 Solution

ID: past-exam-of-the-chemistry-course-of-the-year-3-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-4/61/a/solution

For fixed electron-electron interaction and electron number, a nondegenerate ground-state electron density determines the external potential up to an additive constant, and hence determines the Hamiltonian and ground-state observables.

Suppose two potentials

v, v^{'}

differing by more than a constant have different ground states

Ψ, Ψ^{'}

but the same density

ρ

E < E^{'} + \int [v (r) - v^{'} (r)] ρ (r) d^{3} r,

E^{'} < E + \int [v^{'} (r) - v (r)] ρ (r) d^{3} r .

Adding produces the contradiction

E + E^{'} < E + E^{'}

. This proves the first Hohenberg-Kohn theorem under the stated nondegeneracy assumptions. A constant potential shift changes energy by

N

times that constant without changing the density.^[ref]

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