Density Functional Theory

Problems:
1. Given a strictly positive one-electron density ρ(r), such that R ρ(r)dr = 1, find an
expression for the one-electron potential V (r) that yields ρ(r) in the ground-state. [2
points]
2. Show that the one-electron density
ρ(r) = 2
π3/2a3(a r )2e− a r2 2
where a is an arbitrary length scale, cannot be the ground-state density of an electron
subjected to a potential that is everywhere finite, but can be realized in an excited
state of such a potential. Identify this potential. (Hint: do the previous problem first.)
[4 points]
3. Show that the exchange-correlation hole hσσ xc 0(r1, r2) satisfies:
(i) Z hσσ xc 0(r1, r2)dr2 = −δσ,σ0, [2 points] and (ii) hσσ xc (r, r) = −ρσ(r). [2 points]
4. Show that the exact exchange energy functional Ex[ρα, ρβ] satisfies the spin-scaling
relation:
Ex
[ρα, ρβ] = 1
2
Ex
[2ρα] + 1
2
Ex
[2ρβ],
where E
x[ρ] is the exact exchange energy functional for a spin-unpolarized system
(with density ρ). [3 points]
5. Show that the LDA (local density approximation) exchange energy functional satisfies
the spin-scaling relation. [2 points]
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