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 Gâteaux 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 ρ_0,n^v,λ(r) of an external potential v(r) , we show that δF^λ[ρ]∕δρ(r)∣_{ρ=ρ0,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]↦E₀^v,λ(n)↦μ_SMⁿ=∂E₀^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_SM^N , 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 μ_SM^N=−I . Finally, the equality μ_DFT=μ_SMⁿ is rigorously derived via an alternative route, where μ_DFT is the Lagrangian multiplier used to constrain the normalization of the density in the traditional DFT approach.