4 The Hartree–Fock method
4.1 Introduction
Here and in the following chapter we treat two different approaches to the many-
electron problem: the Hartree–Fock theory and the density functional theory. Both
theories are simplifications of the full problem of many electrons moving in a poten-
tial field. In fact, the physical systems we want to study, such as atoms, molecules
and solids, consist not only of electrons but also of nuclei, and each of these particles
moves in the field generated by the others. A first approximation is to consider the
nuclei as being fixed, and to solve the Schrödinger equation for the electronic sys-
tem in the field of the static nuclei. This approach, called the Born–Oppenheimer
approximation, is justified by the nuclei being much heavier than the electrons so
that they move at much slower speeds. It remains then to solve for the electronic
structure.
The Hartree–Fock methodcan be viewed as a variational method in which the
wave functions of the many-electron system have the form of an antisymmetri-
sed product of one-electron wave functions (the antisymmetrisation is necessary
because of the fermion character of the electrons). This restriction leads to an effect-
ive Schrödinger equation for the individual one-electron wave functions (called
orbitals) with a potential determined by the orbitals occupied by the other elec-
trons. This coupling between the orbitals via the potentials causes the resulting
equations to become nonlinear in the orbitals, and the solution must be found iter-
atively in a self-consistency procedure. The Hartree–Fock (HF)procedure is close
in spirit to the mean-field aproach in statistical mechanics.
We shall see that in this variational approach, correlations between the electrons
are neglected to some extent. In particular, the Coulomb repulsion between the
electrons is represented in an averaged way. However, the effective interaction
caused by the fact that the electrons are fermions, obeying Pauli’s principle, and
hence avoid each other if they have the same spin, is accurately included in the HF
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