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

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