4.10 Nuclear quantum states

While this chapter has so far dealt only with electronic states in station-

ary environments, nuclear motion, if undisturbed and considered over long

periods of time, will also develop into stationary states, governed by the

time-independent Schr¨dinger equation. The knowledge of such nuclear

o

rotational-vibrational states is useful in connection with infrared and Ra-

man spectroscopy. We shall assume the Born“Oppenheimer approximation

(discussed in Section 4.4) to be valid, i.e., for each nuclear con¬guration

the electronic states are pure solutions of the time-independent Schr¨dinger

o

equation, as if the nuclei do not move, and thus the electrons provide a poten-

tial ¬eld for the nuclear interactions. The electrons have been factored-out

of the complete nuclear-electronic wave function, and the electronic degrees

of freedom do not occur in the nuclear Schr¨dinger equation

o

2

’ ∇2 + V (r 1 , . . . r N ) Ψ = EΨ, (4.70)

i

2mi

i

where i = 1, . . . , N enumerates the nuclei, mi is the nuclear mass, Ψ is

a function of the nuclear coordinates and V is the interaction potential

function of the nuclei, including the in¬‚uence of the electrons. For every

electronic state there is a di¬erent potential function and a di¬erent set of

solutions.

The computation of eigenstates (energies and wave functions) is in princi-

ple not di¬erent from electronic calculations. Since there is always a strong

repulsion at small distances between nuclei in molecules, exchange can be

safely neglected. This considerably reduces the complexity of the solution.

This also implies that the spin states of the nuclei generally have no in¬‚u-

ence on the energies and spatial wave functions of the nuclear eigenstates.

However, the total nuclear wave function of a molecule containing identical

nuclei must obey the symmetry properties of bosons or fermions (whichever

is applicable) when two identical nuclei are exchanged. This leads to sym-

metry requirements implying that certain nuclear states are not allowed.

For isolated molecules or small molecular complexes, the translational

motion factors out, but the rotational and vibrational modes couple into

108 Quantum chemistry: solving the time-independent Schr¨dinger equation

o

vibrational-rotational-tunneling (VRT) states. While the energies and wave

functions for a one-dimensional oscillator can be computed easily by nu-

merical methods, as treated in Section 4.3, in the multidimensional case

the solution is expressed in a suitable set of basis functions. These are

most easily expressed as functions of internal coordinates, like Euler angles

and intramolecular distances and angles, taking symmetry properties into

account. The use of internal coordinates implies that the kinetic energy op-

erator must also be expressed in internal coordinates, which is not entirely

trivial. We note that the much more easily obtained classical solution of

internal vibrational modes corresponds to the quantum solution only in the

case that rotational and vibrational modes are separable, and the vibration

is purely harmonic.

The complete treatment of the VRT states for molecular complexes is be-

yond the scope of this book. The reader is referred to an excellent review by

Wormer and van der Avoird (2000) describing the methods to compute VRT

states in van der Waals complexes like argon-molecule clusters and hydrogen-

bonded complexes like water clusters. Such weakly bonded complexes often

havemultiple minima connected through relatively low saddle-point regions,

thus allowing for e¬ective tunneling between minima. The case of the wa-

ter dimer, for which highly accurate low-frequency spectroscopic data are

available, both for D2 O (Braly et al., 2000a) and H2 O (Braly et al., 2000b),

has received special attention. There are eight equivalent global minima,

all connected by tunneling pathways, in a six-dimensional intermolecular

vibration-rotation space (Leforestier et al., 1997; Fellers et al., 1999). The

comparison of predicted spectra with experiment provides an extremely sen-

sitive test for intermolecular potentials.

5

Dynamics of mixed quantum/classical systems

5.1 Introduction

We now move to considering the dynamics of a system of nuclei and elec-

trons. Of course, both electrons and nuclei are subject to the laws of quan-

tum mechanics, but since nuclei are 2000 to 200 000 times heavier than

electrons, we expect that classical mechanics will be a much better approx-

imation for the motion of nuclei than for the motion of electrons. This

means that we expect a level of approximation to be valid, where some of

the degrees of freedom (d.o.f.) of a system behave essentially classically and

others behave essentially quantum-mechanically. The system then is of a

mixed quantum/classical nature.

Most often the quantum subsystem consists of system of electrons in a

dynamical ¬eld of classical nuclei, but the quantum subsystem may also be

a selection of generalized nuclear coordinates (e.g., corresponding to high-

frequency vibrations) while other generalized coordinates are supposed to

behave classically, or describe the motion of a proton in a classical environ-

ment.

So, in this chapter we consider the dynamics of a quantum system in a

non-stationary potential. In Section 5.2 we consider the time-dependent po-

tential as externally given, without taking notice of the fact that the sources

of the time-dependent potential are moving nuclei, which are quantum par-

ticles themselves, feeling the interaction with the quantum d.o.f. Thus we

consider the time evolution of the quantum system, which now involves

mixing-in of excited states, but we completely ignore the back reaction of

the quantum system onto the d.o.f. that cause the time-dependent poten-

tial, i.e., the moving nuclei. In this way we avoid the main di¬culty of

mixing quantum and classical d.o.f.: how to reconcile the evolution of prob-

ability density of the quantum particles with the evolution of the trajectory

109

110 Dynamics of mixed quantum/classical systems

of the classical particles. The treatment in this approximation is applicable

to some practical cases, notably when the energy exchange between classical

and quantum part is negligible (this is the case, for example, for the motion

of nuclear spins in a bath of classical particles at normal temperatures), but

will fail completely when energy changes in the quantum system due to the

external force are of the same order as the energy ¬‚uctuations in the classi-

cal system. How the quantum subsystem can be properly embedded in the

environment, including the back reaction, is considered in Section 5.3.

As we shall see in Section 5.2, the e¬ect of time-dependent potentials is

that initially pure quantum states evolve into mixtures of di¬erent states.

For example, excited states will mix in with the ground state as a result

of a time-dependent potential. Such time dependence may arise from a

time-dependent external ¬eld, as a radiation ¬eld that causes the system to

“jump” to an excited state. It may also arise from internal interactions, such

as the velocity of nuclei that determine the potential ¬eld for the quantum

system under consideration, or from thermal ¬‚uctuations in the environ-

ment, as dipole ¬‚uctuations that cause a time-dependent electric ¬eld. The

wave functions that result do not only represent additive mixtures of dif-

ferent quantum states, but the wave function also carries information on

phase coherence between the contributing states. The mixed states will in

turn relax under the in¬‚uence of thermal ¬‚uctuations that cause dephasing

of the mixed states. The occurrence of coherent mixed states is a typical

quantum behavior, for which there is no classical analog. It is the cause

of the di¬culty to combine quantum and classical treatments, and of the

di¬culty to properly treat the back reaction to the classical system. The

reason is that the wave function of a dephased mixed state can be viewed as

the superposition of di¬erent quantum states, each with a given population.

Thus the wave function does not describe one trajectory, but rather a prob-

ability distribution of several trajectories, each with its own back reaction

to the classical part of the system. If the evolution is not split into several

trajectories, and the back reaction is computed as resulting from the mixed

state, one speaks of a mean-¬eld solution, which is only an approximation.

When the quantum system is in the ground state, and all excited states

have energies so high above the ground state that the motions of the “clas-

sical” degrees of freedom in the system will not cause any admixture of

excited states, the system remains continuously in its ground state. The

evolution now is adiabatic, as there is no transfer of “heat” between the

classical environment and the quantum subsystem. In that case the back

reaction is simply the expectation of the force over the ground-state wave

function, and a consistent mixed quantum-classical dynamics is obtained.

5.1 Introduction 111