Density Functional Theory and the Kohn–Sham LCAO Equations - Molecular Modelling - page 341

theory. The first keynote paper contains two fundamental theorems. The first theorem is
what mathematicians call an existence theorem ; it proves that the ground-state energy of any
electronic system is determined by a functional of the electron density. The theorem means
that in principle we only need to know the electron density in three-dimensional space
and not the full wavefunction in order to calculate any ground-state property including the
energy.
The term functional is mathematical jargon; a function is a mapping from one set of
numbers to another (Figure 20.1) whilst a functional is a mapping from a set of functions
to a set of numbers.

Function

Set of numbers Set of numbers

Set of numbers

Figure 20.1 A function

The proof of the theorem is quite simple, but will not be reproduced here. The paper
is so important that you should see at least the abstract (Hohenberg and Kohn 1964). The
authors use square brackets [...] to denote functionals.

This paper deals with the ground state of an interacting electron gas in an external potential
v ( r ). It is proved that there exists a universal functional of the density, F [ n ( r )], independent
of v ( r ), such that the expression E ≡ ∫ v ( r ) n ( r )d r + F [ n ( r )] has as its minimum value the
correct ground state energy associated with v ( r ). The functional F [ n ( r )] is then discussed for
two situations: (1) n ( r ) = n 0 + n ( r ), n / n 0 << 1, and (2) n ( r ) = ϕ( r / r 0 ) with ϕ arbitrary
and r 0 →∞ . In both cases F can be expressed entirely in terms of the correlation energy
and linear and higher order electronic polarizabilities of a uniform electron gas. This approach
also sheds some light on generalized Thomas-Fermi methods and their limitations. Some new
extensions of these methods are presented.

The second theorem gives a variation principle for the density functionals; it states (in
chemical language) that

ε el [ P ( r ) ]

≥

ε el [ P 0 ( r ) ] (20.13)

where P 0 is the true density for the system and P any other density obeying

P ( r ) dτ = P 0 ( r ) dτ = N (20.14)

where N is the number of electrons. The difficulty is that the Hohenberg-Kohn theorems
give us no clue as to the nature of the density functional, nor how to set about finding it.