variational approach might yield a way to calculate the energy and electron density
(the electron density, in turn, could be used to calculate other properties). Recall
that in wavefunction theory, the Hartree–Fock variational approach (Sec-
tion 5.2.3.4) led to the HF equations, which are used to calculate the energy and
the wavefunction. An analogous variational approach led (1965) to the Kohn–Sham
equations [ 30 ], the basis of current molecular DFT calculations. If we had an
accurate molecular electron density functionrand if we knew the correct energy
functional, we could (assuming the functional were not impossibly complicated) go
straight from the electron density function to the molecular energy, courtesy of the
functional. Unfortunately we do not a priori have an accurater, and we certainly do
not have the correct energy functional, this latter fact being the key problem in
density functional theory. The Kohn–Sham approach to DFT mitigates these two
problems.
The two basic ideas behind the KS approach are: (1) To express the molecular
energy as a sum of terms, only one of which, a relatively small term, involves the
“unknown” functional. Thus one hopes that even moderately large errors in this
term will not introduce large errors into the total energy. (2) To use an initial guess
of the electron densityrin the KS equations (analogous to the HF equations) to
calculate an initial guess of the KS orbitals and energy levels (below); this initial
guess is then used to iteratively refine these orbitals and energy levels, in a manner
similar to that used in the HF SCF method. The final KS orbitals are used to
calculate an electron density that in turn is used to calculate the energy.
The Kohn–Sham Energy
The strategy here is to separate the electronic energy of our molecule into a portion
which can be calculated accurately without using DFT, and a relatively small term
which requires the elusive functional. A key idea in this approach is the concept of a
fictitiousnoninteracting reference system, defined as one in which the electrons do
not interact and in which (this is very important) the ground state electron density
distribution given byrris exactly the same as that in our real ground state system:
rr¼r 0. Noninteracting electrons are readily treated exactly, and the deviations
from the behavior of real electrons are swept into a small term involving a
functional with which we have to grapple. We are talking here about theelectronic
energy of the molecule; the total internal “frozen-nuclei” energy can be found later
by adding in the trivial-to-calculate internuclear repulsions, and the 0 K total
internal energy by further adding the zero-point energy from the normal-mode
vibrations, as explained in Section 2.2.
The ground state electronic energy of the real molecule is the sum of the electron
kinetic energies, the nucleus-electron attraction potential energies, and the
electron–electron repulsion potential energies:
E 0 ¼hiT½r 0 þhiVNe½r 0 þhiVee½r 0 *(7.7)7.2 The Basic Principles of Density Functional Theory 451