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 me- chanical 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 me- chanics 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 deter- mined 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 non- adiabatic 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 tra- jectory thus evolves, roughly speaking, on an average of the potential surfaces as- sociated with each quantum state, weighted according to the instantaneous popu- lations 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-de- pendent 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 as- sociated 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-approx- imately-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 so- lution 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.