Expectation values in density-functional theory, and kinetic contribution to the exchange-correlation energy

Bauer's expression for the difference between the expectation values of an arbitrary quantum-mechanical operator with the physical interacting ground-state wave function and its corresponding noninteracting Kohn-Sham wave function is obtained here by a very simple and general derivation via the constrained-search formulation of density-functional theory. Our proof does not require v-representability or a coupling-constant integration. We show that the following expression of Bass for the kinetic part of the exact exchange-correlation energy ${\mathit{E}}_{\mathrm{xc}}$[n]: ${\mathit{T}}_{\mathrm{xc}}$[n]=${\mathit{a}}_{0}$ \ensuremath{\partial}${\mathit{E}}_{\mathrm{xc}}$[n]/\ensuremath{\partial}${\mathit{a}}_{0}$, where ${\mathit{a}}_{0}$=${\mathrm{\ensuremath{\Elzxh}}}^{2}$/${\mathit{me}}^{2}$ is the Bohr radius, can also be derived by a constrained-search proof and is equivalent to the following expression of Levy and Perdew: ${\mathit{T}}_{\mathrm{xc}}$[n]=(\ensuremath{\partial}${\mathit{E}}_{\mathrm{xc}}$[${\mathit{n}}_{\ensuremath{\lambda}}$]/\ensuremath{\partial}\ensuremath{\lambda})${\mathrm{\ensuremath{\Vert}}}_{\ensuremath{\lambda}=1}$-${\mathit{E}}_{\mathrm{xc}}$[n], where ${\mathit{n}}_{\ensuremath{\lambda}}$(x,y,z)=${\ensuremath{\lambda}}^{3}$n(\ensuremath{\lambda}x,\ensuremath{\lambda}y,\ensuremath{\lambda}z). When an approximate ${\mathit{E}}_{\mathrm{xc}}$ is employed, the two expressions will yield the same result for the corresponding ${\mathit{T}}_{\mathrm{xc}}$ if certain coordinate scaling relations, involving the electronic charge, are satisfied. Moreover, the Levy-Perdew relation has the advantage that it is applied at full electronic charge (e) and full electronic mass (m). Corresponding relations are also exhibited for the correlation hole. We discuss the high- and low-density limits of ${\mathit{T}}_{\mathrm{xc}}$, generalize our results to spin-density functional theory, and present numerical estimates of ${\mathit{T}}_{\mathrm{xc}}$ for atoms, evaluated within the local spin density, gradient expansion, and generalized gradient approximations.