has been derived. In Eq. (17.112) on page 479 a correction for the fermion

or boson character of the particle is given in the form of a repulsive or attrac-

tive short-range potential. As shown in Fig. 17.6, the exchange correction

potential for nuclei can be neglected in all but very exceptional cases.

76 From quantum to classical mechanics: when and how

Exercises

3.1 Check the expansions (3.113) and (3.114).

3.2 Give an analytical expression for the Feynman-Hibbs potential in

the approximation of (3.115) for a Lennard-Jones interaction.

3.3 Evaluate both the full integral (3.112) and the approximation of the

previous exercise for a He“He interaction at T = 40 K. Plot the

results.

3.4 Apply (3.120) to compute the partition function and the Helmholtz

free energy of a system of N non-interacting harmonic oscillators and

prove the correctness of the result by expanding the exact expression

(from (17.84) on page 472).

4

Quantum chemistry: solving the time-independent

Schr¨dinger equation

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4.1 Introduction

As has become clear in the previous chapter, electrons (almost) always be-

have as quantum particles; classical approximations are (almost) never valid.

In general one is interested in the time-dependent behavior of systems con-

taining electrons, which is the subject of following chapters.

The time-dependent behavior of systems of particles spreads over very

large time ranges: while optical transitions take place below the femtosecond

range, macroscopic dynamics concerns macroscopic times as well. The light

electrons move considerably faster than the heavier nuclei, and collective

motions over many nuclei are slower still. For many aspects of long-time

behavior the motion of electrons can be treated in an environment considered

stationary. The electrons are usually in bound states, determined by the

positions of the charged nuclei in space, which provide an external ¬eld for

the electrons. If the external ¬eld is stationary, the electron wave functions

are stationary oscillating functions. The approximation in which the motion

of the particles (i.e., nuclei) that generate the external ¬eld, is neglected,

is called the Born“Oppenheimer approximation. Even if the external ¬eld

is not stationary (to be treated in Chapter 5), the non-stationary solutions

for the electronic motion are often expressed in terms of the pre-computed

stationary solutions of the Schr¨dinger equation. This chapter concerns the

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computation of such stationary solutions.

Thus, in this chapter, the Schr¨dinger equation reduces to a time-indepen-

o

dent problem with a stationary (i.e., still time-dependent, but periodic)

solution. Almost all of chemistry is covered by this approximation. It is not

surprising, therefore, that theoretical chemistry has been almost equivalent

to quantum chemistry of stationary states, at least up to the 1990s, when

77

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

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the scope of theory in chemistry slowly started to be broadened to include

the study of more complex dynamic behavior.

For completeness, in the last section of this chapter attention will be given

to the stationary quantum behavior of nuclei, rather than electrons. This

includes the rotational and vibrational steady state behavior of molecules,

which is useful in spectroscopic (infrared and Raman) studies, in the predic-

tion of spectroscopic behavior by simulations, or in the use of spectroscopic

data to evaluate force ¬elds designed for simulations.

4.2 Stationary solutions of the TDSE

The general form of the time-dependent Schr¨dinger equation (TDSE) is

o

‚Ψ ˆ

i = HΨ, (4.1)

‚t

where the usual (already simpli¬ed!) form of the Hamiltonian is that of

(2.72). If the Hamiltonian does not contain any explicit time dependence

and is only a function of the particle coordinates and a stationary external

potential, the TDSE has stationary solutions that represent bound states:

i

Ψn (r, t) = ψn (r) exp ’ En t , (4.2)

where ψn (r) and En are solutions of the eigenvalue equation

ˆ

Hψ(r) = Eψ(r). (4.3)

The latter is also called the time-independent Schr¨dinger equation

o

The spatial parts of the wave functions are stationary in time, and so is

the probability distribution Ψ— Ψn for each state.

n

In this chapter we shall look at ways to solve the time-independent Schr¨-

o

dinger equation, (4.3), assuming stationary external ¬elds. In chapter 5 we

consider how a quantum system behaves if the external ¬eld is not stationary,

for example if the nuclei move as well, or if there are external sources for

¬‚uctuating potentials.

There are several ways in which ab initio solutions of the time-independent

Schr¨dinger equation can be obtained. In quantum physics the emphasis is

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often on the behavior of a number of quantum particles, which are either

bosons, as in helium-4 liquids, or fermions as electrons in (semi)conductors

or in helium-3 liquids. In chemistry the main concern is the structure and

properties of single atoms and molecules; especially large molecules with

many electrons pose severe computational problems and elude exact treat-

ment.

4.3 The few-particle problem 79

Before considering methods to solve the many-electron problem, we shall

look into the methods that are available to ¬nd the stationary solution for

one or a few interacting quantum particles. Then we consider the question

whether it will be possible to separate the nuclear motion from the elec-

tronic motion in atoms and molecules: this separation is the essence of the

Born“Oppenheimer approximation. When valid, the electronic motion can

be considered in a stationary external ¬eld, caused by the nuclei, while the

nuclear motion can be described under the in¬‚uence of an e¬ective potential

caused by the electrons.

4.3 The few-particle problem

Let us ¬rst turn to simple low-dimensional cases. Mathematically, the SE

is a boundary-value problem, with acceptable solutions only existing for

discrete values of E. These are called the eigenvalues, and the corresponding

solutions the eigenfunctions. The boundary conditions are generally zero

values of the wave function at the boundaries,1 and square-integrability of

the function, i.e., ψ — ψ(x) dx must exist. As any multiple of a solution is

also a solution, this property allows to normalize each solution, such that

the integral of its square is equal to one.

Since any Hamiltonian is Hermitian (see Chapter 14), its eigenvalues E are

real. But most Hamiltonians are also real, except when velocity-dependent

potentials as in magnetic interactions occur. Then, when ψ is a solution,

also ψ — is a solution for the same eigenvalue, and the sum of ψ and ψ — is

a solution as well. So the eigenfunctions can be chosen as real functions.

Often, however, a complex function is chosen instead for convenience. For

example, instead of working with the real functions sin mφ and cos mφ,

one may more conveniently work with the complex functions exp(imφ) and

exp(’imφ). Multiplying a wave function by a constant exp(ia) does not

change any of the physical quantities derived from the wave function.

Consider a single quantum particle with mass m in a given, stationary,

external potential V (x). We shall not treat the analytical solutions for

simple cases such as the hydrogen atom, as these can be found in any text

book on quantum physics or theoretical chemistry. For the one-dimensional

case there are several ways to solve the time-independent SE numerically.

1 In the case of periodic boundary conditions, the wave function and its derivatives must be

continuous at the boundary.

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

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