Essentials of Computational Chemistry

256 8 DENSITY FUNCTIONAL THEORY
Note that the ‘difficult’ terms T and Vee have been lumped together in a term Exc,
typically referred to as the exchange-correlation energy. This is something of a misnomer,
insofar as it is less than comprehensive – the term includes not only the effects of quantum
mechanical exchange and correlation, but also the correction for the classical self-interaction
energy (discussed in Section 8.1.2) and for the difference in kinetic energy between the
fictitious non-interacting system and the real one.
If we undertake in the usual fashion to find the orbitals χ that minimize E in Eq. (8.15),
we find that they satisfy the pseudoeigenvalue equations
where the Kohn–Sham (KS) one-electron operator is defined as
and
h KS
i =−1
2 ∇2 i −
nuclei
k
h KS
i χi = εiχi (8.17)
Zk
|ri − rk| +
Vxc = δExc
δρ
ρ(r ′ )
|ri − r ′ | dr′ + Vxc
(8.18)
(8.19)
Vxc is a so-called functional derivative. A functional derivative is analogous in spirit to more
typical derivatives, and Vxc is perhaps best described as the one-electron operator for which
the expectation value of the KS Slater determinant is Exc.
Note that because the E of Eq. (8.14) that we are minimizing is exact, the orbitals χ must
provide the exact density (i.e., the minimum must correspond to reality). Further note that
it is these orbitals that form the Slater-determinantal eigenfunction for the separable noninteracting
Hamiltonian defined as the sum of the Kohn–Sham operators in Eq. (8.18), i.e.,
N
i=1
h KS
i |χ1χ2 ···χN〉 =
N
εi|χ1χ2 ···χN〉 (8.20)
so there is internal consistency in the Kohn–Sham approach of positing a non-interacting
system with a density identical to that for the real system. It is therefore justified to use
the first term on the r.h.s. of Eq. (8.15) to compute the kinetic energy of the non-interacting
electrons, which turns out to be a large fraction of the kinetic energy of the actual system.
As for determination of the KS orbitals, we may take a productive approach along the
lines of that developed within the context of MO theory in Chapter 4. Namely, we express
them within a basis set of functions {φ}, and we determine the individual orbital coefficients
by solution of a secular equation entirely analogous to that employed for HF theory, except
that the elements Fµν are replaced by elements Kµν defined by
Kµν =
φµ|− 1
2 ∇2 −
nuclei
k
i=1
Zk
|r − rk| +
ρ(r ′ )
|r − r ′ | dr′
+ Vxc|φν
(8.21)