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 Gateaux functional derivative of the universal density functional, deltaF(lambda)[rho]/deltarho(r)(rho=rho0), at ground-state densities with arbitrary normalizations ((rho(0)(r))=n is an element of R+) and an electron-electron interaction strength lambda, 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 rho(0,n)(nu,lambda)(r) of an external potential nu(r), we show that deltaF(lambda)[rho]/deltarho(r)\(nu,lambda)(rho=rho0,n)=mu(SM)(n)-nu(r), where the constant mu(SM)(n) is given by the following chain of dependences: rho(0,n)(nu,lambda)(r) bar right arrow [nu] bar right arrow E-0(nu,lambda)(n) bar right arrow mu(SM)(n) = partial derivativeE(0nu,lambda)(k)/partial derivativek\(k=n). Here [nu] is the class of the external potential nu(r) up to a real constant, and mu(SM)(n) 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 mu(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 mu(SM)(N) =-I. Finally, the equality mu(DFT)=mu(SM)(n) is rigorously derived via an alternative route. where mu(DFT) is the Lagrangian multiplier used to constrain the normalization of the density in the traditional DFT approach.