(m/M )1/4 ; they also show that the ¬rst and third order in the expansion

vanish. The zero-order approximation assumes that the nuclear mass is in-

¬nite, and therefore that the nuclei are stationary and their role is reduced

to that of source of electrical potential for the electrons. This zero-order or

clamped nuclei approximation is usually meant when one speaks of the B“O

approximation per se. When nuclear motion is considered, the electrons ad-

just in¬nitely fast to the nuclear position or wave function in the zero-order

B“O approximation; this is the adiabatic limit. In this approximation the

nuclear motion causes no changes in the electronic state, and the nuclear

motion “ both classical and quantum-mechanical “ is governed by an e¬ec-

tive internuclear potential resulting from the electrons in their “stationary”

state.

The e¬ect of the adiabatic approximation on the energy levels of the

10 Internet site: http://www.zori-code.com.

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

o

hydrogen atom (where e¬ects are expected to be most severe) is easily eval-

uated. Disregarding relativistic corrections, the energy for a single electron

atom with nuclear charge Z and mass M for quantum number n = 1, 2, . . .

equals

1 2μ

E=’ Z hartree, (4.44)

2n2 m

where μ is the reduced mass mM/(m + M ). All energies (and hence spec-

troscopic frequencies) scale with μ/m = 0.999 455 679. For the ground state

of the hydrogen atom this means:

E(adiabatic) = ’0.500 000 000 hartree,

E(exact) = ’0.499 727 840 hartree,

adiabatic error = ’0.000 272 160 hartree ,

= ’0.714 557 kJ/mol.

Although this seems a sizeable e¬ect, the e¬ect on properties of molecules

is small (Handy and Lee, 1996). For example, since the adiabatic correction

to H2 amounts to 1.36 kJ/mol, the dissociation energy D0 of the hydrogen

molecule increases by only 0.072 kJ/mol (on a total of 432 kJ/mol). The

bond length of H2 increases by 0.0004 a.u. or 0.0002 ˚and the vibrational

A

’1 ) decreases by about 3 cm’1 . For heavier atoms the

frequency (4644 cm

e¬ects are smaller and in all cases negligible. Handy and Lee (1996) conclude

that for the motion of nuclei the atomic masses rather than the nuclear

masses should be taken into account. This amounts to treating the electrons

as “following” the nuclei, which is in the spirit of the BO-approximation.

The real e¬ect is related to the quantum-dynamical behavior of moving

nuclei, especially when there are closely spaced electronic states involved.

Such e¬ects are treated in the next chapter.

4.5 The many-electron problem of quantum chemistry

Traditionally, the main concern of the branch of theoretical chemistry that is

called quantum chemistry is to ¬nd solutions to the stationary Schr¨dinger

o

equation for a system of (interacting) electrons in a stationary external ¬eld.

This describes isolated molecules in the Born“Oppenheimer approximation.

There are essentially only two radically di¬erent methods to solve Schr¨- o

dinger™s equation for a system of many (interacting) electrons in an external

¬eld: Hartree“Fock methods with re¬nements and Density Functional The-

ory (DFT). Each requires a book to explain the details (see Szabo and

4.6 Hartree“Fock methods 99

Ostlund, 1982; Parr and Yang, 1989), and we shall only review the princi-

ples of these methods.

Statement of the problem

We have N electrons in the ¬eld of M point charges (nuclei). The point

charges are stationary and the electrons interact only with electrostatic

Coulomb terms. The electrons are collectively described by a wave function,

which is a function of 3N spatial coordinates r i and N spin coordinates ωi ,

which we combine into 4N coordinates xi = r i , ωi . Moreover, the wave

function is antisymmetric for exchange of any pair of electrons (parity rule

for fermions) and the wave functions are solutions of the time-independent

Schr¨dinger equation:

o

ˆ

HΨ = EΨ (4.45)

Ψ(x1 , . . . , xi , . . . , xj , . . . , ) = ’Ψ(x1 , . . . , xj , . . . , xi , . . . , ), (4.46)

N N M N

2 zk e2 e2

ˆ

H=’ ∇2 ’ + . (4.47)

i

2m 4πµ0 rik 4πµ0 rij

i=1 i=1 k=1 i,j=1;i<j

By expressing quantities in atomic units (see page xvii), the Hamiltonian

becomes

1 zk 1

ˆ

H=’ ∇2 ’ + . (4.48)

i

2 rik rij

i i i<j

k

ˆ

Note that H is real, which implies that Ψ can be chosen to be real (if Ψ is

a solution of (4.45), then Ψ— is a solution as well for the same energy, and

so is (Ψ).)

4.6 Hartree“Fock methods

The Hartree“Fock description of the wave function is in terms of products of

one-electron wave functions ψ(r) that are solutions of one-electron equations

(what these equations are will be described later). The one-electron wave

functions are built up as a linear combination of spatial basis functions:

K

ψi (r) = cμi φμ (r). (4.49)

μ=1

If the set of spatial basis functions would be complete (requiring an in¬nite

set of functions), the one-electron wave function could be exact solutions of

the one-electron wave equation; in practise one selects a ¬nite number of

appropriate functions, generally

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

o

“Slater-type” functions that look like the 1s, 2s, 2p, ... hydrogen atom

functions, which are themselves for computational reasons often composed

of several local Gaussian functions. The one-electron wave functions are

therefore approximations.

The one-electron wave functions are ortho-normalized:

—

ψi |ψj = ψi (r)ψj (r) dr = δij , (4.50)

and are completed to twice as many functions χ with spin ± or β:

χ2i’1 (x) = ψi (r)±(ω), (4.51)

χ2i (x) = ψi (r)β(ω). (4.52)

These χ-functions are also orthonormal, and are usually called Hartree“Fock

spin orbitals.

In order to construct the total wave function, ¬rst the N -electron Hartree

product function is formed:

ΨHP = χi (x1 )χj (x2 ) . . . χk (xN ), (4.53)

but this function does not satisfy the fermion parity rule. For example, for

two electrons:

ΨHP (x1 , x2 ) = χi (x1 )χj (x2 )

= ’χj (x1 )χi (x2 ),

while the following antisymmetrized function does:

1

Ψ(x1 , x2 ) = 2’ 2 [χi (x1 )χj (x2 ) ’ χj (x1 )χi (x2 )]

1 χi (x1 ) χj (x1 )

=√ . (4.54)

χi (x2 ) χj (x2 )

2

In general, antisymmetrization is obtained by constructing the Slater deter-

minant:

· · · χk (x1 )

χi (x1 ) χj (x1 )

· · · χk (x2 )