Computational Physics

56 The Hartree–Fock method

are filled. Of course, it is not clear a priori that the lowest energy of the system is
found by filling the lowest states of the Fock spectrum because the energy is not
simply a sum over the Fock eigenvalues. However, in practical applications this
turns out to be the case.
The Hartree–Fock theory is the cornerstone of electronic structure calculations
for atoms and molecules. There exists a method, configuration interaction, which
provides a systematic way of improving upon Hartree–Fock theory; it will be
described briefly in Section 4.10. In solid state physics, density functional theory
is used mostly instead of Hartree–Fock theory (see Chapter 5).
The exchange term in (4.31) is a direct consequence of the particle exchange-
antisymmetry of the wave function. It vanishes for orthogonal stateskandl,so
pairs with opposite spin do not feel this term. The self-energy problem with the
Hartree potential mentioned at the end of Section 4.3.1 appears to be solved in the
Hartree–Fock equations: the self-energy term in the Hartree energy is cancelled by
the exchange contribution as a result of the antisymmetry.
The exchange contributionlowersthe Coulomb interaction between the elec-
trons, which can be viewed as a consequence of the fact that exchange keeps
electrons with the same spin apart; see the discussion below Eq. (4.28). The depend-
ence of this change in Coulomb energy on the electron density can be estimated
using a simple classical argument. Suppose that in an electron gas with average
densityn, each electron occupies a volume which is not accessible to other elec-
trons with like spin. This volume can be approximated by a sphere with radius
rc∝n−^1 /^3. Comparing the Coulomb interaction per volume for such a system with
one in which the electrons are distributed homogeneously throughout space, we
obtain

EC≈n^2

[∫∞

rc

r^2 dr

1

r

−

∫∞

0

r^2 dr

1

r

]

=−n^2

∫rc

0

r^2 dr

1

r

∝−n^2 rc^2 ∝n^4 /^3.

(4.33)

One of the two factorsnin front of the integral comes from the average density
seen by one electron, and the second factor counts how many electrons per volume
experience this change in electrostatic energy. Then^4 /^3 dependence of the exchange
contribution is also found in more sophisticated derivations [7] and we shall meet
it again when discussing the local density approximation in the next chapter.

*4.5.2 Derivation of the Hartree–Fock equations

The derivation of the Fock equation consists of performing a variational calcula-
tion for the Schrödinger equation, where the subspace to which we shall confine
ourselves is the space of all single Slater determinants like Eq. (4.27). We must