sometimes expressed in a way that may at first sight seem less relevant to calculat-
ing energies, namely that the nuclear potential (below) determines the ground-state
electron density, or that there is a one-to-one correspondence between the energy
and the electron density.
The second Hohenberg–Kohn theorem [ 28 ] is the DFT analogue of the wave-
function variation theorem that we saw in connection with the ab initio method
(Section 5.2.3.3): it says that any trial electron density function will give an energy
higher than (or equal to, if it were exactly the true electron density function) the true
ground state energy. In DFT molecular calculations the electronic energy from a
trial electron density is the energy of the electrons moving the under the potential of
the atomic nuclei. This nuclear potential is called the “external potential”, presum-
ably because the nuclei are “external” if we concentrate on the electrons. This
nuclear potential is designatedv(r), and the electronic energy is denoted byEv¼
Ev[r 0 ] (meaning “theEvfunctional of the ground state electron density”). The
second theorem can thus be stated
Ev½rtE 0 ½r 0 (7.6)wherertis a trial electronic density andE 0 [r 0 ] is the true ground state energy,
corresponding to the true electronic densityr 0. The trial density must satisfy the
conditions
R
rt(r)dr¼n, wherenis the number of electrons in the molecule (this is
analogous to the wavefunction normalization condition; here the number of electrons
in all the infinitesimal volumes must sum to the total number in the molecule) and
rt(r)0 for allr(the number of electrons per unit volume can’t be negative). This
theorem tells us that any value of the molecular energy we calculate from the
Kohn–Sham equations (below, a set of equations analogous to the Hartree–Fock
equations, obtained by minimizing energy with respect to electron density) will be
greater than or equal to the true energy. This is actually true only if the functional
used were exact; see below. The Hohenberg–Kohn theorems were originally proved
only for nondegenerate ground states, but have been shown to be valid for degenerate
ground states too [ 29 ]. The functional of the inequality (7.6) is the correct, exact
energy functional (the prescription for transforming the ground state electron density
function into the ground state energy). The exact functional is unknown, soactual
DFT calculations use approximate functionals, and are thusnotvariational: they can
give an energy below the true energy. Being variational is a nice characteristic of a
method, because it assures us that any energy we calculate is an upper bound to the
true energy. However, this is not an essential feature of a method: Møller-Plesset and
practical configuration interaction calculations (Sections 5.4.2 and 5.4.3) are not
variational, but this is not regarded as a serious problem.
7.2.3.2 The Kohn–Sham Energy and the KS Equations
The first Kohn–Sham theorem tells us that it is worth looking for a way to calculate
molecular properties from the electron density. The second theorem suggests that a
450 7 Density Functional Calculations