Download Mixed quantum–classical dynamics

Faraday Discuss., 1998, 110, 407È419
Mixed quantum–classical dynamics
John C. Tully
Departments of Chemistry, Physics and Applied Physics, Y ale University, New Haven,
CT 06520, USA
We present a uniÐed derivation of the mean-Ðeld (Ehrenfest) and surfacehopping approaches to mixed quantumÈclassical dynamics that elucidates
the underlying approximations of the methods and their strengths and
weaknesses. We then report a quantitative test of alternative mixed
quantumÈclassical methods against accurate quantum mechanical calculations for a simple one-dimensional curve-crossing model. We conclude
that, for this model, surface-hopping with the adiabatic representation is
decidedly superior to either mean-Ðeld or surface-hopping with the diabatic
representation.
Introduction
It is currently not computationally feasible to carry out accurate quantum mechanical
or semiclassical calculations of the dynamics of molecular processes that involve a large
number of atoms (e.g., [10). In some cases simpliÐcations such as reduced dimensionality, harmonic bath or short time scales permit quantum mechanical solutions to be
obtained. Frequently, however, these approximations are not valid. In such cases, the
method of choice has been conventional molecular dynamics (MD).1 MD has severe
limitations as well, of course. The need for accurate multidimensional force Ðelds
remains the primary focus. But as the methodology for “ on the Ñy Ï ab initio calculation
of forces improves,2 we need to turn our attention to the more fundamental limitations
of MD.
Conventional MD is based on two underlying approximations. The Ðrst is the BornÈ
Oppenheimer separation of electronic and atomic motion, reducing the dynamics to
atomic motion on a single adiabatic potential energy surface.3 The second is the treatment of the atomic motion by classical mechanics. There are a huge number of applications for which one or both of these approximations is invalid. Processes such as
electron transfer, dynamics at metal surfaces, radiationless processes in molecules or
solids, and photoinduced chemistry usually involve more than one potential energy
surface, with transitions among them. Quantized vibrational levels, zero-point motion
and tunneling through reaction barriers require quantum mechanical description of
atomic motion. Mixed quantumÈclassical dynamics methods have been developed to
address both of these problems. The strategy is to retain a multi-dimensional classical
mechanical treatment for most of the atoms, while designating a few crucial degrees of
freedom to be computed quantum mechanically.
The crucial issue in mixed quantumÈclassical dynamics is self-consistency. The
quantum mechanical degrees of freedom must evolve correctly under the inÑuence of the
surrounding classical motions. In turn, the classical degrees of freedom must respond
correctly to quantum transitions. It is this second requirement that is most challenging,
and this is the major focus of this paper. There are a number of standard approaches
407
408
Mixed quantumÈclassical dynamics
that accurately describe the dynamics of a quantum system interacting with a classical
one. Notable are the RedÐeld approach4,5 and the classical path method.6,7 But these
methods do not properly describe the back reaction of the quantum system on the
classical one. Two approaches have emerged that attempt to treat the interactions
between quantum and classical systems in a self-consistent way, viz. mean-Ðeld and
surface-hopping.
The mean-Ðeld method (sometimes called Ehrenfest or eikonal) is based on a meanÐeld separation of classical and quantum motions.8 The approximations underlying this
method are easily stated and will be examined below. The method is invariant to the
choice of quantum representation (adiabatic or diabatic), it usually provides accurate
quantum transition probabilities, and it properly conserves total (quantum plus
classical) energy. It does not obey microscopic reversibility, however, and it su†ers from
the deÐciency of all mean-Ðeld methods ; it does not describe the correlation between
classical and quantum motions.
The surface-hopping approach was developed to introduce classicalÈquantum correlation.9 The approximations on which surface-hopping is based are not as transparent
as for the mean-Ðeld method, and this has led to some misunderstanding. But as shown
below, surface-hopping can be derived from a multi-conÐguration expansion of the
SchroŽdinger equation with systematic approximations analogous to those of the Ehrenfest method. Surface-hopping has the advantage that a given trajectory can bifurcate
properly into di†erent branches, each subject to a particular quantum state and
weighted by the amplitude of the state ; i.e., classicalÈquantum correlations are included.
Surface-hopping also usually provides accurate quantum transition probabilities, conserves total energy and, depending on the hopping algorithm, satisÐes microscopic
reversibility either rigorously or approximately. But surface-hopping is not invariant to
the choice of quantum representation, the hopping algorithm is not unique, there are
applications for which it is not as accurate as the mean-Ðeld method, and it is usually
more computationally demanding.
In the next section we present a derivation of the mean-Ðeld mixed quantumÈ
classical method. In the section following we develop a parallel derivation of the surfacehopping method and argue that the adiabatic representation is more natural and likely
more accurate than the diabatic representation. We then report a quantitative test of
mean-Ðeld and surface-hopping methods for a simple one-dimensional curve-crossing
model for which accurate quantum solutions have been obtained. All of the methods
acceptably reproduce the quantum transition probabilities. But, for this model, only
surface-hopping based on the adiabatic representation adequately describes the backreaction of the quantum transitions on the classical trajectories. At low (thermal) classical velocities, both mean-Ðeld and diabatic surface-hopping are less accurate. The
results are summarized in the Ðnal section.
Mean-Ðeld (Ehrenfest) approach
The Ehrenfest method8,10h17 can be derived as a classical limit of the time-dependent
Hartree or time-dependent self-consistent Ðeld method (TDSCF).18h22 TDSCF is a
mean-Ðeld method, i.e., it is based on factorization of the total wave function into the
product of fast and slow particle parts,
CP
W(r, R, t) \ N(r, t)X(R, t)exp
i
Å
D
t
E (t@)dt@
r
(1)
where throughout this paper we denote the fast (quantum) variables by r and the slow
(to become classical) variables by R. N(r, t) and X(R, t) are taken to be normalized at
every time t with respect to integration over r and R, respectively. In order to simplify
J. C. T ully
409
the appearance of the Ðnal equations, we have introduced a phase factor E (t),
r
E (t) \
r
PP
N*(r, t)X*(R, t)H (r, R)N(r, t)X(R, t)dr dR .
r
(2)
The Hamiltonian operator that governs the entire system is
Å2
Å2
; m~1Z2 ] V (r, R)
; M~1Z2 [
b rb
rR
a Ra
2
2
b
a
Å2
(3)
\[
; M~1Z2 ] H (r ; R)
r
a Ra
2
a
M and m are the masses of slow particle, a, and fast particle b, respectively. V (r, R)
a
b
rR
includes all inter-particle interactions, fastÈfast, slowÈslow and fastÈslow ; and H (r ; R)
r
is the Hamiltonian of the fast system for slow particles Ðxed at positions R. We specify
the internal phase factors of the two wave functions by23
H\[
iÅ
iÅ
P
P
X*(R, t)
dX(R, t)
dR \ E
dt
(4)
N*(r, t)
dN(r, t)
dr \ E (t)
r
dt
(5)
This unsymmetrical choice of phase factors anticipates our unequal treatment of fast
and slow variables, below. The phase convention is arbitrary, however, and the present
development is entirely equivalent to standard TDSCF.21 Substituting eqn. (1) into eqn.
(3), multiplying from the left by X*(R, t) and integrating over R, using eqn. (4), yields an
e†ective SchroŽdinger equation for the fast variables, r :
GP
H
dN(r, t)
Å2
X*(R, t)V (r, R)X(R, t)dR N(r, t)
(6)
\[
; m~1Z2 N(r, t) ]
rR
b rb
dt
2
b
Similarly, multiplying from the left by N*(r, t) and integrating over r, using eqn. (5), gives
an e†ective SchroŽdinger equation for the slow variables, R :
iÅ
GP
H
dX(R, t)
Å2
N*(r, t)H (r, R)N(r, t)dr X(R, t)
(7)
\[
; M~1Z2 X(R, t) ]
r
a Ra
dt
2
a
Eqn. (6) and (7) are the basic equations of the mean-Ðeld TDSCF method ; the fast
particles move in the average Ðeld of the slow particles, and vice versa. Feedback
between the fast and slow degrees of freedom is incorporated in an average manner in
both directions.
The Ehrenfest method is obtained by taking a classical limit of eqn. (7). Of many
possible ways to obtain a classical limit,10h17 we follow the procedure described by
Messiah24 because it can be directly extended to the multi-conÐguration case to derive
surface-hopping, as demonstrated below. This procedure is closely related to methods
described by Micha11 and Gerber et al.21 Following Messiah, we factor the slowparticle wave function into amplitude and phase terms,
iÅ
C
X(R, t) \ A(R, t)exp
D
i
S(R, t)
Å
(8)
where A(R, t) and S(R, t) are taken to be real-valued. Substituting eqn. (8) into eqn. (7)
and separating real and imaginary terms gives
dS
1
] ; M~1(Z S)2 ]
Ra
dt
2 a
a
CP
D
1
Z2 A
N*(r, t)H (r, R)N(r, t)dr \ ; M~1 Ra
r
2 a
A
a
(9)
410
Mixed quantumÈclassical dynamics
1
dA
(10)
] ; M~1Z A É Z S ] ; M~1AZ2 S \ 0
Ra
Ra
a Ra
2 a
dt
a
a
Eqn. (9) and (10) are entirely equivalent to the original SchroŽdinger equation, eqn. (7).
Note that eqn. (10) does not contain Å. The classical limit is obtained by setting Å ] 0 on
the right-hand side of eqn. (9).24
CP
D
dS
1
N*(r, t)H (r, R)N(r, t)dr \ 0
(11)
] ; M~1(Z S)2 ]
r
Ra
dt
2 a
a
This is the HamiltonÈJacobi equation,25 and is entirely equivalent to NewtonÏs equations, where p is the classical momentum of particle a,
a
dp
a\[Z
N*(r, R, t)H (r, R)N(r, R, t)dr
(12)
r
a
dt
CP
D
Eqn. (10) and (11) describe a Ñuid of non-interacting multi-dimensional classical particles, i.e., a swarm of independent trajectories moving in the average potential of the fast
particles, with eqn. (10) expressing continuity of Ñux. The slow particles move via classical mechanics on a potential energy surface given by the expectation value of the
fast-particle Hamiltonian, H ; i.e., on the mean-Ðeld potential. S(t) is the “ classical
r
action Ï,
S(t) \
P
t
L (t@)dt@
(13)
and
(14)
R0 \ M~1Z S
a Ra
where L (t) is the classical Lagrangian.
Eqn. (12) does not completely deÐne the classical limit, however. Eqn. (6), which
deÐnes the fast-particle wave function, N(r, t), involves the slow-particle wave function,
X(R, t). Following the usual procedure, we replace X(R, t) in eqn. (6) by a delta function
at the classical trajectory position, R(t) :
iÅ
dN(r, R, t)
\ H (r, R)N(r, R, t)
r
dt
(15)
Eqn. (12) and (15) deÐne the mean-Ðeld Ehrenfest method. Since R appears explicitly in
the equation of motion for the quantum particles, the dependence of the wave function
N(r, R, t) on R is indicated in eqn. (12) and (15). It is straightforward to show that the
Ehrenfest method allows transfer of energy between quantum and classical coordinates
such that total energy is conserved.23
There has been discussion in the literature about whether eqn. (12) should be used to
deÐne the Ehrenfest trajectory, or whether it is more accurate to apply the gradient
operator inside the integral, i.e., to use the following equation of motion :26
P
dp
a \ [ N*(r, R, t)[Z H (r, R)]N(r, R, t)dr
(16)
Ra r
dt
As shown elsewhere,23 however, if N(r, R, t) is a solution of eqn. (15), then eqn. (12) and
(16) are entirely equivalent.
As prescribed by eqn. (12) or (16), the classical particles evolve subject to a single
e†ective potential corresponding to an average over quantum states. Thus, as with any
mean-Ðeld approach, correlations are neglected. This deÐciency may be particularly
severe when one is interested in a low-probability channel. In such cases, the Ehrenfest
path will be quite similar to the major channel trajectory, and will likely be a poor
representation of the desired low-probability path. Proper description of correlation
J. C. T ully
411
between quantum and classical motion requires a distinct classical path for each
quantum state ; i.e., surface-hopping. A second deÐciency of the mean-Ðeld method is
that it violates microscopic reversibility. This is clear from the following illustration.
Consider a two-state case where the transition probability between states 1 and 2 is
relatively small. The e†ective potential governing motion in the forward direction that
starts in pure state 1 and then develops small amplitude in state 2 will be not too
di†erent from the adiabatic potential of state 1. In contrast, the e†ective potential for the
reverse trajectory that begins in state 2 will be similar to that of adiabatic potential
energy surface 2. Thus the forward and reverse paths are governed by very di†erent
forces, violating microscopic reversibility.
One advantage of the mean-Ðeld (Ehrenfest) method is that it can be applied without
choosing basis functions by direct numerical propagation of the wave packet N(r, R, t),
using eqn. (15), along the classical trajectory obtained self-consistently from eqn. (12) or
(16). The wave function N(r, R, t) can be expanded in terms of basis functions, if desired,
but the resulting solution will be independent of the basis set if a complete set has been
utilized. Denoting the basis functions by U (r, R), we can expand N(r, R, t) as
j
i t
H dt@
(17)
N(r, R, t) \ ; c (t)U (r, R)exp [
jj
j
j
Å
j
where the basis functions are assumed orthonormal but are otherwise unspeciÐed, i.e.,
they could be adiabatic or diabatic functions. The matrix elements H are
ij
A P
H (R) \
ij
P
B
U*(r, R)H U (r, R)dr
i
r j
(18)
Substituting eqn. (18) into eqn. (15) gives the following coupled di†erential equations for
the amplitudes, c (t),
j
i
i t
dc
j \ [ ; c (t) ; R0 É da ] H exp [
(19)
(H [ H )dt@
a ji Å ji
ii
jj
i
Å
dt
a
iEj
where the nonadiabatic couplings d a (R) are deÐned by
ij
C
d a (R) \
ij
D C P
P
MU (r, R)[Z U (r, R)]Ndr
i
Ra j
D
(20)
Thus, if the quantum wave packet is expanded in a basis, the time-varying amplitudes
c (t) along the mean-Ðeld trajectory are given by eqn. (19). As shown in the next section,
j
eqn.
(19) also deÐnes the quantum-state amplitudes in the surface-hopping method.
There are a number of approximations to eqn. (19) that can be invoked. Notable are the
LandauÈZener approximation27,28 and a menu of useful and accurate approximations
for di†erent situations formulated by Nakamura and Zhu.29 The emphasis of the
present paper, however, is treatment of the feedback of the quantum system on the
classical paths, not on the quantum transition probabilities per se. We will assume
throughout that eqn. (15) or (19) has been solved accurately by numerical integration
along each trajectory. For multi-atom MD simulations this usually involves negligible
additional computational e†ort, at least if relatively few quantum states are included.
Surface hopping
Surface-hopping is based on a multi-conÐguration expansion of the total wave function.
The mean-Ðeld product wave function of eqn. (1) is replaced by a sum over states
W(r, R, t) \ ; U (r, R)X (R, t)
i
i
j
(21)
Mixed quantumÈclassical dynamics
412
with a di†erent wave function X (R, t) describing the evolution of the slow coordinates R
i
for each fast particle quantum state i. We will assume that the fast particle basis functions U (r, R) are orthonormal and are speciÐed in advance, i.e., an adiabatic or diabatic
i
representation is employed. If a complete set of basis functions were used, the wave
function of eqn. (21) would approach the exact solution. In practice, however, the summation will be truncated. The slow particle wave functions are neither orthogonal nor
normalized ; the integral over R of o X (R, t) o2 is the population of state i at time t.
i
Substituting eqn. (21) into the time-dependent SchroŽdinger equation, multiplying from
the left by U*(r, R) and integrating over r gives a set of coupled di†erential equations for
the X (R, t), j
j
dX (R, t)
Å2
j
iÅ
\[
; M~1Z2 X (R, t) ] ; H (R)X (R, t)
ji
i
a Ra j
dt
2
i
a
Å2
(22)
] ; M~1Da (R)X (R, t) [ Å2 ; M~1d a (R) É Z X (R, t)
a ji
i
a ji
Ra i
2
ai
a,iEj
where H and d a (R) were deÐned previously and
ij
ji
Da (R) \ [
ij
P
MU (r, R)[Z2 U (r, R)]Ndr
i
Ra j
(23)
Surface-hopping is the classical analogue of the multi-conÐguration expression, eqn. (22).
We use the word analogue rather than limit here because surface-hopping is not a rigorous classical limit. As a result, there are several variations of surface-hopping with di†erent hopping algorithms. The intent of surface-hopping is to improve on the Ehrenfest
mean-Ðeld method in situations where multiply-branched classical paths are required for
an accurate description. In some cases surface-hopping has been shown to do this very
successfully. We shall examine one such case in the next section. However, there are
situations for which it is less accurate than the Ehrenfest method, as well.30,31
In this paper we derive surface-hopping by an extension of the methods we have
used above to derive the mean-Ðeld Ehrenfest method. We express the slow-particle
wave functions as
C
D
i
X (R, t) \ A (R, t)exp S (R, t)
j
j
Å j
(24)
where we no longer require A (R, t) and S (R, t) to be real-valued. Substituting this into
j
j
eqn. (22) gives
dS
1
j ] ; M~1(Z S )2 ] H
jj
Ra j
2 a
dt
a
Å2
i
Å2
Z2 A
M~1(A /A )Da exp (S [ S )
\;
M~1 Ra j [ ;
a
i j ji
j
a
A
2
Å i
2
j
a,i
a
i
] Å2 ; M~1d a É (Z A )A~1 exp (S [ S )
a ji
a i j
j
Å i
a,i
1
dA
j ] ; M~1Z A É Z S ] ; M~1A Z2 S
a Ra j Ra j
2 a j Ra j
dt
a
a
i
i
] ; A ; M~1d a É Z S ] H exp (S [ S ) \ 0
a ji Ra i Å ji
j
i
Å i
a
iEj
C
C
GC
D C
D
D
DH
(25)
(26)
J. C. T ully
413
As before, eqn. (25) and (26) are completely equivalent to the original coupled SchroŽdinger equations, eqn. (22). However, we have separated the terms in a creative way,
putting the diagonal H terms in eqn. (25) and the o†-diagonal couplings H in eqn.
jj
ji
(26). By doing so, we can take the classical Å ] 0 limit of eqn. (25) to obtain a HamiltonÈ
Jacobi equation describing motion on a single diagonal potential energy surface, H ,
jj
dS
1
j ] ; M~1(Z S )2 ] H \ 0
(27)
jj
Ra j
dt
2 a
a
We can consider S to be deÐned by eqn. (27). The difficult parts of the problem are then
j
contained in eqn. (26). In contrast to the mean-Ðeld case, eqn. (26) still contains Å, and A
j
is not real-valued. The physical picture emerging from eqn. (26) and (27) is clear ; motion
evolves on each (adiabatic or diabatic) potential energy surface, with transport of Ñux
between potential surfaces governed by the term in eqn. (26) that involves the nonadiabatic coupling d a and/or the o†-diagonal terms of the Hamiltonian, H . We can
ji
ji
make this more quantitative by substituting eqn. (13) and (14) into eqn. (26),
1
dA
j ] ; M~1Z A É Z S ] ; M~1A Z2 S
a
a
j
R
j
a
R
2 a j Ra j
dt
a
a
i
i t
(H [ H )dt@ \ 0
(28)
] ; A ; R0 É da ] H exp [
ia ji Å ji
ii
jj
i
Å
a
iEj
The last term in eqn. (28) is seen to be essentially identical to eqn. (19). Thus, identifying
the amplitude A with c , the transitions between potential energy surfaces in surfacej
hopping are governed
byj eqn. (19).
A practical algorithm for obtaining direct solutions to eqn. (27) and (28) has not been
developed due to difficulties associated with the non-locality of the coupling terms in
eqn. (28). The source and sink terms on any potential energy surface are a†ected by the
distributions of trajectories on other potential surfaces. This coupling between phase
space distributions is discussed by Martens and Fang.32 Surface-hopping methods eliminate the computational difficulty of dealing with coupled distributions of trajectories by
imposing an independent trajectory approximation. The most accurate methods numerically integrate eqn. (19) along each trajectory to obtain the time-varying amplitudes c (t)
corresponding to each quantum state, i.e., the source and sink terms of eqn. (28).j A
stochastic algorithm is then devised to switch trajectories from one potential energy
surface to another. This algorithm must be designed so that, at any instant in time, the
fraction of trajectories assigned to any potential energy surface is equal, at least approximately, to the relative population o c (t) o2 of that state. Variations of surface-hopping
di†er mainly in the explicit hoppingj algorithm employed.9,33h42 The algorithm we
employ in the numerical tests of the next section is the “ fewest switches Ï algorithm.36
This is a variationally-based hopping algorithm that maintains, at all times, the correct
populations o c (t) o of each state, as determined by numerical integration of eqn. (19)
(subject to the icaveat below), and does so with the minimum number of hops. SpeciÐcally, for a two-state case, the probability of a switch from state 1 to state 2 during an
incremental time interval *t is given by
GC
D C P
DH
P
\ [ d[log o c (t) o2]/dt
(29)
1?2
1
If this probability is less than zero, no hop occurs. The caveat mentioned above is that
hops may occasionally be designated to states that are energy forbidden. In such cases,
hops are aborted and the fewest switches algorithm does not exactly reproduce the
populations o c (t) o . Such situations are a manifestation of quantum tunneling, and
i
would not be expected
to be described by a classical mechanical theory. They appear to
occur surprisingly often, however, and proper treatment of these events will require
further study.30,31
414
Mixed quantumÈclassical dynamics
Each hop is accompanied by a change in the component of velocity in the direction
of the local nonadiabatic coupling vector in order to conserve the total energy of the
quantumÈclassical system. This has been justiÐed in a number of ways.23,40,43h45 Thus
an ensemble of initially identical trajectories will branch into many paths, each evolving
on di†erent potential energy surfaces for di†erent periods of time. Note that, since the
amplitudes of each state are determined from eqn. (19), phase information (quantum
coherence, interferences, etc.) is maintained in surface-hopping through the complex
amplitudes, c (t).
i
As discussed above, the quantum mean-Ðeld and resulting classical Ehrenfest
methods do not satisfy microscopic reversibility. This can introduce signiÐcant quantitative errors as well as failure to approach correct thermal populations. An additional
consequence of failure to satisfy microscopic reversibility is that methods for treating
infrequent events cannot be employed, at least not in a rigorous way.46 The standard
method for treating infrequent processes, such as barrier crossings, in molecular
dynamics simulations is to begin trajectories at the top of the barrier and to exploit
microscopic reversibility to integrate both forwards and backwards in time. The issue of
microscopic reversibility in surface-hopping is not so obvious. The original version of
surface-hopping,9 in which hops occur only at particular prescribed locations, satisÐes
microscopic reversibility rigorously. There are two requirements for a mixed quantumÈ
classical dynamics to satisfy microscopic reversibility. First, the forward and reverse
classical paths must be identical. Second, the weightings or amplitudes of these paths
must also be identical. For the original surface-hopping, the Ðrst criterion is clearly
satisÐed ; trajectories hop up or hop down at exactly the same locations, and follow the
same adiabatic potential energy surfaces elsewhere. The second criterion is also satisÐed,
provided that the hopping algorithm is a symmetrical one. Thus, if the hopping probability is based on an approximate expression, such as LandauÈZener,27h29 then a symmetrized velocity must be used in the expression. If one uses the velocity on the ground
state to determine an upward hop, and the velocity on the excited state to determine a
downward hop, then microscopic reversibility will be violated. If the hopping probability is determined by integrating eqn. (19) along the trajectory, the velocity R0 that
a the
appears in eqn. (19) must be chosen symmetrically, i.e., it must be the same in
forward and reverse directions. More sophisticated versions of surface-hopping allow
the locations of hops to be determined “ on the Ñy Ï, using the amplitudes c (t) of eqn. (19).
It has proved difficult to do this in a way that produces identically equalj weightings for
forward and reverse trajectories that hop at the same location. Thus these methods do
not rigorously obey microscopic reversibility. However, in general, deviations from
microscopic reversibility are quantitatively much less severe for surface-hopping
methods than for the Ehrenfest method.
The derivation of surface-hopping presented above is valid for any representation,
i.e., adiabatic or diabatic. Eqn. (27) forces trajectories to evolve classically on the diagonal potential energy surfaces. Eqn. (28) corrects for this restriction by introducing transitions between states but because of the independent trajectory approximation of
surface-hopping, this is not done rigorously. Therefore, it is important to choose a representation which minimizes the number of transitions. The adiabatic representation does
this at low energies. At high energies, a diabatic representation may be more appropriate, but in this case trajectories are much less sensitive to the quantum forces. Thus it
appears that the adiabatic representation will usually be superior. The model calculations reported in the next section support this conclusion. The potential energy curves
on which these model calculations were based are shown in Fig. 1. The solid and dashed
curves are the adiabatic and diabatic potential energy curves, respectively. Without any
calculations, it is clear that a particle with kinetic energy slightly above 20 kJ mol~1 can
surmount the energy barrier on the lower adiabatic potential energy curve. However, 26
kJ mol~1 is required to surmount the barrier on the diabatic potential energy curve.
J. C. T ully
415
Fig. 1 Adiabatic (solid) and diabatic (dotted) potential energy curves for the one-dimensional
curve-crossing model
Motion on adiabatic potential surfaces can, at least in principle, approximate a diabatic
trajectory by trajectories that exhibit multiple hops between adiabatic potential energy
surfaces. The converse is not true ; no sequence of hops between diabatic surfaces can
adequately approximate an adiabatic path as the diabatic potential energy surfaces lie
between the adiabatic ones.
Curve-crossing model
We present here a comparison of mixed quantumÈclassical dynamics methods as
applied to a simple model for which accurate full-quantum calculations have been
carried out. The model is the same as one employed previously36 to test the fewest
switches surface-hopping algorithm. Here we use it to contrast the results of mean-Ðeld
(Ehrenfest) with adiabatic and diabatic surface-hopping. The model represents a typical
avoided crossing between two adiabatic potential energy curves. Thus there are two
electronic states and one nuclear degree of freedom, denoted by x. The two-state interaction matrix in the diabatic representation is given by :
V (x) \ 26.25 exp(1.6x), x O 0,
11
V (x) \ 52.5 [ 26.25 exp([1.6x), x [ 0
11
V (x) \ [V (x)
22
11
V (x) \ 13.125 exp([x2)
12
(30)
Distances are in Bohr radii (a ) and energies in kJ mol~1. The mass of the particle is
0 (solid curves) and diabatic (dotted curves) are shown
taken to be 2000 u. The adiabatic
in Fig. 1. Surface-hopping calculations employed 5000 trajectories at each energy. Trajectories were started with x \ [10 a on the lower quantum state. The velocity of a
0
trajectory was not reversed when the hopping
algorithm called for an energy forbidden
hop. The particle was incident from the left.
Fig. 2 shows the computed probability for transmission to the right, ending on the
lower energy adiabatic state. (The adiabatic and diabatic energies are equal asymptotically.) The points are the results of accurate quantum wave packet propagation
reported previously.36 The solid curve is the result of surface-hopping carried out in the
adiabatic representation. The dashed curve (superimposed on the solid curve except in
416
Mixed quantumÈclassical dynamics
Fig. 2 Joint probability for transmission from left to right and population of the lower energy
(adiabatic or diabatic) quantum state. Trajectories were started with x \ [10 a on the lower
0 Solid curve
energy quantum state. Points are the results of the quantum wavepacket benchmark.
is the result of surface-hopping with the adiabatic representation. Dotted curve is the result of
surface-hopping with the diabatic representation. Dashed curve is the mean-Ðeld (Ehrenfest) result.
Dot-dashed curve is the result of the LandauÈZener approximation, assuming a constant velocity
trajectory with the velocity measured relative to the energy of the crossing of the diabatic curves
(26.25 kJ mol~1).
the upper left) is the mean-Ðeld result. The dotted curve is the result of surface-hopping
carried out in the diabatic representation. Finally, the dotÈdashed line is the result of the
LandauÈZener approximation. All of the mixed quantumÈclassical results are in reasonable agreement with the quantum mechanical benchmark.
Agreement at superthermal energies is not too difficult to achieve. As discussed
above, both surface-hopping and mean-Ðeld employ eqn. (19) for the quantum state
amplitudes. At the higher energies the trajectory passes right through the interaction
region without being a†ected very much by quantum transitions. Therefore all the
methods give about the same result. This is underscored by the fact that even the
LandauÈZener approximation using a constant velocity trajectory is in fairly good
agreement.
Molecular dynamics trajectories for thermal processes involving many degrees of
freedom are much more sensitive to small changes in forces. Turning points often occur
within the interaction region. Thus it is important to examine the performance of the
mixed quantumÈclassical methods at low energies. Fig. 3 displays the same results as
Fig. 2, but showing only the low energy region. Di†erences between the methods are
now apparent. The threshold energy is incorrect for diabatic surface-hopping for the
reason discussed in the previous section. Only the mean-Ðeld and adiabatic surfacehopping results are in acceptable agreement with the quantum points.
The probability of reÑection back to the left, on the lower energy curve, is shown in
Fig. 4. The incorrect threshold for diabatic surface-hopping is again apparent. Adiabatic
surface-hopping again agrees well with the full quantum results, even in the region from
33 to 53 kJ mol~1. Below 53 kJ mol~1 there is insufficient energy to populate the
excited state asymptotically. However, near x \ 0 the excited state energy dips down to
33 kJ mol~1, opening up the possibility for transient trapping in the potential energy
well. Remarkably, the adiabatic surface-hopping describes this resonance e†ect quite
well. (The energy width of the wave-packet in the benchmark calculation was about 1 kJ
mol~1, e†ectively smoothing out any sharp resonance features that might exist.)
J. C. T ully
417
Fig. 3 Same as Fig. 2, with a di†erent energy scale
The mean-Ðeld results are not plotted in Fig. 4. This is because the mean-Ðeld
method cannot describe both reÑection and transmission at the same energy ; there
exists only one mean-Ðeld trajectory for a given set of initial conditions. Thus, above 10
kJ mol~1, the mean-Ðeld method gives 100% transmittance. Below 10 kJ mol~1, it gives
100% reÑection. In addition, above 10 kJ mol~1 the mean Ðeld trajectory has a velocity
intermediate between that appropriate for the motion on the lower and upper potential
energy curves. For multidimensional MD simulations, the fact that the mean-Ðeld trajectory continues to evolve on a combination of potential energy surfaces even long after it
leaves the interaction region is a serious deÐciency.
Conclusion
We have presented a partial appraisal of mean-Ðeld and surface hopping approaches to
mixed quantumÈclassical dynamics. The parallel derivation of the two approaches sheds
Fig. 4 Joint probability for reÑection and population of the lower energy quantum state. Points
are the results of the quantum wave packet benchmark. Solid curve is the result of surfacehopping with the adiabatic representation. Dotted curve is the result of surface-hopping with the
diabatic representation. The mean-Ðeld probability is zero above 10 kJ mol~1.
Mixed quantumÈclassical dynamics
418
light on their underlying approximations and their strengths and weaknesses. Application to a simple avoided crossing model demonstrates a clear quantitative advantage to
adiabatic surface-hopping. The advantage of using the adiabatic representation rather
than a diabatic representation with surface-hopping is likely to be sustained for most
applications. The advantage of adiabatic surface-hopping over mean-Ðeld, however, is
certainly system dependent. In particular, recent applications of surface-hopping to
problems involving long interaction times or repeated entrances into the interaction
region have shown surface-hopping to be deÐcient.30,31,47 This is not surprising, since
the independent trajectory approximation will result after a signiÐcant time in loss of
correct phase coherence. On the other hand, it has also been shown30,31 that the meanÐeld method can be very accurate when applied to nearly harmonic systems. Thus
mean-Ðeld and surface-hopping appear complementary. It should be emphasized,
however, that the avoided crossings represent perhaps the most commonly encountered
situation in multi-dimensional MD, appearing frequently both in electron transfer and
in tunneling systems.48
This paper has focused on the self-consistency between quantum and classical
degrees of freedom in mixed quantumÈclassical dynamics. But no mixed quantumÈ
classical method can be expected to be accurate for all situations. Serious problems can
arise when similar degrees of freedom are treated by di†erent methods. In addition, the
replacement of classical for quantum equations of motion for any atoms can introduce
signiÐcant errors, just as for ordinary MD on a single potential energy surface. Attempts
have been made to address one manifestation of this, the neglect of decoherence e†ects,
by phenomenological introduction of decoherence into surface-hopping.49 A number of
recent semiclassical developments also appear very promising.50h52
This work was supported by the US National Science Foundation, grant number CHE9707798.
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Paper 8/01824C ; Received 5th March, 1998