(y(r, t))y( r, 0) ) 1, the overlap between the H
I
N -
i
A 0.8 oc 5 0.5 g v
i
0.3
t
0
-. 0
1
2
3
4
5
6
time (psec) Fig. 1.Theprobability for separation of H and I (cage exit of H) versus time.
0
hA-a 0.1
0.2 0.3 time (psec)
0.4
Fig. 3. Overlap of hydrogen wavepacket with initial state versus time.
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Volume 173, number 4
CHEMICAL PHYSICS LETTERS
wavepacket al time t and the initial state. The results show several sharp peaks for times t< 0.25 ps. These are due to phase-coherent vibrations of the H between the surrounding Xe walls. These phase-coherent vibrations decay with time, because of dephazing and energy relaxation due to the H/Xc collisions. The large mass difference between H and the Xe (or I) atoms brings about a relatively late decay of the phase coherence. This result could be of considerable interest for ultrafast pulse spectroscopy in such systems. In conclusion, photodissociation of hydrides at low temperature solids and with relatively low excess energy is subject to very large quantum effects. Indeed, the separation of the photoproducts can be purely by tunneling as found in the case studied here. Zeropoint effects associated with the motion of the light product can also be very large. Calculations that incorporate quantum effects are thus essential for physically correct descriptions of such processes. This research was supported by Grant No. 8& 00082 of the US-Israel Binational Science Foundation. RBG gratefully acknowledges support in the framework of the Saerree K. and Louis P. Fiedler Chair in Chemistry at The Hebrew University. The Fritz Haber Research Center is supported by the Minerva Gesellschaft ftir die Forschung, mbH, Munich, FRG. References [ 11V.E. Bondybey and L.E. Brus, J. Chem. Phys. 62 (1975) 620; L.E. Brus and V.E. Bondybey, 1. Chem. Phys. 65 (1976) 71.
396
12 October 1990
[2 JH. Frei and G.C. Pimentel, J. Chem. Phys. 78 ( 1983) 3698. [ 31 M.E. Fajardo and V.A. Apkarian, J. Chem. Phys. 85 (1986) 5660:Chem. Phys. Letters 134 (1987) 51. [4] M.E. Fajardo, R. Whitnall, J. Feld, W. Lawrence, L. Weideman and V.A. Apkarian, Laser Chem. 9 (1988) I. [ 5 ] M. Mandich, P. Beeken and G.W. Flynn, 5. Chem. Phys. 77 (1982) 702. [ 61 W. Lawrence, F. Okada and V.A. Apkatian, Chem. Phys. Letters I50 (1988) 339. [ 71 R. Schriver, M. Chergui, H. Kuntz, V. Stepanenko and N. Schwentner, J. Chem. Phys. 91 (1989) 4128. [8] R. Alimi, R.B. Gerber and V.A. Apkarian, J. Chem. Phys. 89 (1988) 174. [9] R.B. Gerber, R. Alimi and V.A. Apkarian, Chem. Phys. Letters 158 (1989) 257. [lo] R. Alimi, A. Brokman and R.B. Gerber, J. Chem. Phys. 9 I (1989) 1611. [ 111 R. Alimi, R.B. Gerber and V.A. Apkarian, J. Chem. Phys., in press. [ 121 M.P. Allen and D.J. Tildesley, Computer simulations of liquids (Clarendon Press, Oxford, 1987). [ 131J.T. Hynes, Ann. Rev. Phys. Chem. 36 ( 1985) 573. [ 141R.B. Gerber, V. Buch and M.A. Ratner, 1. Chem. Phys. 77 ( 1982) 3022. [ 151D. Thirmulai, E.J. Bruskin and B.J. Berne, J. Chem. Phys. 83 (1985) 230. [ 161R.B. Gerber, R. Kosloff and M. Berman, Computer Phys. Rept. 5 (1986) 61. [ 171R.N. Bamet, U. Landman and A. Nitzan, J. Chem. Phys. 89 (1988) 2242. [ 181R. Alimi, R.B. Gerber, A.D. Hammerich, R. Kosloff and M.A. Ratner, J. Chem. Phys., submitted for publication. [ 191R. Alimi and R.B. Gerber, to be published. [20] E. Heller, J. Chem. Phys. 62 (1975) 1544; S.Y. Lee and E. Heller, J. Chem. Phys. 76 (1982) 3035. [21] R. Heather and H. Metiu, J. Chem. Phys. 84 ( 1986) 3250. [22] R.D. Coalson and M. Karplus, Chem. Phys. Letters 90 (1982) 301. [23] K. Kay, to be published. [24] I. Kucarand H.-D. Meyer, J. Chem. Phys. 90 ( 1989) 5566.
CHEMICAL PHYSICS LETTERS
12 October 1990
Quantum effects in molecular reaction dynamics in solids: photodissociation of HI in solid Xe R.B. Gerber ‘sband R. Alimi a ’ Department of Physical Chemistry and The Fritz Haber Research Center for Molecular Dynamics, The Hebrew University, Jerusalem 9 I904, Israel ’ Department of Chemistry, Universityo/California. Irvine. CA 9.2717, USA
Received 2 1 May 1994 in final form 24 July 1990
The photodissociation of a molecular HI impurity in a host Xe crystal is studied theoretically with focus on the role of quantum effects. The method used treats the H atom by timedependent wavepackets, the heavy atoms classically. Comparison is made with fully classical molecular dynamics simulations. Important differences are found between the quantum and the classical behavior, due to tunneling and zero-point energy effects for the H atom. The results demonstrate the desirability of using mixed quantum/classical molecular dynamics methods in simulating chemical processes in condensed matter at low temperatures.
The experimental study of molecular reactions in low-temperature rare-gas solids has been a topic of considerable activity in recent years [l-7]. This is part of the motivation for the theoretical interest in these systems, which are appealing also because they offer a framework for studying chemical dynamics in solvents of relatively simple structure and of known interaction potentials. These advantages are important in search for first-principles understanding of molecular reaction dynamics in condensed phases. There have been several theoretical studies of simple photochemical reactions of molecular impurities in rare-gas solids [g-l 11. The method employed in these studies were many-particle classical trajectory (molecular dynamics) simulations [ 121. That indeed has also been the main theoretical tool so far for gaining insight into the microscopic details of chemical processes in liquids [ 131. This raises the important question as to what is the role of quantum effects in such processes. Such effects are expected to be specially important at low temperatures, and should in general be most dramatic for processes in which very light atoms are involved. The present paper explores the nature and role of quantum effects in photochemical reaction in a low temperature solid in which a hydrogen atom is involved. The specific process studied is photodissociation of an HI im-
purity in a Xe host crystal. This system was chosen because it has already been the topic of experimental work [ 61, and because it was extensively studied theoretically by classical molecular dynamics simulations [8]. The process considered here is the photodissociation of HI caused by electronically exciting this molecule in its site within the host Xe crystal from the ground state ‘E+ to the purely repulsive excited state ‘II. Description of the process requires knowledge of the interaction potentials for both the ground and the excited electronic state. We assume that the potential function of the system, both before and after electronic excitation, can be modelled as a sum of pairwise interactions between the atoms of the system. The pair potentials for Xe-Xe, Xe-I, Xe-H and H-I were obtained by fitting gas-phase spectroscopic and scattering data, and are described in ref. [ 81. For the H-I pair interaction, the gas-phase ‘C+ and ‘II potential curves were used respectively for the ground and for the excited state. The crudest approximation in the potential functions used is the assumption that the electronic ground state interaction between each Xe atom and the I-IImolecule can be written as a sum of atom-atom potentials. We tested the consequences of this assumption, which simplifies the present calculations, in the framework of classical
0009-2614/90/$03.50 0 1990 - Elsevier Science Publishers B.V. (North-Holland)
393
Volume 173,number 4
CHEMICALPHYSICS LETTERS
MD simulations. It was found not to affect the main, semiquantitative aspects of the dynamics. In view of the large mass separation between H and the heavier atoms in this system, it is expected that the main quantum effects will be associated with the H atom. We estimate that already around 5 K (the temperature of the simulations) the behavior of Xe and I in processes of the type considered here is nearly classical. We therefore adopt a method that describes the H atom by time-dependent wavepackets, and the Xe and I atoms by classical trajectories - mixed quantal/classical molecular dynamics [ 14-l 71. This approach was used in several recent studies of processes somewhat related to the present topic [ 14- 17] , The coupling between the quantum and classical degrees of freedom in this approach is introduced by means of the mixed quantum/classical version of the time-dependent self-consistent field (TDSCF) approximation [ 14,161. Strong support for the validity of this approximation is that the method was tested very recently against exact quantum mechanical calculations for photodissociation of HI in the collinear cluster XeHI, and found to be of excellent accuracy [ 181. The scheme is as follows. Let V(r, R) be the potential function of the system, r the H atom position vector, R a collective label for the heavy particle coordinates. If R( t) is the trajectory function of the heavy particles, then VQ(r, t) = V(r, R(t) ) is used as the time-dependent potential for the H atom motion. The time evolution of the H atom wavefunction is determined from iA
Wr, t)
~=[T,+V,(r,f)lyl(r,t), at
(1)
where T, is the H atom kinetic energy operator. As the potential for the motions of the classical particles the method employs the mean field: V,(R,t)=(~(r,1)IY(r,R)ldu(r,t)).
(2)
Eq. ( I ) for t,~(r, I), and Newton’s equations of motion for R(t), with the potential of eq. (2), must be solved self-consistently, In solving eq. (1) we employed the following algorithm, which proved very efficient. v(r, t) was expanded in a basis set, that consists of two types of functions g, (r, t ) and &(r): vtr, t)=
1 a;(t)g,(r,
f) + C
bj(O$ji(r)
.
(3)
The gi( r, t) are basis functions that move with time. 394
12October 1990
Specifically, each g,(r, t) was modelled as a Gaussian wavepacket, the center of which moves according to a classical trajectory ri( t) for the H atom. The classicaltrajectories were obtained from classicalMD simulations of the system. The gi( r, t) thus span the wavefunction of the H in the classically allowed region. The 4,(r) are time-independent basis states, chosen to efficiently span the classically forbidden regions of w( r, t). They are practically most important in representing the events in which the H atom tunnels from its initial site to some final site, following photodissociation. The @j(r) were thus taken as localized functions along the tunneling paths. The Schrijdinger equation ( 1) is solved numerically for the unknown coefficients ai( bj( t) of both the moving and the “static” basis states. The difficulties from the nonorthogonality of the basis states did not prove a major obstacle in this work. Thepower ofthe algorithm is due to: (I) the ef$ciency with which the moving part of the basis set spans the classically allowed part of configuration space (an advantage stemmingfrom the fact that the gi(V,t) move with the classical trajectoriesfor the same system); (2) the fact that the tunneling paths for this system occur along narrow “tubes” in conjig-urationspace, and can be easily covered with a few basis states #,{qjin the classically forbidden region. Details of the method, and aspects of its convergence, etc., will be published elsewhere [ 191. The use of moving basis sets has been proposed by Heller [ 201 and developed by several authors [21-241. It should be noted that the present, still restricted version of the method cannot allow for wavepacket spreading. Each Gaussian represents now a peak of the wavepacket, and with the basis used here, there is no way for each peak to “broaden” or to get “narrower”. At long times this could affect the validity of eq. (2). This is not a problem in the present case because of the short time scale. This weakness can, however, be easily repaired by extending the basis to cope with the problem more generally. Around each center (trajectory) one should put several Gaussians of different widths, each with an independent coefficient (that varies with time). This should work in principle since Gaussians having the same center but different widths are linearly independent. In the system studied, the HI center-of-mass occupies before photolysis a substantial site in the fee
CHEMICAL PHYSICS LETTERS
Volume 173, number 4
Xe lattice. A slab of 108 Xe atoms (with the HI in it) was taken to represent the solid, periodic boundary conditions imposed at the ends. With such a slab, the classical MD simulations of the same system were converged with respect to size [ 8 1. Before photodissociation, the system was propagated on the ground state potential surface, the classical particles equilibrated to a temperature of 5 K, and the H vibrations described by their zero-point motions. Photolysis was assumed to occur by an extremely fast pulse: The wavefunction of the H atom was then used as an initial state for propagation on the excited potential surface, likewise the coordinates of the classical particles were integrated on the new potential from the initial values at the instant of the pulse (by the Franck-Condon principle). The calculations reported here are for photodissociation (photon) energy of 4.5 eV (corresponding to translational energy of 1.5 eV for H produced from HI). The results can be summarized as follows: For this excess energy, at very low temperature ( w 2 K), with extensive initial site sampling, we found that classically there is essentially zero probability for the separation of the photoproducts, and for exit of the H atom from the reagent case. At the same excess energy and temperature, the quantum result is very different. Fig. 1 shows P(t), the probability of finding H outside the reagent case. The build-up of probability with time is appreciable, and translated into macroscopic terms predicts a measurable yield of H and I separation. The mixed quantum/classical MD simulations thus predict a purely quantum, tunneling photoseparation process in these conditions. Fig. 2
lime (psec) Fig. 2. Kinetic energy of H atom after photolysis. The solid and dashed lines are the quantum and classical results respectively.
shows the kinetic energy of the H atoms that remain in the cage, comparing classical MD with the mixed quantum/classical MD scheme. The initial rapid fluctuations in the kinetic energy are due to the collisions of the H with the surrounding walls. Initially, the classical and quantum results are semiquantitatively in accord. For t> 0.25 ps, a large difference is seen, the energy of the quantum calculations reaching a limit larger by m0.15 eV than the classical result. This difference was found to be dw to the zeropoint motion of the H atom in its site, after it underwent energy relaxation by collisions with the Xe atoms. The “violation” of the zero-point motion by the classical calculation leads to a difference that is of major importance for a system of this type. This should imply e.g. very large differences between the classical and the quantum calculations for the vibrational spectrum of the H atom. Finally, we consider fig. 3, which shows the time dependence of I(y(r, t))y( r, 0) ) 1, the overlap between the H
I
N -
i
A 0.8 oc 5 0.5 g v
i
0.3
t
0
-. 0
1
2
3
4
5
6
time (psec) Fig. 1.Theprobability for separation of H and I (cage exit of H) versus time.
0
hA-a 0.1
0.2 0.3 time (psec)
0.4
Fig. 3. Overlap of hydrogen wavepacket with initial state versus time.
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wavepacket al time t and the initial state. The results show several sharp peaks for times t< 0.25 ps. These are due to phase-coherent vibrations of the H between the surrounding Xe walls. These phase-coherent vibrations decay with time, because of dephazing and energy relaxation due to the H/Xc collisions. The large mass difference between H and the Xe (or I) atoms brings about a relatively late decay of the phase coherence. This result could be of considerable interest for ultrafast pulse spectroscopy in such systems. In conclusion, photodissociation of hydrides at low temperature solids and with relatively low excess energy is subject to very large quantum effects. Indeed, the separation of the photoproducts can be purely by tunneling as found in the case studied here. Zeropoint effects associated with the motion of the light product can also be very large. Calculations that incorporate quantum effects are thus essential for physically correct descriptions of such processes. This research was supported by Grant No. 8& 00082 of the US-Israel Binational Science Foundation. RBG gratefully acknowledges support in the framework of the Saerree K. and Louis P. Fiedler Chair in Chemistry at The Hebrew University. The Fritz Haber Research Center is supported by the Minerva Gesellschaft ftir die Forschung, mbH, Munich, FRG. References [ 11V.E. Bondybey and L.E. Brus, J. Chem. Phys. 62 (1975) 620; L.E. Brus and V.E. Bondybey, 1. Chem. Phys. 65 (1976) 71.
396
12 October 1990
[2 JH. Frei and G.C. Pimentel, J. Chem. Phys. 78 ( 1983) 3698. [ 31 M.E. Fajardo and V.A. Apkarian, J. Chem. Phys. 85 (1986) 5660:Chem. Phys. Letters 134 (1987) 51. [4] M.E. Fajardo, R. Whitnall, J. Feld, W. Lawrence, L. Weideman and V.A. Apkarian, Laser Chem. 9 (1988) I. [ 5 ] M. Mandich, P. Beeken and G.W. Flynn, 5. Chem. Phys. 77 (1982) 702. [ 61 W. Lawrence, F. Okada and V.A. Apkatian, Chem. Phys. Letters I50 (1988) 339. [ 71 R. Schriver, M. Chergui, H. Kuntz, V. Stepanenko and N. Schwentner, J. Chem. Phys. 91 (1989) 4128. [8] R. Alimi, R.B. Gerber and V.A. Apkarian, J. Chem. Phys. 89 (1988) 174. [9] R.B. Gerber, R. Alimi and V.A. Apkarian, Chem. Phys. Letters 158 (1989) 257. [lo] R. Alimi, A. Brokman and R.B. Gerber, J. Chem. Phys. 9 I (1989) 1611. [ 111 R. Alimi, R.B. Gerber and V.A. Apkarian, J. Chem. Phys., in press. [ 121 M.P. Allen and D.J. Tildesley, Computer simulations of liquids (Clarendon Press, Oxford, 1987). [ 131J.T. Hynes, Ann. Rev. Phys. Chem. 36 ( 1985) 573. [ 141R.B. Gerber, V. Buch and M.A. Ratner, 1. Chem. Phys. 77 ( 1982) 3022. [ 151D. Thirmulai, E.J. Bruskin and B.J. Berne, J. Chem. Phys. 83 (1985) 230. [ 161R.B. Gerber, R. Kosloff and M. Berman, Computer Phys. Rept. 5 (1986) 61. [ 171R.N. Bamet, U. Landman and A. Nitzan, J. Chem. Phys. 89 (1988) 2242. [ 181R. Alimi, R.B. Gerber, A.D. Hammerich, R. Kosloff and M.A. Ratner, J. Chem. Phys., submitted for publication. [ 191R. Alimi and R.B. Gerber, to be published. [20] E. Heller, J. Chem. Phys. 62 (1975) 1544; S.Y. Lee and E. Heller, J. Chem. Phys. 76 (1982) 3035. [21] R. Heather and H. Metiu, J. Chem. Phys. 84 ( 1986) 3250. [22] R.D. Coalson and M. Karplus, Chem. Phys. Letters 90 (1982) 301. [23] K. Kay, to be published. [24] I. Kucarand H.-D. Meyer, J. Chem. Phys. 90 ( 1989) 5566.