Density Functional Theory

Density Functional Theory

Homework 2

Problems:
1. Given a strictly positive one-electron density (r), such that
R
(r)dr = 1, nd an
expression for the one-electron potential V (r) that yields (r) in the ground-state. [2
points]
2. Show that the one-electron density
(r) =
2
3=2a3 (
r
a
)2e??r2
a2
where a is an arbitrary length scale, cannot be the ground-state density of an electron
subjected to a potential that is everywhere nite, but can be realized in an excited
state of such a potential. Identify this potential. (Hint: do the previous problem rst.)
[4 points]
3. Show that the exchange-correlation hole h0
xc (r1; r2) satises:
(i)
Z
h0
xc (r1; r2)dr2 = ??;0 ; [2 points] and (ii) h
xc (r; r) = ??(r): [2 points]
4. Show that the exact exchange energy functional Ex[; ] satises the spin-scaling
relation:
Ex[; ] =
1
2
Ex[2] +
1
2
Ex[2];
where Ex[] is the exact exchange energy functional for a spin-unpolarized system
(with density ). [3 points]
5. Show that the LDA (local density approximation) exchange energy functional satises
the spin-scaling relation. [2 points]
1