4.5 Self-consistency and exchange: Hartree–Fock theory 55

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(a) (b) (c)

Figure 4.1. The Hartree–Fock spectrum. The figure shows how the levels are filled

for (a) the ground state of an even number of electrons, (b) the ground state of an

odd number of electrons and (c) an excited state in the spectrum of (a). Note that

the spectrum in (c) does not correspond to the ground state; see Section 4.5.3.

Instead it corresponds to the restricted approximation, in which the same set of

energy levels is available for electrons with both spins.

the same as the third, with two spin-orbital labelskandlinterchanged and a minus
sign in front resulting from the antisymmetry of the wave function – it is called the
exchange term. Note that this term is nonlocal: it is an operator acting onψk,but
its value atris determined by the value assumed byψkat all possible positionsr′.
A subtlety is that the eigenvalueskof the Fock operator are not the energies of
single electron orbitals, although they are related to the total energy by

E=

1

2

∑

k

[k+〈ψk|h|ψk〉]. (4.32)

InSection 4.5.3we shall see that the individual levelskcan be related to excitation
energies within some approximation.
It is clear that(4.31)is a nonlinear equation, which must be solved by a self-
consistency iterative procedure analogously to the previous section. Sometimes the
name ‘self-consistent field theory’ (SCF) is used for this type of approach. The self-
consistency procedure is carried out as follows. Solving (4.31) yields an infinite
spectrum. To find the ground state, we must take the lowestNeigenstates of this
spectrum as the spin-orbitals of the electrons. These are theψlwhich are then used
to build the new Fock operator which is diagonalised again and the procedure is
repeated over and over until convergence is achieved. Figure 4.1(a) and (b) gives a
schematic representation of the Hartree–Fock spectrum and shows how the levels