Functional derivative of the universal density functional in Fock space

Within the framework of zero-temperature Fock-space density-functional theory (DFT), we prove that the G\^ateaux functional derivative of the universal density functional, δF^λ[ρ]∕δρ(r)_ρ=ρ₀, at ground-state densities with arbitrary normalizations (⟨ρ₀(r)⟩=n∊R₊) and an electron-electron interaction strength λ, is uniquely defined, but is discontinuous when the number of electrons n becomes an integer, thus providing a mathematically rigorous confirmation for the ``derivative discontinuity'' initially discovered by Perdew et al. [Phys. Rev. Lett. 49, 1691 (1982)]. However, the functional derivative of the exchange-correlation functional is continuous with respect to the number of electrons in Fock space; i.e., there is no ``derivative discontinuity'' for the exchange-correlation functional at an integer electron number. For a ground-state density ρ0,n^v,λ(r) of an external potential v(r), we show that ${{δ}{F}^{{λ}}[{ρ}]∕{δ}{ρ}({r}){}}_{{ρ}={{ρ}}0,n^{v,{λ}}}={{μ}}SMⁿ{-}v({r})$, where the constant ${{μ}}SMⁿ$ is given by the following chain of dependences: ${{ρ}}0,n^{v,{λ}}({r}){↦}[v]{↦}{E}₀^{v,{λ}}(n){↦}{{μ}}SMⁿ={{∂}{E}₀^{v,{λ}}(k)∕{∂}k{}}ₖ₌ₙ$. Here $[v]$ is the class of the external potential $v({r})$ up to a real constant, and ${{μ}}SMⁿ$ is the chemical potential defined according to statistical mechanics. At an integer electron number $N$, we find that there is no freedom of adding an arbitrary constant to the value of the chemical potential ${{μ}}SMN$, whose exact value is generally not the popular preference of the negative of Mulliken's electronegativity, ${-}1/2(I+A)$, where $I$ and $A$ are the first ionization potential and the first electron affinity, respectively. In addition, for any external potential converging to the same constant at infinity in all directions, we resolve that μSMN=-I. Finally, the equality μDFT=μSMⁿ is rigorously derived via an alternative route, where μDFT is the Lagrangian multiplier used to constrain the normalization of the density in the traditional DFT approach.