Essentials of Computational Chemistry

8.3 KOHN–SHAM SELF-CONSISTENT FIELD METHODOLOGY 255
8.3 Kohn–Sham Self-consistent Field Methodology
The discussion above has emphasized that the density determines the external potential,
which determines the Hamiltonian, which determines the wave function. And, of course,
with the Hamiltonian and wave function in hand, the energy can be computed. However, if
one attempts to proceed in this direction, there is no simplification over MO theory, since the
final step is still solution of the Schrödinger equation, and this is prohibitively difficult in most
instances. The difficulty derives from the electron–electron interaction term in the correct
Hamiltonian. In a key breakthrough, Kohn and Sham (1965) realized that things would
be considerably simpler if only the Hamiltonian operator were one for a non-interacting
system of electrons (Kohn and Sham 1965). Such a Hamiltonian can be expressed as a sum
of one-electron operators, has eigenfunctions that are Slater determinants of the individual
one-electron eigenfunctions, and has eigenvalues that are simply the sum of the one-electron
eigenvalues (see Eq. (7.43) and surrounding discussion).
The crucial bit of cleverness, then, is to take as a starting point a fictitious system of
non-interacting electrons that have for their overall ground-state density the same density
as some real system of interest where the electrons do interact (note that since the density
determines the position and atomic numbers of the nuclei (see Eq. (8.2)), these quantities
are necessarily identical in the non-interacting and in the real systems). Next, we divide the
energy functional into specific components to facilitate further analysis, in particular
E[ρ(r)] = Tni[ρ(r)] + Vne[ρ(r)] + Vee[ρ(r)] + T [ρ(r)] + Vee[ρ(r)] (8.14)
where the terms on the r.h.s. refer, respectively, to the kinetic energy of the non-interacting
electrons, the nuclear–electron interaction (Eq. (8.3)), the classical electron–electron repulsion
(Eq. (8.4)), the correction to the kinetic energy deriving from the interacting nature of
the electrons, and all non-classical corrections to the electron–electron repulsion energy.
Note that, for a non-interacting system of electrons, the kinetic energy is just the sum
of the individual electronic kinetic energies. Within an orbital expression for the density,
Eq. (8.14) may then be rewritten as
E[ρ(r)] =
+
N
i
χi|− 1
2 ∇2 i |χi
nuclei Zk
− χi|
|ri − rk|
k
|χi
N
i
χi| 1
2
ρ(r ′ )
|ri − r ′ | dr′
|χi + Exc[ρ(r)]
(8.15)
where N is the number of electrons and we have used that the density for a Slaterdeterminantal
wave function (which is an exact eigenfunction for the non-interacting system)
is simply
N
ρ = 〈χi|χi〉 (8.16)
i=1