Computational Chemistry

7.2.3 Current DFT Methods: The Kohn–Sham Approach.............

7.2.3.1 Functionals. The Hohenberg–Kohn Theorems

Nowadays DFT calculations on molecules are based on the Kohn–Sham approach,
the stage for which was set by two theorems published by Hohenberg and Kohn in
1964 (proved in Levine [ 27 ]). The first Hohenberg-Kohn theorem [ 28 ] says that all
the properties of a molecule in a ground electronic state are determined by the
ground state electron density functionr 0 (x,y,z). In other words, givenr 0 (x,y,z) we
can in principle calculate any ground state property, e.g. the energy,E 0 ; we could
represent this as

r 0 ðx;y;zÞ!E 0 (7.2)

The relationship (7.2) means thatE 0 is afunctionalofr 0 (x,y,z). Afunctionis
a rule that transforms a number into another (or the same) number:

2!

x^3

8

1!

x^3

1

Afunctionalis a rule that transforms a function into a number:

fðxÞ¼x^3 """""!

R 2

0 fðxÞdxx^4

4

2

0

¼ 4 (7.3)

The functional

R 2

0 fðxÞdxtransforms the functionx

(^3) into the number four. We
designate the fact that the integral is a functional off(x) by writing
Z 2
0
fðxÞdx¼F½fðxފ (7.4)
A functional is a function of a “definite” (cf. the definite integral above)
function.
The first Hohenberg–Kohn theorem, then, says that any ground state property of
a molecule is a functional of the ground state electron density function, e.g. for the
energy
E 0 ¼F½r 0 Š¼E½r 0 Š (7.5)
The theorem is “merely” anexistence theorem: it says that a functionalFexists,
but does not tell us how to find it; this omission is the main problem with DFT. The
significance of this theorem is that it assures us that there is in principle a way to
calculate molecular properties from the electron density. Thus we can infer that
approximate functionals will give at least approximate answers. The theorem is
7.2 The Basic Principles of Density Functional Theory 449