d

d d

B

dA

d

A

d

d

d

T d

T T

d

d

d d

E ”E0

d

d splitting 2C

d

energy

d

d d

d c c

d

d d

B dA B

d

d d

d

d

E nuclear coordinate

Figure 5.1 Two crossing diabatic states A and B. Precise solution in the crossing

region (right) yields two adiabatic non-crossing states A and B . Time dependence

may cause transitions in the crossing region.

One method in this category, the Car“Parrinello method, also referred to

as ab initio molecular dynamics (see Section 6.3.1), has proved to be very

successful for chemically reactive systems in the condensed phase.

In cases where excited states are relevant, adiabatic dynamics is not suf-

¬cient and the separation between quantum and classical d.o.f. is no longer

trivial. Now we are fully confronted with the question how to treat the evo-

lution into a multitude of trajectories and how to evaluate the back reaction

of the quantum system onto the classical d.o.f.

Consider the case that the quantum system develops into a mixture of two

“pure” states. This could easily happen if the trajectory arrives at a point

where the quantum system is degenerate or almost degenerate, i.e., where

two states of the quantum system cross or nearly cross (see Fig. 5.1). When

there is a small coupling term H12 = H21 = C between the two states, the

hamiltonian in the neighborhood of the crossing point will be:

’ 1 ”E0 C

2

H= , (5.1)

1

C 2 ”E0

and the wave functions will mix. The eigenvalues are

1

E1,2 = ± (”E0 )2 + C 2 . (5.2)

4

112 Dynamics of mixed quantum/classical systems

At the crossing point (”E0 = 0), the adiabatic solutions are equal mixtures

of both diabatic states, with a splitting of 2C. Then there will be essentially

two trajectories of the classical system possible, each related to one of the

pure states. The system “choses” to evolve in either of the two branches.

Only by taking the quantum character of the “classical” system into account

can we fully understand the behavior of the system as a whole; the full wave

function of the complete system would describe the evolution. Only that full

wave function will contain the branching evolution into two states with the

probabilities of each. In that full wave function the two states would still be

related to each other in the sense that the wave functions corresponding to

the two branches are not entirely separated; their phases remain correlated.

In other words, a certain degree of “coherence” remains also after the “split-

ting” event, until random external disturbances destroy the phase relations,

and the two states can be considered as unrelated. The coherence is related

to reversibility: as long as the coherence is not destroyed, the system is

time-reversible and will retrace its path if it returns to the same con¬gura-

tion (of the “classical” d.o.f.) where the splitting originally occurred. Such

retracing may occur in small isolated systems (e.g., a diatomic molecule in

the gas phase) if there is a re¬‚ection or a turning point for the classical

d.o.f., as with the potential depicted in Fig. 5.2; in many-particle systems

such revisiting of previously visited con¬gurations becomes very unlikely.

If in the mean time the coherence has been destroyed, the system has lost

memory of the details of the earlier splitting event and will not retrace to its

original starting point, but develop a new splitting event on its way back.

If we would construct only one trajectory based on the expectation value

of the force, the force would be averaged over the two branches, and “

assuming symmetry and equal probabilities for both branches (Fig. 5.1)

after the splitting “ the classical d.o.f. would feel no force and proceed

with constant velocity. In reality the system develops in either branch A,

continuously accelerating, or branch B, decelerating until the turning point

is reached. It does not do both at the same time. Thus, the behavior

based on the average force, also called the mean-¬eld treatment, is clearly

incorrect. It will be correct if the system stays away from regions where

trajectories may split up into di¬erent branches, but cannot be expected to

be correct if branching occurs.

In Section 5.3 simulations in a mixed quantum-classical system with back

reaction are considered. The simplest case is the mean-¬eld approach (Sec-

tion 5.3.1), which gives consistent dynamics with proper conservation of

energy and momentum over the whole system. However, it is expected to

be valid only for those cases where either the back reaction does not notice-

5.1 Introduction 113

E

U

A B

q

T

G

L

nuclear coordinate

Figure 5.2 Two crossing diabatic states G and E, developing into two adiabatic

states U (upper) and L (lower). After excitation from G to E (reaching point A) the

system either stays on the adiabatic level U or crosses diabatically to L, depending

on coupling to dynamical variables in the crossing region. If it stays in U, it reaches

a turning point B and retraces its steps, if in the meantime no dephasing has taken

place due to external ¬‚uctuations. With dephasing, the system may cross to G on

the way back and end up vibrating in the ground state.

ably in¬‚uence the classical system, or the nuclei remain localized without

branching. An approximation that catches the main de¬ciency of the mean-

¬eld treatment is the surface-hopping procedure (Section 5.3.3), introduced

by Tully (1990). The method introduces random hops between di¬erent

potential energy surfaces, with probabilities governed by the wave function

evolution. So the system can also evolve on excited-state surfaces, and in-

corporate non-adiabatic behavior. Apart from the rather ad hoc nature of

the surface-hopping method, it is incorrect in the sense that a single, be

it stochastic, trajectory is generated and the coherence between the wave