Computational Physics

44 The Hartree–Fock method

approach. There exist several methods that improve on the approximations made
in the HF method.
The Hartree–Fock approach is very popular among chemists, and it has also
been applied to solids. In this chapter, we give an introduction to the Hartree–
Fock method and apply it to simple two-electron systems: the helium atom and the
hydrogen molecule. We describe the Born–Oppenheimer approach and independent
particle approaches (of which HF is an example) in a bit more detail in the next
section. In Section 4.3 we then derive the Hartree method for a two-electron system
(the helium atom). In Section 4.3.2, a program for calculating the ground state of
the helium atom is described.
In Sections 4.4 and 4.5 the HF method for systems containing more than two
electrons is described in detail, and in Section 4.6 the basis functions used for
molecular systems are described. In sections 4.7 and 4.8 some details concerning
the implementation of the HF method are considered. In Section 4.9, results of the
HF method are presented, and in Section 4.10 the configuration interaction (CI)
method, which improves on the HF method is described.

4.2 The Born–Oppenheimer approximation and the independent-particle

method

The Hamiltonian of a system consisting ofNelectrons andKnuclei with charges
Znreads

H=

∑N

i= 1

p^2 i

2 m

+

∑K

n= 1

Pn^2

2 Mn

+

1

4 π 0

1

2

∑N

i,j=1;i
=j

e^2

|ri−rj|

−

1

4 π 0

∑K

n= 1

∑N

i= 1

Zne^2

|ri−Rn|

+

1

4 π 0

1

2

∑K

n,n′=1;n
=n′

ZnZn′e^2

|Rn−Rn′|

. (4.1)

The indexirefers to the electrons andnto the nuclei,mis the electron mass,
andMnare the masses of the different nuclei. The first two terms represent the
kinetic energies of the electrons and nuclei respectively; the third term represents
the Coulomb repulsion between the electrons and the fourth term the Coulomb
attraction between electrons and nuclei. Finally, the last term contains the Coulomb
repulsion between the nuclei. The wave function of this system depends on the
positionsriandRnof the electrons and nuclei respectively. This Hamiltonian looks
quite complicated, and in fact it turns out that if the number of electrons and nuclei
is not extremely small (typically smaller than four), it is impossible to solve the
stationary Schrödinger equation for this Hamiltonian directly on even the largest
and fastest computer available.