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 $v,v^{\prime }$ differing by more than a constant have different ground states $\Psi ,{\Psi }^{\prime }$ but the same density $\rho$. Strict Rayleigh-Ritz variational principle inequalities giveAdding produces the contradiction $E+E^{\prime }<E+E^{\prime }$. 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]