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58 CHAPTER 6. DENSITY FUNCTIONAL THEORY
6.2 Hohenberg-Kohn theorems

It is self-evident that the external potential in principle determines all the properties of the sys-
tem: this is the normal approach to quantum mechanical problems, by solving the Schr¨odinger
Chapter 6 equation for the eigenstates of the system.
The first Hohenberg-Kohn theorem demonstrates that the density may be used in place
of the potential as the basic function uniquely characterizing the system. It may be stated
Density functional theory as: the ground-state density n(r) uniquely determines the potential, up to an arbitrary
constant.
In the original Hohenberg-Kohn paper, this theorem is proven for densities with non-
1 degenerate ground states. The proof is elementary, and by contradiction. Suppose there
This chapter deals with the foundation of modern density functional theory as an exact
approach (in principle) to systems of interacting particles. In our case, electrons, i.e., existed two potentials differing by more than a constant, yielding the same density. These
would have two different ground-state wavefunctions, Ψ 1 and Ψ 2 . Consider Ψ 2 as a trial
fermions interacting via the Coulomb repulsion.
wavefunction for potential v ext,1 (r). Then, by the variational principle,
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
6.1 One electron %Ψ 2 |T + V ee + V ext,1 |Ψ 2 & ≥ %Ψ 1 |T + V ee + V ext,1 |Ψ 1 &. (6.5)
But since both wavefunctions have the same density, this implies
To illustrate the basic idea behind density functional theory, we first formulate the problem
ˆ
ˆ
ˆ
ˆ
of one electron as a density functional problem. We know that the ground-state energy can %Ψ 2 |T + V ee |Ψ 2 & ≥ %Ψ 1 |T + V ee |Ψ 1 &. (6.6)
be written as a density functional instead of as a wavefunction functional. This functional is But we can always swap which wavefunction we call 1 and which we call 2, which reverses
1 1 3 |∇n| 2 1 3 this inequality, leading to a contradiction, unless the total energies of the two wavefunctions
E[n] = d r + d r v(r) n(r) (6.1) are the same, which implies they are the same wavefunction by the variational principle and
8 n
3
!
in any number of dimensions. We must minimize it, subject to the constraint that d r n(r) = the assumption of non-degeneracy. Then, simple inversion of the Schr¨odinger equation, just
as we have done several times for the one-electron case, yields
1. Using Lagrange multipliers, we construct the auxillary functional:
N N
- 1 - 2 1 - 1
H[n] = E[n] − µ N (6.2) v ext (r i ) = − ∇ Ψ/Ψ + (6.7)
i
i=1 2 i=1 2 j!=i |r i − r j |
3
!
where N = d r n(r). Minimizing this yields This determines the potential up to a constant.
δH δT VW Exercise 24 Finding the potential from the density
= + v(r) − µ = 0 (6.3)
δn(r) δn(r) For the smoothed exponential, φ(x) = C(1+|x|) exp(−|x|), find the potential for which this
is an eigenstate, and plot it. Is it the ground state of this potential?
Using the derivative from Ex. 5, we find the Schr¨odinger equation for the density:
 2 2  An elegant constructive proof was found later by Levy, which automatically includes degen-
∇ |∇n|
− 2 erate states. It is an example of the constrained search formalism. Consider all wavefunctions

 + + v(r) n(r) = µn(r) (6.4)
4 8n
Ψ which yield a certain density n(r). Define the functional
with boundary conditions that n(r) and ∇n(r) → 0 at the edges. We can identify the @ @ ˆ @ @
@
Lagrange multiplier by integrating both sides over all space at the solution. Since the integral F[n] = min%Ψ @ T + V ee@ Ψ& (6.8)
ˆ @
Ψ→n
of any Laplacian vanishes (with the given boundary conditions), we see that µ = E. where the search is over all antisymmetric wavefunctions yielding n(r). Then, for any n(r),
ˆ
ˆ
Exercise 23 Checking equation for density any wavefunction minimizing T + V ee is a ground-state wavefunction, since the ground-state
Show that the ground-state density for a particle in a box satisfies Eq. (6.4). energy is simply > 1 ?
3
E = min F[n] + d r v ext (r) n(r) , (6.9)
1 c !2000 by Kieron Burke. All rights reserved.
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