Model studies of nonadiabatic dynamics

Transcription

1 JOURNAL OF CHEMICAL PHYSICS VOLUME 109, NUMBER SEPTEMBER 1998 Moel stuies of nonaiabatic ynamics Daniela Kohen a) Lucent Technologies, Bell Laboratories, Murray Hill, New Jersey Frank H. Stillinger Lucent Technologies, Bell Laboratories, Murray Hill, New Jersey an Princeton Materials Institute, Princeton University, Princeton, New Jersey John C. Tully Chemistry Department, Yale University, New Haven, Connecticut Receive 23 January 1998; accepte 19 June 1998 Mixe quantum-classical methos are applie to an increasingly challenging series of moel problems, an their accuracy is examine. The moels involve one light an one heavy egree of freeom, an exhibit substantial nonaiabatic behavior. In all of the moels the coupling between the light an heavy particles is linear harmonic. In aition, ifferent external potentials are applie to the heavy particle only. The energies of the light particle quantum states, as a function of the position of the heavy particle, efine a sequence of groun an excite Born Oppenheimer potential energy curves. Because the light particle experiences a purely harmonic potential, the potential energy curves are parallel an equally space for all of the moels. In aition, the nonaiabatic couplings among potential energy curves persist for all times ue to the nonvanishing linear coupling between light an heavy particles. The moel problems were use to test two strategies for carrying out mixe quantum-classical ynamics in systems involving nonaiabatic transitions: mean fiel an surface hopping. The moel calculations reporte here suggest that, in cases where linear couplings ominate, the mean fiel mixe quantum-classical metho isplays useful accuracy an is robust to the introuction of anharmonic heavy-particle interactions. The moel calculations also reveal special situations in which the surface hopping approximation is inappropriate American Institute of Physics. S I. INTRODUCTION Many ynamical phenomena in physics an chemistry involve couple motions of particles with wiely iffering masses. Electronic an nuclear egrees of freeom, respectively, in single atoms an molecules, provie a vivi illustration, which the Born Oppenheimer separation scheme 1 exploits to a goo avantage. The transfer of light protons along hyrogen bons between pairs of heavy electronegative atoms e.g., N, O, F, Cl constitutes another class of examples. 2 The interaction of liqui 3 He or 4 He with highatomic-weight solis, both crystalline an amorphous, presents further illustrations. 3 The presence of extreme mass ratios invites the use of a mixe quantum an classical escription. 4 This is particularly appropriate for escribing the virtually classical motions of electron-bearing heavy atoms or molecules that form nonreactive insulating meia. That is precisely the basis of the molecular ynamics simulation metho, for which the Born Oppenheimer groun-electronic-state energy surface provies the potential energy function for numerical integration of Newton s equations. 5,6 Assuming that electronic excitation remains absent, methos are available when require to generate quantum corrections to the strict classical ynamical escription of the heavier particles On the other a Current aress: Department of Chemistry, University of California, Irvine, Irvine, California han, there are methos that can accurately treat the electron ynamics even if electronic excitations occur but provie that their impact on the motion of the heavier atoms is negligible Difficulties begin to arise in those cases where electronic or other light particle excitations out of the groun state become significant an these, in turn, influence the behavior of the heavier particles. One encounters this situation, for example, in photochemical reactions involving nonraiative transitions, an in oxiation-reuction or proton transfer reactions in liqui solutions. Accurate escription of structure an ynamics in liqui metals an reactions at metal surfaces are also problematic, since a continuum of excite conuction electrons must be involve. The intrinsic power of molecular ynamics computer simulations to illuminate physical an chemical phenomena has lent consierable weight to extening its traitional classical version as escribe above, to cover cases with electronic excitation. Inee several approximation schemes for realizing this goal treating the quantum an classical egrees of freeom self-consistently have been propose for reviews, see Refs. 15 an 16. These inclue both mean fiel an surface hopping approaches. But in spite of an extene effort no universally acceptable approximation strategy has yet emerge. The present paper reports results of a moest stuy intene to contribute some further unerstaning to this active research area. In particular, our /98/109(12)/4713/13/$ American Institute of Physics

2 4714 J. Chem. Phys., Vol. 109, No. 12, 22 September 1998 Kohen, Stillinger, an Tully efforts coul be regare as an attempt to unerstan to what egree the use of some of these methos is appropriate when the evolution of a heavy plus light particle system is stuie using the particle coorinates irectly rather than some collective coorinates. For simple moel problems it woul be more appropriate to base a quantum-classical separation on normal moe coorinates in orer to minimize the effect of the quantum coorinates on the classical ones, an vice versa. In realistic many particle simulations a normal moe separation is not practical, however, an the bare particle must generally be use. This introuces coupling between classical an quantum coorinates, even in harmonic systems, an this coupling can be problematical for mixe quantum-classical ynamics. The simplest moel potentials that represent this situation are those of linearly couple harmonic an anharmonic oscillators. In this context nonaiabatic transitions may occur between the aiabatic Born Oppenheimer light-mass states simply because the bare particle coorinates o not correspon to the normal moes of the system. The following section argues in favor of stuying simple tractable moels of nonaiabatic effects, an introuces three such moels, each involving a couple pair of particles moving in one imension uner the influence of an external potential. Section III provies computational etails about the respective ynamical investigations for each of these three moels. Section IV presents our numerical results for the three. The final Sec. V iscusses the implications of our results for future theoretical an simulational activity. Let n (x,x) an E n be the exact eigenfunctions an eigenvalues for the time-inepenent wave equation corresponing to the Hamiltonian H in operator form, H n x,xe n n x,x subject to given bounary conitions. The general timeepenent solution to this quantum-mechanical problem has the form 2 x,x,t n A n exp i E nt n x,x 3 with constant A n selecte to satisfy initial conitions. One must keep in min that this last equation must be moifie to inclue an appropriate integral over continuum states when they exist for a case uner consieration. The Born Oppenheimer BO approximation replaces the full quantum problem Eq. 2 by a sequential pair of single-particle problems. The first examines the light particle motion in the presence of a stationary heavy particle i.e., fixe at position X, H 0 xx j xx j 0 X j xx, where H (0) (xx) simply contains X as a parameter, H 0 xx p2 2m vx,x. The secon utilizes j (0) (X) as a supplement to V(X) for motion of the heavy particle, 4 5 II. SIMPLE MODELS H 1 j X l, j X 1 l, j l, j X, where now 6 Each of the three cases to be examine involves a pair of spinless particles confine to move in one imension only. The lighter particle of the pair mass m, position x, momentum p interacts with the heavier particle mass M, position X, momentum P via potential v(x,x). At the same time the heavier particle experiences an external potential V(X). The two-particle Hamiltonian consequently has the form H p2 P2 2m 2M vx,xvx. The three cases to be analyze are istinguishe by the corresponing assignments of v an V, to be specifie below. Reliance on one-imensional moels to illuminate threeimensional phenomena obviously must be viewe with caution. Nevertheless this strategy offers some avantages. In particular, some one-imensional moels can be solve analytically in close form see moel I below, while others that are not fully solvable may still permit a more thorough analysis than o their more complicate three-imensions counterparts. Furthermore, various contributing effects in one imension can usually be easily isolate from moel to moel, thereby simplifying the task of interpretation. We view the stuy of nonaiabatic effects in the oneimensional context as a helpful precursor to the evelopment of theoretical techniques that are broaly applicable to three-imensional systems. 1 H j 1 X P2 2M VX j 0 X. The outcomes of this sequential approximation are estimates for the exact eigenfunctions n x,x j xx l, j X an for the exact eigenvalues 1 E n l, j 9 for the full two-particle Hamiltonian H. In principle these estimates coul be inserte into Eq. 3 to approximate time epenence of the two-particle ynamics. For all three moels to be examine, the interaction v(x,x) was assume to epen only on relative separation x X, an in particular was chosen to be harmonic, vx,x 1 2kx X Consequently the light an heavy particles are boun together, forming a kin of harmonic hyrogen atom. As a result of this extra simplification, the first stage of the BO sequential approximation, Eqs. 4 5, leas to the explicit results 43 j xxn j exp 1 2s 2 H j s, 0 j j 1 2 0, 12 11

3 J. Chem. Phys., Vol. 109, No. 12, 22 September 1998 Kohen, Stillinger, an Tully 4715 where H j is the Hermite polynomial, N j is the normalization constant, smk/ 2 1/4 xx, 13 an (0) is the harmonic frequency for the light particle oscillating about the fixe heavy particle, (0) (k/m) 1/2. Note that m is larger than the reuce mass for the pair of particles, mm/(mm); consequently (0) is smaller than (k/) 1/2, the natural oscillation frequency for the pair in free space. Our three moels are now istinguishe by the following V(X) choices: x,x,t0n j xxexp i k 0X XX Here N is a normalizing constant, X 0 the initial mean position of the heavy particle, k 0 is the initial momentum, an the with parameter is assigne values typically aroun 20 times the inverse of the initial momentum, unless otherwise note. The BO functions j l, j form a complete set for all three moels, an coul be use for propagation in time. Inee one of our objectives is to monitor time epenence of the light particle quantum-state probabilities VX 1 2KX 2, moel I, VXexp2X2 expx, moel II, 14 where a i t j c* j,i tc j,i t, 16 VXAX 4 BX 2, moel III. The first is fully harmonic, an therefore fully solvable by separation into inepenent normal moes. The secon involves a Morse oscillator potential for the heavy particle, an can be viewe as a one-imensional analog of an atomsubstrate bining potential of the type often encountere in surface physics an chemistry. 44 The thir introuces quartic anharmonicity, an if A0, B0 it can represent bistable motion. Because the (0), Eq. 12, are equally space an inepenent of X for the harmonic v choice, Eq. 10, the aiabatic potential surfaces for motion of the heavy particle are parallel an equally space for all three moels. Note that sets of parallel potentials also originate when the system is chosen such that in the iabatic representation it is a triiagonal Hamiltonian matrix where all the iagonal terms are ientical functions of the classical coorinates an the offiagonal couplings are properly chosen constants, all inepenent of the classical subsystem coorinate. III. SOLUTION PROCEDURE Our objective for each of the three moels is to etermine the precise quantum evolution of an initial wave packet state, an to compare these results with preictions of two specific approximation schemes Ehrenfest mean fiel an fewest switches surface hopping 35. The initial state is selecte to conform to the BO picture reflecting the intention to use the heavy an light particle coorinates as the relevant coorinates. For the moel potentials that are explore below the use of approximately separable linear combinations of the bare coorinates might simplify our efforts. But espite this possibility this alternative was rejecte in orer to retain generality, given that as the complexity of the system increases it becomes more ifficult to ientify such combinations. In particular, the light particle is place in one of the H (0) eigenstates, j, an the heavy particle in a translating Gaussian wave packet. Consequently, we set c j,i t X * j,i X x i *xxx,x,t 17 that inicate transitions between the aiabatic potential surfaces. In aition to following the a i (t) we have foun it useful to examine the time epenence of the heavy particle mean position an momentum, Xtx X *txt, Ptx X *tpt We present these results below as phase space iagrams, i.e., as curves in the X, P plane. Moel I has the avantage that the fully quantum problem is solvable in terms of inepenent normal moes see Appenix. This is use when examining the system time evolution. First, the trivial time evolution in the normal moes basis set is calculate an then transforme into the BO basis set to obtain the a i mentione above. Note that these normal moe quantum states iffer from the BO states by an amount epenent on the mass ratio m/m; as the ratio goes to zero the two representations converge to one another. However when the ratio is nonzero the chosen initial state Eq. 15 cannot be a proper normal moe eigenstate, so that the subsequent evolution inevitably involves transitions between the BO surfaces. Therefore this moel is a very simple case to stuy the extent to which the mixe quantumclassical approximation schemes uner consieration are suitable to represent the system evolution using the particle coorinates explicitly. To calculate the full quantum evolution for moels II an III fast Fourier transform techniques are use. The proceure use to solve the two-imensional time epenent Schröinger equation is that of Kosloff an Kosloff 45 generalize to n light particle states the n accessible aiabatic states. Tooso(x,X,t) is expane in the j (xx) basis; it becomes

4 4716 J. Chem. Phys., Vol. 109, No. 12, 22 September 1998 Kohen, Stillinger, an Tully where 1 X,t x,x,t 0X,t 2 X,t... n X,t, 20 where the ot represents a time erivative an j,i x j i X. 27 It can be shown, base on conservation of energy arguments, that the effective force is then given by the following expression: i X,t x i *xxx,x,t 21 is a wave function evolving on V i (X)V(X) i (0) (X), an the Hamiltonian becomes a noniagonal matrix, H i, j X x i *xxhx,x j xx. 22 Thus the two-imensional time evolution problem has been transforme into n-couple one-imensional time epenent Schröinger equation. Note that the coupling occurs because the BO basis functions, j (xx), are not eigenvectors of the P operator inclue in H see Eq. 1. Asthem/Mratio goes to zero the coupling vanishes, an as in the case of moel I transitions cease to occur. The precise full quantum results will be compare with results generate by the two approximate methos mentione above, Ehrenfest mean fiel an the fewest switches surface hopping. These are two well-establishe approximation schemes that are esigne to take avantage of the simplicity of classical ynamics for the heavier particle, while presuming to account for nonaiabatic quantum effects of the lighter particle in a self-consistent manner. These methos are escribe below. The Ehrenfest mean fiel approximation postulates classical ynamics for the heavier particle, subjecte to an effective force F eff efine as F eff X V x *H 0, 23 where (xxt), the light particle wave function in the presence of the classically moving heavy particle, evolves accoring to the following time epenent Schröinger equation, i t xxth0 x,xtxxt. 24 These equations efine the metho an can be use irectly. Alternatively the time-epenent solution for the motion of the light particle can formally be expresse xxt j c j t j xx, 25 F eff V X j V X j c j 0 2 j X j Ẋ c j 0 2 j X 0 j c j 2 j,lj c j *c l c l *c j j 0 l 0 j,l. t The first term on the right-han sie in this last equation is the force ue to the external potential acting irectly on the heavy particle, the secon is an aiabatic contribution to F eff, while the thir term comprises nonaiabatic effects light particle quantum number changes. Observe that the effective force interpolates among the forces efine separately on each of the accessible states. Fewest switches surface hopping is an intrinsically stochastic metho in which the heavier particle executes finitetime-interval classical motion on istinct surfaces. Equation 26, the time epenent Schröinger equation for the light particle (xxt) in terms of the light particle aiabatic basis set, is also use here, but the motion of the classical heavy particle evolves on only one aiabatic potential surface at each instant of time, with instantaneous hopping between states accoring to the fewest switches algorithm. Thus, in contrast to the metho escribe above, the heavier particle is subjecte to a force that is simply F X V j 0 30 except for occasional elta-function impulses require to conserve total energy when hops occur. Note that this metho, unlike the mean-fiel metho, is efine upon a basis set which in this case is chosen to be the light particle aiabatic basis set, j (xx). The fewest switches algorithm minimizes the number of state switches while maintaining a statistical istribution in an ensemble of trajectories that closely reprouces the light particle state populations. To o so, instantaneous switching probabilities from the occupie level i to all the others states j uring the time interval t to t are calculate as using the previously introuce light particle BO basis functions, j (xx). These c j s are the coefficients that will be use to compute the light particle quantum-state probabilities that will be compare with the results of the fully quantum calculations. Clearly, the equations for the c j s time evolution can be obtaine using Eq. 24, g k, j b j,k c k 2, where b is efine by see Eq. 26 c j 2 t k j b j,k iċ j j 0 c j iẋ k j,k c k, 26 2Ẋ kj Rec j *c k j,k, 33

5 J. Chem. Phys., Vol. 109, No. 12, 22 September 1998 Kohen, Stillinger, an Tully 4717 an compare to a uniform ranom number, between 0 an 1 to etermine if a switch is to occur. For example, if j1, a switch to state 2 will occur if g 1,2. A switch to state 3 will occur if g 1,2 g 1,2 g 1,3, etc. In aition, for a hop to occur the kinetic energy of the classical particle must be large enough to compensate for the loss in potential energy that might be involve in the transition. To unerstan how the algorithm works note that if the system has only two levels the switching probability is the rate at which the quantum light particle population probability is increasing on the surface where the classical particle might hop. In the absence of forbien hops, i.e., the appearance of nonzero amplitues in states that are energy forbien, the fewest switches algorithm statistically partitions trajectories correctly among ifferent potential energy surfaces accoring to the probabilities c j 2. The treatment of forbien hops is iscusse below. The approximate mixe quantum-classical methos involve solving both the time epenent Schröinger equation 26 an the Newton equations. Both were solve numerically using the Runga Kutta Gill metho. 46 IV. NUMERICAL RESULTS A. Moel I: Harmonic oscillator This moel consists of harmonic potentials in both the heavier an lighter particle coorinates, with a coupling that coul be viewe as bilinear expaning the square of (xx), Hx,X p2 P2 2m 2M 1 2 kxx2 1 2 KX2. 34 In this section some representative results for this moel will be escribe. As mentione above the evolution of a system uner this Hamiltonian can be etermine trivially by transforming the problem into its separable normal moes, see Appenix. This will be use to calculate the exact fully quantum evolution of the system. But we reiterate that the motions of the heavy an light particles will be viewe as such, rawing upon the Born Oppenheimer representation. Figure 1a isplays the potential surfaces, the BO eigenvalues plus the heavy particle external potential see Eq. 12, V n X 1 2 KX 2 n 1 2k/m, V n X X 2 n These equations inicate choices for some of the parameters use in our numerical calculations. In aition the masses m1 an M10 were assigne, an for convenience we use atomic units. The with parameter was chosen to be 0.15 in Eq. 15 an is such that the results o not significantly change when is smaller. Initially the groun state is the only populate aiabatic state, an X 0 an k 0 the large ot in the Fig. 1a are the initial conitions of the heavy particle in the mixe quantum-classical methos or the center of the wave packet in the fully quantum calculations. The ratio of masses is such that this example falls clearly in a regime were the BO approximation shoul not hol an transitions between the levels must occur. The separation of the parallel aiabatic states assures that when a transition occurs the amount of energy transferre from the quantum egree of freeom to the classical one affects the ynamics of the heavy particle. This problem then may represent a challenge to any of the approximation methos. First we concentrate on the mean fiel approximation. Figure 1b shows the phase space iagram. The soli line correspons to the solution of the fully quantum problem X is plotte vs P. The mean fiel results X is plotte vs P are essentially ientical to the fully quantum results. The otte line is the quantum aiabatic approximation solution, i.e., no transitions X is plotte vs P. Thus exact quantum an mean fiel versions are ientical, even though as shown by the contrast with the aiabatic approximation, transitions occur to a great extent. This result can be unerstoo with the ai of the Ehrenfest theorem 47 which states that for any Hamiltonian of the form Hq p q 2 2m Vq, the mean values q an p q are given by t q p q m ; t p q V q When applie to the present completely harmonic moel, Eq. 34, we obtain t t x p m ; X P M ; t pkxx; t PkKXkx Alternatively, invoking the mean fiel approximation for the heavy particle motion, an then taking the classical limit, prouces the following Newton equations of motion: t X P M ; P t X 1 2 KX2 x *H 0 KX x * H0 X KkXkx, 41 where the fact that is the solution of the time epenent Schröinger equation involving H (0) has been applie to obtain the secon to the last equation. The ynamics of the quantum egree of freeom can be examine applying the Ehrenfest theorem to H (0) see Eq. 4, this leas to t x p m ; t pkxx, 42 where here as in the rest of the paper the represents integration over the quantum egrees of freeom. Note how the equations of motion, Eqs. 41 an 42, are ientical to the solution of the fully quantum problem, Eqs. 39 an 40 with X X an P P, explaining why the mean fiel approximation is exact while calculating the evolution of the heavy particle. This is true only because the coupling is bi-

6 4718 J. Chem. Phys., Vol. 109, No. 12, 22 September 1998 Kohen, Stillinger, an Tully FIG. 1. Numerical example for moel I: Harmonic oscillator. a Potential surfaces. The small circle inicates the initial position of the heavy particle in the mixe quantum-classical calculations or the center of the wave packet in the fully quantum calculations. b Mean fiel results: Phase space iagram. The soli line correspons to the solution of the fully quantum problem X is plotte vs P. For this case the mean fiel results X vs P are ientical to the fully quantum results. The otte line correspons to the quantum aiabatic approximation solution X is plotte vs P. The figure shows the first 20 time units of the evolution. c Mean fiel results: Evolution of populations. The soli line is the numerically exact groun state population an the broken line is the result of the mean fiel approximation, both are plotte as a function of time. Only the groun state population is shown for clarity although there is population transfer to the first an secon excite states. Surface hopping results: Phase space iagram. The soli line correspons to the solution of the fully quantum problem, the broken line to the surface hopping results average X is plotte vs average P an the otte line to the quantum aiabatic approximation solution. The first 20 time units of the evolution are shown. e Surface hopping results: Evolution of populations. The soli line is the numerically exact groun state population, the broken line the population calculate as the fraction of trajectories on each surface an the otte line the population calculate as the average over all the ifferent trajectories of c i 2. The last two are results of the surface hopping approximation. As in c only the groun state population is shown for clarity. linear an the potentials quaratic in both coorinates, otherwise eviations occur as will be seen below. As a curiosity it is har to envision a practical irect use of what follows note that if the role of x an X are interchange while applying the mean fiel metho the heavier particle is treate by quantum mechanics an the lighter by classical mechanics their evolution in phase space woul still be escribe exactly. Even for harmonic moels, the mean-fiel mixe quantum-classical metho is exact only in reproucing the first moments, X an P. Higher moments are not compute exactly. This is apparent when comparing the time epenence of the light particle quantum-state probabilities in Fig. 1c. While the curves resemble each other, they are not ientical. Overall the performance of the mean fiel metho is remarkably goo for moel I. The harmonic oscillator problem is peculiar an therefore it is interesting to stuy to what

7 J. Chem. Phys., Vol. 109, No. 12, 22 September 1998 Kohen, Stillinger, an Tully 4719 egree this result survives when the harmonicity is somehow relaxe. Clearly, there are circumstances where the metho will eventually stop performing well. In particular, as has been escribe in etail elsewhere 15,35,42,48 this metho cannot escribe accurately the evolution of minority reaction channels if these channels involve potential energy surfaces that iffer consierably from the one experience by the majority channel. Note that this is precisely the class of problems that the metho of surface hopping was esigne to aress. 35 It woul be of little use to moify the potential just in orer to see the mean fiel approximation fail; it seems more informative instea to concentrate on a specific class of potentials. Here, as mentione earlier, we will focus on the class of potentials that are harmonic in the separation between x an X but let the V(X) change the potential energy surfaces are then constraine to be parallel an monitor the egree to which the given approximation hols. Now we focus on the performance of the secon approximation metho, the fewest switches surface hopping algorithm. The results shown correspon to averaging 1000 trajectories, all starting at (X 0,k 0 ) but with a ifferent stochastic history of hops. Figure 1 shows the resulting phase space iagrams. The ifferent lines have a similar meaning to those in Fig. 1b although here the classical quantities represent an average over all the trajectories. Note how the first two curves iffer significantly, especially at longer times. Figure 1e shows the population evolutions. The soli line is the exact result, the broken line the population calculate as the fraction of trajectories on each surface, an the otte line the population calculate as the average over all the ifferent trajectories of c i *c i. At early times, the populations compute by surface-hopping are very similar to the mean fiel results of Fig. 1c. At longer times, the results of surface hopping show ampe oscillations in the populations an an inwar spiraling in the phase space iagram. This can be attribute to the fact that ifferent trajectories exhibit a ifferent history of ranom hops. Over a perio of time this causes the ensemble to lose coherence an therefore sprea in phase space. A secon factor that can introuce error in surface hopping is associate with forbien hops. With the algorithm employe here, hops inuce by the evolving quantum populations are teste for energetic feasibility. In the present calculation the classical particles were starte at a turning point an immeiately quantum population began to evelop on the upper surface. On the other han some time elapses before the classical particle reaches a position where it is energetically possible for it to begin to evolve on the upper energy surface. As a consequence the fewest switches algorithm fails to statistically partition trajectories correctly among ifferent potential energy surfaces. This will not appear to be the major source of error in this case, however. A relate issue is that of the treatment of a forbien surface hop. It has been argue 49 that the velocity of the classical particle shoul be reverse after such an occurrence. More recently, Müller an Stock 48 have shown that it may be more accurate to continue motion in the same irection. The reason behin the unsatisfactory performance of the surface hopping algorithm for longer times is not relate to the precise nature of the potential in moel I. Whenever the potential involves a coupling that oes not vanish outsie a localize scattering region, surface hopping will introuce unphysical loss of coherence. In aition, this class of interactions also encourages the presence of regions where the hops are classically forbien but encourage by quantum mechanics. This situation escribes all of the moels stuie in this work; they involve heavy an light particles connecte by a spring an therefore couple for all times. The only ifference between them is the profile of the scattering region. This fact plus the explicit exclusion see earlier of the class of potentials were the surface hopping metho is known to outperform the mean fiel metho motivates us to focus on the performance of the mean fiel approximation. B. Moel II: Morse oscillator A In relaxing the harmonic restriction in the form of V(X) an obvious choice is that of the Morse oscillator, Hx,X p2 P2 2m 2M 1 2 kxx2 exp2x2 expx. 43 In this section some representative results for this moel will be escribe. Figure 2a isplays the potential surfaces, V n Xexp2X2 expx n 1 2k/m, V n X60exp0.6X2 exp0.3x n Also shown in the equation above is the choice of some of the parameters use in this calculation. As before, the masses are m1 an M10, an initially the groun aiabatic state is the only one populate in the lighter particle egree of freeom. The anharmonicity of the potential causes the quantum wave packet to eform, losing its Gaussian shape as time progresses, unlike moel I. Therefore, when using this mixe quantum-classical metho one trajectory is not sufficient. It is necessary to perform ynamics for a collection of trajectories iffering in the initial position an momentum of the heavy particle. To choose the initial conitions we have use a Wigner transform. 50 The Wigner transform carries a ensity operator in the coorinate representation (X,X) (X)(X) to phase space; X,P 1 YXYX,XXY exp2ipy/. 46 From this (X,P) the set of ifferent initial conitions for the collection of trajectories is chosen. The number of ifferent initial conitions is such that the results are substantially inepenent of it, an this number epens not only on the potential surface but also on the initial conitions an length of the ynamics.

8 4720 J. Chem. Phys., Vol. 109, No. 12, 22 September 1998 Kohen, Stillinger, an Tully FIG. 2. Numerical example for moel II: Morse potential A. a Potential surfaces. The initial wave packet, 0 (X,t0) see text, is also shown. b Phase space iagram. The soli line correspons to the solution of the fully quantum problem X is plotte vs P, the broken line to the mean fiel results ensemble average X vs ensemble average P an the otte line to the quantum aiabatic approximation solution X is plotte vs P. The figure shows the first 20 time units of the evolution. c Evolution of populations. The soli lines are the numerically exact populations an the broken lines are the result of the ensemble average mean fiel approximation, both are plotte as a function of time. Only the groun state population is shown for clarity although there is population transfer to the first an secon excite states. Figure 2b shows phase space iagrams. The soli line correspons to the solution of the fully quantum problem solve numerically as escribe in Sec. III, the broken line to the ensemble average mean fiel results an the otte line to the quantum aiabatic approximation solution. Figure 2c shows the time epenence of the light particle quantum-state probabilities. Even though the results are not ientical, the metho performance is still quite goo. We emphasize that this result, although representative, was obtaine keeping in min the goal of fining the limitations of the mean fiel metho for this class of potentials. Its success motivate the selection of moel III escribe in the next subsection. C. Moel III: Double well The mean fiel metho cannot properly escribe situations for which the wave packet bifurcates into two or more istinct paths governe by ifferent interactions. Moel III introuces quaratic anharmonicity that can represent bistable motion if A0 an B0, Hx,X p2 P2 2m 2M 1 2 kxx2 AX 4 BX In what follows some representative results for this moel will be escribe, an again the conitions were chosen as to maximize the chances of the mean fiel metho to fail. Figure 3a isplays the potential surfaces, V n XAX 4 BX 2 n 1 2k/m, V n XX 4 12X 2 n , that are accessible. As in the previous moels, the masses are m1 an M10. The initial state is the groun aiabatic state in the lighter particle egree of freeom; an the heavier particle egree of freeom is a Gaussian wave packet in the fully quantum calculation, while in the mixe classical-quantum calculations it is escribe by a collection of X 0 an k 0 the initial conitions for the set of trajectories chosen from a Wigner istribution as escribe in the previous subsection. Figure 3b shows phase space iagrams. The soli line correspons to the solution of the fully quantum problem solve numerically as escribe in Sec. III, the broken line to the ensemble average mean fiel results an the otte line to the quantum aiabatic approximation solution. Figure 3c shows the time epenence of the light particle quantum-state probabilities. Note that the phase space trajectory, Fig. 3b, spirals inwars approaching an unsymmetric position of x about 0.5. Quantum mechanically, this is ue to trapping of the fraction of the wave packet that has been excite to the n1 level. The initial conitions of the wave packet have been chosen so that there is sufficient energy for most of the wave packet to traverse the barrier in the groun (n0) state, but not in the first excite (n1). Tunneling through the excite state is sufficiently slow that the asymmetry in position persists over the time scale of the calculation. It is interesting to note that the mean fiel approximation reprouces this effect quite well, at least in an average way. There is a sprea of initial energies of the inepenent mean fiel trajectories, etermine from the Wigner istribution. For the conitions of Fig. 3, trajectories in the low energy tail of this istribution get trappe by the potential barrier slightly higher that the groun state barrier ue to mixing with the excite state. The high energy majority traverse the barrier. The phase space asymmetry in the meanfiel result of Fig. 3 is thus largely etermine by the with of the initial momentum istributions, not by the probability of excitation to the n1 state as in the fully quantum case.

9 J. Chem. Phys., Vol. 109, No. 12, 22 September 1998 Kohen, Stillinger, an Tully 4721 D. Moel II revisite: Morse oscillator B an C The potential surfaces involve in the first example in this subsection are isplaye in Fig. 4a, V n X10exp1.0X2 exp0.5x n FIG. 3. Numerical example for moel III: Double well. a Potential surfaces. The initial wave packet, 0 (X,t0) see text, is also shown. b Phase space iagram. Same as Fig. 2. c Evolution of populations. Same as Fig. 2. Nevertheless, on average, the mean fiel metho quite accurately reprouces the asymmetry of the full quantum calculation, at least for the initial conitions chosen here. Even though the results are not perfect the performance of the mean fiel metho is still quite goo, espite the fact that uring the evolution a small but significant portion of the population is in the excite state. Note that at the energy we selecte, the potential oes not permit exploring large regions of space. This may imply that even when the etaile evolution is not being escribe properly, the evolution of the average quantities that are being measure is accurate. Therefore, in the next section moel II will be re-examine but in a case were the evolution is not confine to a restricte region of space. Note that these iffer from the case shown above in Eq. 45. The initial conitions were chosen so that a small but nonnegligible fraction of population might be transiently trappe in the well of the first excite potential curve. The masses are m1 an M10, an the initial state is the groun aiabatic state in the lighter particle egree of freeom, an in the heavier particle egree of freeom it is a Gaussian wave packet in the fully quantum calculation, an in the mixe classical-quantum calculations it is escribe by a set of X 0 an k 0 as alreay state. The initial state in the heavier particle egree of freeom, 0 (X), is also shown in Fig. 4a, its baseline representing total energy of the system in this calculation. Note that outsie the well region the total energy is less than V 1 (X). Figure 4b shows phase space iagrams. The soli line correspons to the solution of the fully quantum problem X vs P, the broken line to the ensemble average mean fiel results, an the otte line to the quantum aiabatic approximation solution. Figure 4c shows the time epenence of the light particle quantum-state probabilities. Note the persistent oscillations clearly shown in the inserts isplaying the th time units in both the populations an in the phase space iagram, resulting from the fact that the coupling between the heavy an light particle oes not vanish when the compoun particle leaves the scattering region. The oscillations correspon to the perioic transfer of energy between the heavy an light particles, an far from the scattering region simply monitor the behavior of the heavier particle in the unperturbe composite particle. The amount of population transfer is rather small, espite our efforts to fin parameters where the mean fiel approximation was likely to fail. This surprising situation is cause by the fact that if the initial momentum is increase the size of the oscillations just escribe increase as well, thus masking any other effects. This as to the fact that increasing the magnitue of the coupling oes not have a significant net effect on the transition rates since k is also proportional to the separation of the levels an therefore inversely proportional to the transition probabilities. Despite this, the metho performance is still better than expecte even though the populations in the ifferent channels evolve in regions that are quite separate spatially. This puzzle can be solve by recognizing that population can be evolving in the upper surface but not be trappe an therefore not in a minority channel. Figures 4 an 4e show 0 (X,t) an 1 (X,t), respectively. The region enclose in the quarilateral figure in 4e is the only portion of the population effectively trappe, an this portion is almost nonexistent. Most of the population in the upper surface behaves as a ghost of the population on the groun state, the behavior of the former merely mirroring that of the

10 4722 J. Chem. Phys., Vol. 109, No. 12, 22 September 1998 Kohen, Stillinger, an Tully FIG. 4. Numerical example for moel II: Morse potential B. a Potential surfaces. The initial wave packet, 0 (X,t0) see text, is also shown. b Phase space iagram. Same as Fig. 2. c Evolution of populations. Same as Fig. 2. The inserts in b an c isplay the evolution between the 190 an 200 time units. Evolution of 0 (X,t). e Evolution of 1 (X,t). The region enclose in the quarilateral is the only portion of the population effectively trappe. latter. This is a consequence of the persistent coupling between the light an heavy particles. resent a collision with a har wall. To encourage transitions low an the inner wall is quite steep; therefore it might rep- Thus, to observe a significant failure the transition rate given the absence of a substantial well, the population is must be somehow increase. This may be accomplishe by initially in the first excite aiabatic state. An alternative increasing the slope of the inner wall an therefore increasing the match between the frequency of the bounce an that kinetic energy, but this woul have resulte in the appear- woul have been to give the heavier particle more initial of the oscillations. This creates a small fraction of the population with a significantly ifferent amount of kinetic energy This final choice of parameters prouces a bifurcation of ance of masking oscillations. than that of the majority, consequently evolving uner inappropriate forces. That is the case if the potential surfaces are the initial wave packet into two scattere portions of quite ifferent velocity. An average path cannot be expecte to V n X0.001exp20.0X2 exp10.0x properly escribe the iniviual pathways, so the mean fiel metho is expecte to break own. The surface hopping n metho was evelope in orer to hanle such bifurcation. as shown in Figs. 5a 5e the figures are equivalent to However, this specific example, in aition to proucing a those in Fig. 4. In this example the well is extremely shal- bifurcation, involves a coupling between light an heavy that

11 J. Chem. Phys., Vol. 109, No. 12, 22 September 1998 Kohen, Stillinger, an Tully 4723 FIG. 5. Numerical example for moel II: Morse oscillator C. a Potential surfaces. The initial wave packet, 1 (X,t0) see text, is also shown. b Phase space iagram. Same as in Fig. 2. c Evolution of population on the groun state. Evolution of population on the secon excite state. Note how when the mean fiel approximation is use the state to which most of the population is transferre is not the correct one. e Evolution of 0 (X,t). The region enclose in the quarilateral is the portion of the population evolving in the minority channel. f Evolution of 1 (X,t). never vanishes. It is likely, therefore, that surface-hopping woul also have ifficulty with this moel, at least reproucing the long time oscillatory behavior. V. DISCUSSION AND CONCLUSIONS We have examine the accuracy of mixe quantumclassical ynamics as applie to an increasingly challenging series of moel problems. The moels involve one light an one heavy egree of freeom, an exhibit substantial nonaiabatic behavior. In all of the moels the coupling between the light an heavy particles is linear harmonic. In aition, an external potential is applie to the heavy particle only. The moels iffer only in the magnitue of the coupling force constant, an the form an magnitue of the external potential. The energies of the light particle quantum states, as a function of the position of the heavy particle, efine a sequence of groun an excite Born Oppenheimer potential energy curves. Because the light particle experiences a purely harmonic potential, the potential energy curves are parallel an equally space for all of the moels. In aition, the nonaiabatic couplings among potential energy curves persists for all times ue to the nonvanishing linear coupling between light an heavy particles. The moel problems were use to test two strategies for carrying out mixe quantum-classical ynamics in systems involving nonaiabatic transitions, mean-fiel an surfacehopping. With the mean-fiel approach the classical motion of the heavy particle is governe by an effective potential

12 4724 J. Chem. Phys., Vol. 109, No. 12, 22 September 1998 Kohen, Stillinger, an Tully energy surface resulting from an average over light particle quantum states, weighte by their time-varying populations. It is well known that this metho is inaequate in cases where the quantum wave packet splits into parts that are governe by substantially ifferent forces. The surfacehopping proceure for stochastically splitting a trajectory into multiple branches was evelope in orer to overcome this ifficulty. Surface-hopping has been shown to outperform the mean-fiel metho in such cases. 51 In the present series of moel stuies, however, the forces associate with each quantum state are ientical since the potential energy curves are parallel. The necessity for splitting trajectories into branches is therefore not apparent. Furthermore, the fact that nonaiabatic transitions continue to occur for arbitrarily long times presents a significant obstacle for surface-hopping methos; the ifferent stochastic histories of iniviual trajectories result in an unphysical loss of quantum coherence at long times. This was emonstrate by application to moel I, where surface hopping prove to be an aequate approximation to the exact quantum results at short times, but was inaequate at long times an clearly inferior to the mean fiel metho. The fact that the forces are ientical on each potential energy curve oes not ensure the success of the mean-fiel metho. First, the classical approximation is certainly not exactly even when motion is confine to a single potential energy surface. Since our interest here is in mixe quantumclassical treatment of systems involving multiple potential surfaces, we selecte moel parameters so that classical mechanics provie an accurate escription of single surface ynamics, as juge by comparison with exact quantum calculations within the aiabatic approximation. Secon, even with parallel potential energy surfaces, the mean-fiel approximation is not exact. The results reporte above help elineate the range of valiity of the metho. For a completely harmonic system, moel I, we have shown in analogy with the stanar Ehrenfest theorem that the mean-fiel classical heavy particle position X an momentum P reprouce exactly the corresponing quantum mechanical expectation values X an P. Higher moments of X an P an quantum state probabilities are not reprouce exactly by the mean-fiel metho, but they are quite accurate. With the aition of anharmonicity into the external potential that acts on the heavy particle, moels II an III, the Ehrenfest corresponence is no longer exact, an the mean-fiel results eviate somewhat more from the exact quantum results. Nevertheless, the mean-fiel metho was of acceptable accuracy for all but one of the moel systems stuie. For the ouble well potential, moel III, we expecte that the mean-fiel metho woul be incapable of correctly escribing the bifurcation of the quantum wave packet into two parts localize, respectively, in the left an right potential wells. However, the fraction of mean-fiel trajectories trappe in each well agree quite accurately with the quantum probabilities as i the mean values of X an P. The only case for which the mean-fiel metho i not provie an acceptable escription was the final parameterization of moel II, for which the quantum wave packet split into a scattere wave packet on the groun state potential curve an a scattere wave packet on the first excite potential. The class of moels consiere here, restricte to linear light heavy coupling, is amittely artificial. The moels were chosen to eluciate the application of mixe quantumclassical ynamics to conense phase systems involving many heavy classical egrees of freeom which are couple to a few light quantum egrees of freeom. In such systems it is impractical to carry out ynamical simulations using normal moe coorinates. Rather, the bare coorinates of the heavy an light coorinates are generally use. In the bare particle coorinate system, the coupling between light an heavy particles will have a linear term. It is reasonable to suppose that in many cases this linear term will ominate the coupling, particularly when the masses of the light an heavy particles o not iffer greatly, e.g., hyrogen versus carbon. The moel calculations reporte here suggest that, in cases where linear couplings ominate, the mean-fiel mixe quantum-classical metho can be of useful accuracy an is robust to the introuction of anharmonic heavy-particle interactions. ACKNOWLEDGMENT We thank Dr. Davi Sholl for valuable iscussions. APPENDIX: NORMAL MODES OF MODEL I The total Hamiltonian is Hx,X p2 P2 2m 2M 1 2 kxx2 1 2 KX2. A1 If the potential is written in matrix form an in term of mass weighte coorinates qmx an QMX it becomes 1 2 qq Q, q A2 where k/m, (kk)/m, an k/mm. Therefore to obtain the normal moes it is necessary to solve the following eigenvalue problem: A3 0. The eigenvalues are an the corresponing iagonalizing coorinates are where A4 q 2 Q, A5 2 q Q, 2 2. The Hamiltonian thus becomes a two-imensional uncouple harmonic problem, with normal moes coorinates an, A6 A7

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