Exact Exchange

We will now describe the exact exchange (EXX) functional
[77,78,79].
A common misconception amongst non-specialists is that HF and EXX are
the same thing. It is true, of course, that both functionals
treat exchange ``exactly'' in the sense that the exchange energy is given in
terms of the orbitals by Equation (3.3). The difference arises from
the fact the orbitals themselves are different - in EXX the orbitals are
eigenstates of a Hamiltonian with a *local* potential, while
in HF they are eigenstates of a Hamiltonian with
a *non-local* potential. This leads to very different results,
notably in the calculation of band structures, where, as we will see
in Chapter 4, HF performs disastrously
while EXX performs very well indeed [79].
EXX also includes treatment of correlation
in the style of the LDA, which is not the case in standard HF. Although LDA
correlation can be added to HF, this does not improve the band structure problem,
the reason for which will become clear in the discussion in Section 5.3.