deviate from exponential behavior for short times, but will develop into an

exponential tail. Its correlation time D’1 is given by the inverse of (5.75);

it is inversely proportional to the correlation time of ωz . The faster ωz

¬‚uctuates, the slower cos φ will decay and the smaller its in¬‚uence on the

reaction rate will be.

We end by noting, again, that in simulations the approximations re-

quired for analytical solutions need not be made; the reaction rates can

be computed by solving the complete density matrix evolution based on

time-dependent perturbations obtained from simulations.

5.2.5 The multi-level system

The two-state case is able to treat the dynamics of tunneling processes in-

volving two nearby states, but is unable to include transitions to low-lying

excited states. The latter are required for a full non-adiabatic treatment of

a transfer process. The extension to the multi-level case is straightforward,

but the analogy with a three-dimensional rotating top is then lost. The ba-

sis functions should be chosen orthogonal, but they need not be solutions of

any stationary Schr¨dinger equation. Nevertheless, it is usually convenient

o

and e¬cient to consider a stationary average potential and construct a set

of basis functions as solutions of the Schr¨dinger equation with that poten-

o

tial. In this way one can be sure that the basis set adequately covers the

required space and includes the ¬‚exibility to include low-lying excited states.

Mavri and Berendsen (1995) found that ¬ve basis functions, constructed by

diagonalization of ¬ve Gaussians, were quite adequate to describe proton

transfer over a hydrogen bond in aqueous solution. They also conclude that

the use of only two Gaussians is inadequate: it underestimates the transfer

rate by a factor of 30! A two-level system can only describe ground-state

tunneling and does not allow paths involving excited states; it also easily

underestimates the coupling term because the barrier region is inadequately

described.

5.3 Embedding in a classical environment 129

In the multi-level case the transfer rate cannot simply be identi¬ed with

the course-grained decay rate of the population, such as ρ11 in the two-level

case (5.58). This is because a “reactant state” or “product state” cannot be

identi¬ed with a particular quantum level. The best solution is to de¬ne a

spatial selection function S(r) with the property that it is equal to one in

the spatial domain one wishes to identify with a particular state and zero

elsewhere. The probability pS to be in that state is then given by

Ψ— ΨS(r) dr = tr (ρS),

pS = (5.78)

with

φ— φm S(r) dr.

Snm = (5.79)

n

5.3 Embedding in a classical environment

Thus far we have considered how a quantum (sub)system develops when it

is subjected to time-dependent perturbing in¬‚uences from classical degrees

of freedom (or from external ¬elds). The system invariably develops into a

mixed quantum state with a wave function consisting of a superposition of

eigenfunctions, even if it started from a pure quantum state. We did not ask

the question whether a single quantum system indeed develops into a mixed

state, or ends up in one or another pure state with a certain probability gov-

erned by the transition rates that we could calculate. In fact that question

is academic and unanswerable: we can only observe an ensemble containing

all the states that the system can develop into, and we cannot observe the

fate of a single system. If a single system is observed, the measurement can

only reveal the probability that a given ¬nal state has occurred. The com-

mon notion among spectroscopists that a quantum system, which absorbs a

radiation quantum, suddenly jumps to the excited state, is equally right or

wrong as the notion that such a quantum system gradually mixes the ex-

cited state into its ground state wave function in the process of absorbing a

radiation quantum. Again an academic question: we don™t need to know, as

the outcome of an experiment over an ensemble is the same for both views.

We also have not considered the related question how the quantum sys-

tem reacts back onto the classical degrees of freedom. In cases where the

coupling between quantum and classical degrees if freedom is weak (as, e.g.,

in nuclear spins embedded in classical molecular systems), the back reaction

has a negligible e¬ect on the dynamics of the classical system and can be

disregarded. The classical system has its autonomous dynamics. This is also

true for a reaction (such as a proton transfer) in the very ¬rst beginning,

130 Dynamics of mixed quantum/classical systems

when the wave function has hardly changed. However, when the coupling

is not weak, the back reaction is important and essential to ful¬ll the con-

servation laws for energy and momentum. Now it ` important whether the

±s

single quantum system develops into a mixed state or a pure state, with very

di¬erent strengths of the back reaction. For example, as already discussed

on page 112, after a “crossing event” the system “chooses” one branch or an-

other, but not both, and it reacts back onto the classical degrees of freedom

from one branch only. Taking the back reaction from the mixed quantum

wave function “ which is called the mean ¬eld back reaction “ is obviously

incorrect. Another example (Tully, 1990) is a particle colliding with a solid

surface, after which it either re¬‚ects back or gets adsorbed. One observes

20% re¬‚ection and 80% absorption, for example, but not 100% of something

in between that would correspond to the mean ¬eld reaction.

It now seems that an academic question that cannot be answered and

has no bearing on observations, suddenly becomes essential in simulations.

Indeed, that is the case, and the reason is that we have been so stupid as

to separate a system into quantum and classical degrees of freedom. If a

system is treated completely by quantum mechanics, no problems of this

kind arise. For example, if the motion along the classical degree of freedom

in a level-crossing event is handled by a Gaussian wave packet, the wave

packet splits up at the intersection and moves with certain probability and

phase in each of the branches (see, for example, Hahn and Stock (2000) for a

wave-packet description of the isomerization after excitation in rhodopsin).

But the arti¬cial separation in quantum and classical parts calls for some

kind of contraction of the quantum system to a single trajectory and an

arti¬cial handling of the interaction.6 The most popular method to achieve

a consistent overall dynamics is the surface hopping method of Tully (1990),

described in Section 5.3.3.

5.3.1 Mean-¬eld back reaction

We consider how the evolution of classical and quantum degrees of freedom

can be solved simultaneously in such a way that total energy and momentum

are conserved. Consider a system that can be split up in classical coordinates

(degrees of freedom) R and quantum degrees of freedom r, each with its

conjugated momenta. The

total Hamiltonian of the system is a function of all coordinates and mo-

menta. Now assume that a proper set of orthonormal basis functions φn (r; R)

6 For a review of various methods to handle the dynamics at level crossing (called conical inter-

sections in more-dimensional cases), see Stock and Thoss (2003).

5.3 Embedding in a classical environment 131

ˆ

has been de¬ned. The Hamiltonian is an operator H in r-space and is rep-

resented by a matrix with elements

ˆ

Hnm (q, p) = n|H(r, R, P )|m , (5.80)

where P are the momenta conjugated with R. These matrix elements can

be evaluated for any con¬guration (R, P ) in the classical phase space. Using

this Hamiltonian matrix, the density matrix ρ evolves according to (5.29) or

(5.37), which reduces to (5.35) for basis functions that are independent of

the classical coordinates. The classical system is now propagated using the

quantum expectation of the forces F and velocities:

™ ˆ

F = P = ’ tr (ρF), with Fnm = n| ’ ∇R H|m , (5.81)

™

R = tr (ρ∇P H). (5.82)

The latter equation is “ for conservative forces “ simply equal to

™

R = ∇P K, (5.83)

™

or V = R for cartesian particle coordinates, because the classical kinetic

energy K is a separable term in the total Hamiltonian.

It can be shown (see proof below) that this combined quantum/classical

dynamics conserves the total energy of the system:

dEtot d

= tr (ρH) = 0. (5.84)

dt dt