Nonadiabatic Molecular Dynamics

JOHN C. TULLY AT&T Bell Laboratories, Murrav Hill, New Jersey 07974

Abstract

Molecular dynamics simulation of mixed quantum-classical systems, in situations where the quanta1 degrees of freedom undergo transitions among states, poses a number of challenging problems. Among the difficulties are bifurcation of trajectories that evolve into different quantum states and proper treatment of quantum coherence. In this article we outline the problems and contrast the ways in which they are addressed by current methods for nonadiabatic molecular dynamics. In the course of this comparison we present a new result, the relationship between the velocity adjustment in the “surface hopping” method and the “Pechukas force,” as well as some new reflections on an old result, oscillatory yields in ion-surface scattering.

Introduction Conventional molecular dynamics, the numerical integration of the classical mechanical equations of motion for a number of interacting atoms, is an extremely powerful and widely employed method for studying gas-phase and condensed-phase molecular changes [ 11. There are many situations, however, where classical mechanics is inadequate for some degrees of freedom. Examples are tunneling or quantized vibrations of light atoms (hydrogen) or explicit inclusion of electronic coordinates for describing electron transfer or localization. If there are only a few such quantum degrees of freedom, it may be feasible to carry out a completely quantum mechanical treatment of their motion simultaneously with a classical mechanical treatment of the remaining degrees of freedom [ 21. Such procedures have been quite successful in cases in which the quantum degrees of freedom remain in a single adiabatic quantum state throughout the course of the simulation. It is more difficult to carry out this strategy in cases in which “nonadiabatic” transitions among quantum states occur. The probabilities of quantum transitions are determined by the paths of the classical particles. The classical paths are influenced, in turn, by the quantum state populations. Thus the theory must, in general, treat both quantum and classical variables in a self-consistent manner. In addition, quantum coherences can be important; amplitude and phase information about the quantum states can be crucial, and it is difficult to properly account for this in a classical mechanical framework. A number of procedures have been proposed for carrying out mixed quantum-classicalmolecular dynamics of systems undergoing nonadiabatic transitions. We give a brief appraisal of some of these methods below, with particular emphasis on quantum coherence and on whether and how classical paths assigned to different quantum states are split into independent branches.

International Journal of Quantum Chemistry: Quantum Chemistry Symposium 25,299-309 (1991) 0 1991 John Wiley & Sons, Inc. CCC 0020-7608/9 1/010299-1 1$04.00 300 TULLY

The Classical Path Method The problem becomes greatly simplified if self-consistencybetween the quantum and classical variables is not required [ 31. This can occur if the energy splittings between involved quantum states are small compared to the kinetic energies of the classical particles. In this case, the trajectory followed by the classical particles is essentially the same, regardless of the populations of the quantum states. Let us assume that this trajectory is known or can be obtained by an entirely classical mechanical calculation. We designate the trajectory by R( t),where t is time and R is the collection of all classical coordinates. We denote the ith quantum state by 4,(R), and the quantum Hamiltonian by XQ(R). The basis functions may be adiabatic states, i.e., eigenfunctions of XQ(R) , or they may be any other convenient “diabatic” wave functions. Both the basis functions &( R) and the Hamiltonian ZQ(R)depend on the classical positions R, which are changing with time along the prescribed trajectory. We can express the wave function describing the quantum system at time t, \k(t),in terms of a linear combination of the states 4,(R),

Substituting Eq. ( 1 ) into the time-dependent Schrodinger equation, we obtain the following expression for the time derivatives of the amplitudes of the quantum states:

I ihCj(t)= C ci(t)Zj(R)- ih.2 ci(t)R.(4(R)IVR4i(R)) (2) i i where

Z,(R) = (4,(R)I‘%Q(RM(R)). (3) Brackets indicate inner products over the space of the quantum coordinates. If an adiabatic basis were chosen, the off-diagonal elements of Z,(R) would vanish and changes in the amplitudes (quantum transitions) would occur only through nonadiabatic coupling terms, (4,(R)lkfm)). Equation (2) can be integrated numerically along any trajectory R to obtain the amplitudes c,( t)of each quantum state and their probabilities I c? (t)I. This non-self-consistent procedure has been employed successfully to describe many high-energy collision processes, including cases in which interference effects due to quantum coherence are exhibited. Interference effects are frequently discussed in semiclassical terms as arising from the confluence of two or more paths with differing quantum phases. It is important to recognize that interference effects such as Stueckelberg oscillations are properly included in the classical path method even though there is only a single trajectory. The quantum-mechanical phase associated with each state is contained in the amplitude c,( t).This point has not always been appreciated, and we will return to it later. NONADIABATIC MOLECULAR DYNAMICS 30 1

The Best Average Path The assumption of the classical path method, that the trajectory is unaffected by the evolution of the quantum system, is rarely acceptable for low-energy or thermal chemical processes. A number of methods have been proposed to overcome this deficiency by allowing the trajectory to respond to changes in the quantum state in a self-consistent manner [ 41. The most successful and widely used of these is the so-called Ehrenfest method: here the “classical” trajectory is determined by an effective potential Ve/,ff-(R)given by the expectation value of energy of the quantum system, VdR) = (*I~Q(R) I *) . (4) A calculation typically proceeds as follows: The trajectory begins on the potential energy surface corresponding to the initial quantum state. As the trajectory is evolved on this potential, Eq. (2) for the amplitudes of the quantum states is integrated simultaneously. As the amplitudes of new quantum states begin to build, the effective potential energy surface begins to deviate from the initial potential according to Eq. (4). The Ehrenfest choice of the effective potential insures that the total energy of the quantum-classical system is conserved throughout the simulation. The trajectory thus evolves, roughly speaking, on an average of the potential surfaces associated with each quantum state, weighted according to the instantaneous populations of each state. This “best average path” procedure can offer significant improvement over methods in which the trajectory is not obtained self-consistently with the quantum evolution. It is related, in fact, to the completely quantum mechanical time-dependent self-consistent-field ( TDSCF)method, but with some degrees of freedom taken in the classical limit. As with TDSCF, the quantum state at any time is described by a single coherent “mixed state.” Thus, even after the trajectory has left the nonadiabatic coupling region, it is still evolving on an average potential energy surface. As has been demonstrated previously, this can be a very severe limitation of the theory in many applications [ 5 1. For example, it is common in a chemically reacting system for the energy of the reaction barrier to depend significantly on the electronic state. A trajectory may proceed towards products if in a low barrier state, but be reflected back to reactants if in a high barrier state. A single trajectory moving on an average mixed-state potential cannot adequately describe both possibilities. As discussed below, in such situations a procedure must be employed in which trajectories split into branches, each governed by the potential energy surface associated with a particular quantum state. The best average path methods exhibit another serious deficiency, namely, they do not satisfy detailed balance. To illustrate how seriously detailed balance can be violated, consider the very commonly encountered two-state case in which the probability of a quantum transition is small. The average potential governing the forward trajectory will then be approximately equal to the potential energy surface associated with the initial state. The time-reversed trajectory will follow-approximately-the final-state potential energy surface. Unless the potential energy surfaces corresponding to the two states are very similar, in which case almost any method will suffice, the best trajectory methods can be very inaccurate. 302 TULLY

The Pechukas Method In 1969, well before application of the best average path methods became fashionable, Pechukas recognized their deficiencies and discovered an elegant solution to them [ 61. He derived a path integral representation of the motion of the heavy particles while the internal quantum state undergoes a transition from state 1 at time toto state 2 at time tl. By stationary phase evaluation of this path integral, he obtained a semiclassical expression for the most important “classical-like’’ path. His result is that the heavy particles evolve according to classical mechanics, subject to the following effective force:

We have written this force in the form utilized by Webster, Rossky, and Friesner ( WRF) [ 71. In Eq. (5), 9(t) is the total wave function defined in Eq. ( 1 ), equal to $l(t)at time to and evolved to time t via Eq. (2). @(t)is the total wave function that equals $*(t) at time t, and is evolved via Eq. (2) backwards in time to time t. Following WRF,we will call Fp(t) the “Pechukas force.” It can be easily shown that a trajectory governed by the force Fp(t) and evolving from state 1 to state 2 will conserve total quantum plus classical energy. It is im- mediately apparent from the symmetry of Eq. (5) with respect to forward and backward propagation that the Pechukas method satisfies detailed balancing exactly. Finally, trajectories corresponding to different final quantum states follow distinct paths, not an average path. The Pechukas method has been criticized on several counts, mainly by Pechukas himself [ 81. One difficulty is that the Pechukas force, evaluated at time t, requires information about future times through the wave function @(t).Thus in general the trajectory must be determined iteratively. This can be tedious, and it has not been demonstrated that a unique converged trajectory exists in all cases. A second difficulty of the Pechukas method, in its original form, is that while it correctly begins and ends with a pure quantum state, the quantum variables are described by a mixed state for extended times. This can be problematic for condensed-phase applications in which a beginning and an end of the interaction cannot be defined unambiguously, and it may be necessary to resolve the mixed state into its component states at any intermediate time. These problems have been addressed by WRF as discussed below. In spite of the criticisms, the Pechukas formalism remains a cornerstone in the theory of mixed quantum-classical systems. In addition, it may well be that most of the criticisms can be surmounted, and that with future improvements it will become the method of choice for nonadiabatic molecular dynamics.

Surface Hopping I In 197 I, Tully and Preston [ 9 ] developed a simpler but more ad hoc method than the Pechukas approach for splitting trajectories into branches, each corresponding to a particular final quantum state. The “surface hopping” method NONADIABATIC MOLECULAR DYNAMICS 303 assigns each trajectory to a particular quantum state at all times; the trajectories do not evolve on mixed-state or average potentials. In the original version of surface hopping, transitions (“hops”) between one state and another occur only at a countable number of assigned locations along the trajectory, e.g., at the position of the minimum energy splitting between two states at an avoided intersection, or at the position of a maximum in the nonadiabatic coupling strength. The justification for imposing such sudden transitions is that transitions are likely to occur with high probability only in regions where potential energy surfaces approach each other closely in energy, so the hops are not drastic in- terruptions of the paths. Furthermore, if the nonadiabatic coupling is localized in space, then the transition will be correspondingly localized, and the resulting trajectory insensitive to the exact location of the hopping point. In other words, if the coupling is very localized, then the changeover of the Pechukas force from that of the initial to the final state will occur rapidly, approximating a step function. A number of variations of surface hopping have been proposed [ 10,111. In the most general formulation, the probability of a state switch or, equivalently, the weighting of each trajectory branch, is determined by the state occupations obtained by evaluating Eq. (2) throughout the region of nonadiabatic coupling [ 51. Thus, like the Pechukas method, knowledge about future times is required. However, since in this case only the ultimate weighting and not the actual path is affected, this poses no additional difficulty. Furthermore, employing approximations for the transition probabilities such as Landau-Zener [ 51, when applicable, allow the probability to be assigned when the hop is made. Since surface hopping assigns transition probabilities according to quantum-state probabilities, not amplitudes, it is frequently stated that the method ignores quantum-phase or coherence effects. This is misleading. As stated above, the quantum-state amplitudes are obtained by numerically integrating Eq. (2) throughout the region of nonadiabatic coupling, i.e., complete coherence is preserved throughout the coupling region. This is true even when approximations such as Landau-Zener are invoked. Landau-Zener is a closed form solution of Eq. (2) for a simplified interaction model throughout the entire coupling region. In some applications of surface hopping, the quantum state has been reinitialized between regions of nonadiabatic coupling. This is not necessary, however, so coherence effects between successive coupling regions can be included in surface hopping, allowing correct description of interference effects such as Stueckelberg oscillations. Since the surface hopping trajectory follows the same path forward and backward in time, detailed balancing is obeyed. A velocity adjustment is required at each hop between nondegenerate states, however, to conserve total quantum plus classical energy. It has been shown by several methods that the velocity adjustment is correctly applied to the component of velocity along the nonadiabatic coupling vector [ 5,9,12 3. We present another demonstration of this, employing the Pechukas force of Eq. (5). Consider, for simplicity, a two-state problem in which a transition occurs between initial quantum state 1 and final state 2. We can express the normalized forward and backward total wave functions in the following form: 304 TULLY

where g( t), (t)and q2( t)are continuous real-valued functions.

where El(t) and E2(t) are the diagonal elements of ZQ[ Eq. (3)] . It can be easily shown by substitution into Eq. (2) that if the coefficients of dl(R) and &( R) in Eq. (6) satisfy Eq. ( 2), then so do those of Eq. (7). Furthermore, @( tl) equals the final state c$~(R), so +(t) of Eq. (7) is the correct backward propagating wave function required in Eq. (5) for the Pechukas force. Substituting Eqs. (6)and (7) into Eq. ( 5 ) , we obtain

We have applied the Hellmann-Feynman theorem to obtain the first two terms of Eq. (9), and the following off-diagonal Hellmann-Feynman expression to obtain the final term

The first term in Eq. (9) is simply the force associated with potential energy surface 1, multiplied by a weighting factor that evolves from unity at time to to zero at tl . NONADIABATIC MOLECULAR DYNAMICS 305

Similarly, the second term is the force on potential 2 times a weighting factor that evolves from zero to unity. The final term is the “transition force” that insures conservation of energy as the transition from the potential El (R) to E2(R) occurs. The transition force is seen from Eq. (9) to be a scalar function times the nonadiabatic coupling, i.e., it is in the direction of the nonadiabatic coupling vector. Taking the limit as the Pechukas time interval approaches zero results in an infinite transition force along the nonadiabatic coupling vector, as required to make a finite velocity adjustment in infinitesimal time. The major limitation of the original surface hopping method is obvious; quantum transitions are assumed to be localized in space and time. There are many situations where this assumption is not valid. For example, systems exhibiting extended regions of nonadiabatic coupling or threshold cases in which trajectories enter regions of strong coupling but do not reach the hopping point. It is desirable to develop methods that will allow for quantum-state changes wherever called for by the changing amplitudes of Eq. (2). Two recently proposed methods that accomplish this are discussed below.

The WRF Method

Webster, Rossky, and Friesner [ 71 have recently presented a method that can be considered a synthesis of the Pechukas and surface hopping approaches. The WRF method limits quantum coherence to short time intervals At, with At chosen to represent a characteristic dephasing time or coherence time of the system. Requiring the transition to occur over a small time interval resolves the main practical difficulties of the Pechukas method. Webster, Rossky, and Freisner show that iteration of the Pechukas force to self-consistency is straightforward when the time interval is small. Furthermore, questions about evolution as a mixed state for a prolonged time are eliminated. The multiple, short time transitions are treated independently by WRF, as in the usual surface hopping approaches. Webster, Rossky, and Friesner have applied their method to simulation of an excess electron in liquid water, represented by 200 water molecules. They dem- onstrate that the method is tractable even for complex condensed-phase systems, and the results of the calculations appear physically reasonable. However, some problems remain. One such problem concerns the choice of the time interval At, which WRF take to be an adjustable parameter. A sound prescription for selecting the time interval is desired. This is particularly important because the cumulative transition probability computed by the WRF method is not independent of the choice of time interval. In fact, as the time interval approaches zero, the transition probability vanishes. This behavior has been shown by WRF to result from quenching of the quantum coherence. It is clear that forcing incoherence over too short a time interval can lead to significant inaccuracy. Quantum Coherence The question is, over what time scale does the quantum evolution remain coherent in a many-degree-of-freedom condensed-phase system? This is a difficult question, 306 TULLY in part because accurate, fully quantum-mechanical benchmark calculations are impractical for systems of the complexity required to address this question. However, we may be able to derive some guidance from experiment. Figure 1 illustrates schematically the adiabatic potential curves relevant to collision of a helium positive ion with the surface of metallic lead. Lead has a relatively narrow d band that lies about 25 eV below the vacuum level, nearly resonant with the ionization potential of the helium atom. Thus near-resonant charge transfer, leading to neutralization of the helium, can occur upon collision of the atom with the surface. If the solid were a rigid, smooth object, then the probability of ion neutralization would exhibit oscillations as a function of incident ion energy. These Stueckelberg oscillations are a result of the coherent evolution of the mixed quantum state. The relative phases are initiated as the atom passes through the strong interaction region (indicated by the box), and compared as the reflected atom passes through the interaction region a second time. The occurrence of striking oscillatory behavior due to this mechanism is well documented in gas-phase atom-atom collisions [ 31. Now consider the real experimental situation, in which the surface is not rigid but composed of vibrating lead atoms. Phase coherence is still established as the incident atom passes through the nonadiabatic coupling region. But the atom then impacts the surface with considerable momentum, causing multiphonon excitations. At the energies of the experiment under discussion, 100 to 2000 eV, essentially every impact leaves the solid in a different quantum state. Similarly, there is van- ishing probability that a trajectory following one adiabatic potential curve would leave the solid in the same quantum state as a trajectory following the other potential

a) He +S+ \ /

I I . *O Rrn INTERNUCLEAR DISTANCE (R) Figure 1. Schematic illustration of near-resonant charge transfer between a helium atom and a metallic lead surface. The region of strong nonadiabatic coupling is indicated by the box. The solid curves are the adiabatic potential energy curves. NONADIABATIC MOLECULAR DYNAMICS 307 curve. The time elapsed between the first and second entrances into the nonadiabatic coupling region is about s, which is longer than is usually assumed for vi- brational dephasing times. One might expect, therefore, that quantum coherence would be destroyed and the ion survival probability would not exhibit oscillations. On the contrary, the experiments show strong Stueckelberg oscillations (Fig. 2) [ 131. Calculations using the simple classical path method assuming a straight-line trajectory are in reasonable qualitative accord with the experiments [ 13,141. This demonstrates that quantum coherences in many-degree-of-freedom systems may be more important than has been generally believed. Of course, interference effects such as shown in this example are not likely to be relevant in typical condensed- phase situations. But coherences extending through a single region of nonadiabatic

0 I000 2000 INCIDENT ENERGY (ev)

Figure 2. Helium ion survival probability as a function of incident energy for helium scattered from a lead surface. Upper: experimental. Lower: calculation. (Reprinted from Ref. 13). 308 TULLY coupling must be properly incorporated in any generally applicable mixed quantum- classical theory.

Surface Hopping I1 The final method we will discuss is a recent proposal by the author, termed “molecular dynamics with electronic transitions” [ 151. The method can be considered an extension of the conventional surface hopping method, and thus is re- ferred to as surface hopping I1 in this article. The method retains the following features of the original surface hopping. Trajectories evolve on a single potential energy surface at any instant in time, not a mixed state or average potential. Tra- jectories make instantaneous transitions (hops) from one state to another. After a hop, the component of classical velocity in the direction of the nonadiabatic coupling vector is adjusted to conserve energy. This is justified, as for the original surface hopping, by several arguments, among them the analysis of the Pechukas force presented above. Surface hopping I1 differs from the original version in an important way; here hops can occur anywhere in space or time among any number of interacting states, as governed by the quantum state amplitudes of Eq. (2). Thus, whereas a single trajectory makes an instantaneous transition from one state to another, an ensemble of trajectories evolves gradually from the initial to the final state, with some trajectories hopping early and some late in the coupling region. The method proceeds as follows. A trajectory begins, as always, on the potential energy surface associated with the initial quantum state. The classical mechanical equations of motion are integrated simultaneously with Eq. (2) for the quantum amplitudes. At every time step of the integration, a decision is made whether to switch to another quantum state. The crux of the method is the probabilistic algorithm that determines when a state switch occurs. As can be seen in Ref. ( 15 ), the algorithm is simple to im- plement, and ensures that for an infinite ensemble of trajectories, the fraction of trajectories assigned to any quantum state i will be equal to I cf I computed from Eq. (2) at every instant of time. Furthermore, the algorithm accomplishes the correct state populations with the minimum number of state switches. This uniquely defines the switching algorithm. The fewest-switches criterion is justified by a number of arguments, none of which are compelling, but in total are at least plausible. Probably the strongest argument is the gratifying agreement between the method and exact quantum dynamics for some selected one-dimensional two-state models, including a case exhibiting strong interference effects. It should be emphasized that propagation of the quantum wave function-integration of Eq. (2)-is carried out coherently forever in surface hopping 11. This is a very different philosophy from that of WRF, who strive to reduce the coherence time as much as possible. However, because trajectories that undergo state switches at a distribution of positions will diverge rapidly in the enormous phase space of a condensed matter simulation, coherence effects damp out in a natural way in surface hopping 11. A method for additional damping of coherence has been proposed [ 15 1, but it may be that the completely coherent limit is entirely acceptable for the majority of condensed-phase applications. NONADIABATIC MOLECULAR DYNAMICS 309

Final Remarks The objective of all of the methods compared here is to provide an accurate and tractable description of the dynamics of mixed quantum-classical systems in which transitions between quantum states occur. In one sense, this goal is unachievable. Quantum mechanics and classical mechanics are fundamentally incompatible dynamical theories. Thus we must settle for a compromise between accuracy and practicality, and it is probably unrealistic to expect that any single procedure will be optimal for all situations. Further study should help to define the ranges of validity of the methods discussed here, and perhaps lead to the development of even better methods.

Bibliography

[I] See, for example, J. Kushick and B. J. Berne, in Statistical Mechanics, Part B, B. J. Berne, Ed. (Plenum, New York, 1977), p. 41. A. Selloni, P. Carnevali, R. Car, and M. Pamnello, Phys. Rev. Lett. 59,823 ( 1987); M. Sprik and M. L. Klein, J. Chem. Phys. 89, 1592 (1988); R. Kosloff, J. Phys. Chem. 92,2087 (1988); P. J. Rossky and J. Schnitker, J. Phys. Chem. 92, 4277 ( 1988). E. E. Nikitin, Theory ofElementary Atomic andMolecular Processes in Gases (Clarendon, Oxford, 1974). W. H. Miller and C. W. McCurdy, J. Chem. Phys. 69, 5163 ( 1978); D. A. Micha, J. Chem. Phys. 78, 7139 (1983); G. D. Billing, Chem. Phys. Lett. 30, 391 (1975); Z. arson, R. B. Gerber, A. Nitzan, and M. A. Ratner, Surf. Sci. 137, 527 ( 1984). [ 51 J. C. Tully, in Dynamics of Molecular Collisions, Part B, W. H. Miller, Ed. (Plenum, New York, 1976), p. 217. [6] P. Pechukas, Phys. Rev. 181, 174 (1969). [ 71 F. Webster, P. J. Rossky, and R. A. Friesner, Comp. Phys. Comm., 63,494 ( 1991 ); F. J. Webster, J. Schnitker, M. S. Friedrichs, R. A. Friesner and P. J. Rossky, Phys. Rev. Lett. 66, 3172 ( 1991 ). [ 81 P. Pechukas, personal communication. [9] J. C. Tully and R. K. Preston, J. Chem. Phys. 55,562 ( 1971 ). [ 101 J. R. Stine and J. T. Muckerman, J. Chem. Phys. 91,459 (1987); N. C. Blais and D. G. Truhlar, J. Chem. Phys. 79, 1334 (1983); G. Parlant and E. A. Gislason, J. Chem. Phys. 91,4416 (1989). [ 1 1 ] We should note that an alternative semiclassical surface hopping method was developed by W. H. Miller and T. F. George, J. Chem. Phys. 56, 5637 ( 1972). Their procedure avoids the problem of velocity adjustment by following trajectories in complex-valued coordinate space to the actual analytically continued intersection points of the potential energy surfaces. Because of the complex- valued semiclassical nature of the trajectories, we have chosen not to include this approach in the current comparison of mixed quantum-classical dynamical methods. [ 121 M. F. Herman and E. Kluk, in Dynamical Processing Condensed Matter, M. W. Evans, Ed. (Wiley, New York, 1985), p. 577. [ 131 J. C. Tully and N. H. Tolk, in Inelastic Ion-Surface Collisions, N. H. Tolk, J. C. Tully, W. Heiland, and C. W. White, Eds. (Academic, New York, 1977), p. 105. [ 141 J. C. Tully, Phys. Rev. B16,4324 ( 1977). [IS] J. C. Tully, J. Chem. Phys. 93, 1061 (1990).

Received April 9, 199 1