1

Ψ(x1 , x2 ), . . . , xN ) = √ . (4.55)

. .

. .

N . .

χi (xN ) χj (xN ) · · · χk (xN )

This has antisymmetric parity because any exchange of two rows (two par-

ticles) changes the sign of the determinant. The Slater determinant is ab-

breviated as Ψ = |χi χj . . . χk .

Thus far we have not speci¬ed how the one-electron wave functions ψi ,

4.6 Hartree“Fock methods 101

and hence χi , are obtained. These functions are solutions of one-dimensional

ˆ

eigenvalue equations with a special Fock operator f (i) instead of the hamil-

tonian:

ˆ

f (i)χ(r i ) = µχ(r i ) (4.56)

with

1 zk

ˆ

f (i) = ’ ∇2 ’ + v HF (i) (4.57)

2i rik

k

Here v HF is an e¬ective mean-¬eld potential that is obtained from the com-

bined charge densities of all other electrons. Thus, in order to solve this

equation, one needs an initial guess for the wave functions, and the whole

procedure needs an iterative approach until the electronic density distribu-

tion is consistent with the potential v (self-consistent ¬eld, SCF).

For solving the eigenvalue equation (4.56) one applies the variational prin-

ciple: for any function ψ = ψ0 , where ψ0 is the exact ground state solution

ˆ ˆ

of the eigenvalue equation Hψ = Eψ, the expectation value of H does not

exceed the exact ground state eigenvalue E0 :

ψ — Hψ dr

ˆ

≥ E0 (4.58)

ψ — ψ dr

The wave function χ is varied (i.e., the coe¬cients of its expansion in ba-

sis functions are varied) while keeping χ— χ dr = 1, until χ— f χ dr is a

minimum.

The electrons are distributed over the HF spin orbitals χ and form a

con¬guration. This distribution can be done by ¬lling all orbitals from

the bottom up with the available electrons, in which case a ground state

con¬guration is obtained. The energy of this “ground state” is called the

Hartree“Fock energy, with the Hartree“Fock limit in the case that the basis

set used approaches an in¬nite set.

But even the HF limit is not an accurate ground state energy because the

whole SCF-HF procedure neglects the correlation energy between electrons.

Electrons in the same spatial orbital (but obviously with di¬erent spin state)

tend to avoid each other and a proper description of the two-electron wave

function should take the electron correlation into account, leading to a lower

energy. This is also the case for electrons in di¬erent orbitals. In fact,

the London dispersion interaction between far-away electrons is based on

electron correlation and will be entirely neglected in the HF approximation.

The way out is to mix other, excited, con¬gurations into the description

of the wave function; in principle this con¬guration interaction (CI) allows

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

o

for electron correlation. In practise the CI does not systematically converge

and requires a huge amount of computational e¬ort. Modern developments

use a perturbative approach to the electron correlation problem, such as the

popular Møller“Plesset (MP) perturbation theory. For further details see

Jensen (1999).

4.7 Density functional theory

In SCF theory electron exchange is introduced through the awkward Slater

determinant, while the introduction of electron correlation presents a major

problem by itself. Density functional theory (DFT) o¬ers a radically di¬er-

ent approach that leads to a much more e¬cient computational procedure.

Unfortunately it is restricted to the ground state of the system. It has one

disadvantage: the functional form needed to describe exchange and correla-

tion cannot be derived from ¬rst principles. In this sense DFT is not a pure

ab initio method. Nevertheless: in its present form DFT reaches accura-

cies that can be approached by pure ab initio methods only with orders of

magnitude higher computational e¬ort. In addition, DFT can handle much

larger systems.

The basic idea is that the electron charge density ρ(r) determines the exact

ground state wave function and energy of a system of electrons. Although

the inverse of this statement is trivially true, the truth of this statement

is not obvious; in fact this statement is the ¬rst theorem of Hohenberg and

Kohn (1964). It can be rigorously proven. An intuitive explanation was

once given by E. Bright Wilson at a conference in 1965:11 assume we know

ρ(r). Then we see that ρ shows sharp maxima (cusps) at the positions of the

nuclei. The local nuclear charge can be derived from the limit of the gradient

of ρ near the nucleus, since at the nuclear position |∇ρ| = ’2zρ. Thus, from

the charge density we can infer the positions and charges of the nuclei. But

if we know that and the number of electrons, the Hamiltonian is known

and there will be a unique ground state solution to the time-independent

Schr¨dinger equation, specifying wave function and energy.

o

Thus the energy and its constituent terms are functionals of the density

ρ:

E[ρ] = Vne [ρ] + K[ρ] + Vee [ρ] (4.59)

where

Vne = ρ(r)vn (r) dr (4.60)

11 Bright Wilson (1968), quoted by Handy (1996).

4.7 Density functional theory 103

is the electron“nuclear interaction, with vn the potential due to the nuclei,

K is the kinetic energy of the electrons, and Vee is the electron“electron

interaction which includes the mutual Coulomb interaction J:

’1

1

J[ρ] = dr 1 dr 2 r12 ρ(r 1 )ρ(r 2 ), (4.61)

2

as well as the exchange and correlation contributions. Now, the second

theorem of Hohenberg and Kohn states that for any density distribution

ρ = ρ (where ρ is the exact ground state density), the energy is never

smaller than the true ground state energy E:

E[ρ ] ≥ E[ρ]. (4.62)

Thus ¬nding ρ and E reduces to applying the variational principle to E[ρ],

i.e., minimizing E by varying ρ(r), while keeping ρ(r) dr = N . Such

a solution would provide the ground state energy and charge distribution,

which is all we want to know: there is no need for knowledge of the detailed

wave function. There is a slight problem, however: the functional form of

the terms in (4.59) is not known!

A practical solution was provided by Kohn and Sham (1965), who consid-

ered the equations that a hypothetical system of N non-interacting electrons

should satisfy in order to yield the same density distribution as the real sys-

tem of interacting electrons. Consider N non-interacting electrons in 1 N 2

(+ 1 for odd N ) orbitals; the total properly antisymmetrized wave funct-

2

ion would be the Slater determinant of the occupied spin-orbitals. For this

system the exact expressions for the kinetic energy and the density are

φ— (’ 1 ∇2 )φi dr,

Ks [ρ] = ni (4.63)

i 2

i=1

ni φ— φi (r).

ρ[r] = (4.64)

i

i=1

Here ni = 1 or 2 is the number of electrons occupying φi . The wave functions

are solutions of the eigenvalue equation

{’ 1 ∇2 + vs (r)}φi = µi φi , (4.65)