Essentials of Computational Chemistry

8.4 EXCHANGE-CORRELATION FUNCTIONALS 257
Indeed, the similarities with HF theory extend well beyond the mathematical technology
offered by a common variational principle. For instance, the kinetic energy and nuclear
attraction components of matrix elements of K are identical to those for F. Furthermore,
if the density appearing in the classical interelectronic repulsion operator is expressed in
the same basis functions used for the Kohn–Sham orbitals, then the result is that the same
four-index electron-repulsion integrals appear in K as are found in F (historically, this made
it fairly simple to modify existing codes for carrying out HF calculations to also perform
DFT computations). Finally, insofar as the density is required for computation of the secular
matrix elements, but the density is determined using the orbitals derived from solution of
the secular equation (according to Eq. (8.16)), the Kohn–Sham process must be carried out
as an iterative SCF procedure.
Of course, there is a key difference between HF theory and DFT – as we have derived it
so far, DFT contains no approximations: it is exact. All we need to know is Exc as a function
of ρ... Alas, while Hohenberg and Kohn proved that a functional of the density must exist,
their proofs provide no guidance whatsoever as to its form. As a result, considerable research
effort has gone into finding functions of the density that may be expected to reasonably
approximate Exc, and a discussion of these is the subject of the next section. We close
here by emphasizing that the key contrast between HF and DFT (in the limit of an infinite
basisset)isthatHFisadeliberatelyapproximate theory, whose development was in part
motivated by an ability to solve the relevant equations exactly, while DFT is an exact
theory, but the relevant equations must be solved approximately because a key operator has
unknown form.
It should also be pointed out that although exact DFT is variational, this is not true once
approximations for Exc are adopted. Thus, for instance, the BPW91 functional described
in Section 8.4.2 predicts an energy for the H atom of −0.5042 Eh, but the exact result is
−0.5. Note that the H atom is a one-electron system for which the Schrödinger solution can
be solved exactly – there is no correlation energy. However, because the BPW91 Exc for
this system slightly exceeds the classical self-interaction energy (third term on the r.h.s. of
Eq. (8.15)), which is 100 percent in error for this one-electron system, the energy is predicted
to be slightly below the exact result. Both exact and approximate DFT are size-consistent.
The Kohn–Sham methodology has many similarities, and a few important differences, to
the HF approach. We will, however, delay briefly a full discussion of how exactly to carry
out a KS calculation, as it is instructive first to consider how to go about determining Exc.
8.4 Exchange-correlation Functionals
As already emphasized above, in principle Exc not only accounts for the difference between
the classical and quantum mechanical electron–electron repulsion, but it also includes the
difference in kinetic energy between the fictitious non-interacting system and the real system.
In practice, however, most modern functionals do not attempt to compute this portion explicitly.
Instead, they either ignore the term, or they attempt to construct a hole function that
is analogous to that of Eq. (8.6) except that it also incorporates the kinetic energy difference
between the interacting and non-interacting systems. Furthermore, in many functionals