Open-system density functional theory may be formulated in terms of ensemble averages arising from interaction with a bath. The system is allowed to exchange particles with the bath and the states in the ensemble average are those corresponding to integer numbers of particles. The weights in the ensemble average are typically equated with time-averaged values of the occupation numbers of the various states comprising the open system. As a result, there are two constraints on the occupation numbers: (1) Their sum must be unity so that the ensemble average is a probability function and (2) The sum of the occupation numbers times the number of particles for the associated state must equal the time-averaged number of particles. By solving explicitly the first constraint we arrive at an expression for the energy having a form structurally equivalent to a Gibbs thermodynamic function. From this form follows both chemical potential equalization and the identification of the functional derivative of the energy for a particular state with respect to its integer-state density as the chemical potential. We highlight a distinction between functional derivatives of the time-averaged and the integer-state energies with respect to integer-state densities. From this we can restate the jump discontinuity behavior of the exchange-correlation functional in terms of integer-state behavior.
For every electronic many-body Hamiltonian with a one-body potential that goes to zero at infinity, there is a maximum number of bound electrons (
Jmax) that can be sustained in the system with an infinite lifetime. At this integer
Jmax, the derivative discontinuity in the exact energy of ensemble density functional theory (DFT) is ill defined. However, we investigate the derivative discontinuity of the energy within the framework of density functional resonance theory [D. L. Whitenack and A. Wasserman, Phys. Rev. Lett. 107, 163002 (2011)], which reduces to ground-state DFT as a coordinate scaling parameter is taken to zero, and find that the exact exchange-correlation potential experiences discontinuous jumps at integer particle numbers, including
Jmax. For integers below
Jmax the jump is purely real because of the real shift in the chemical potential. At
Jmax, the jump has a nonzero imaginary component reflecting the metastability of the system upon addition of one more electron. In addition, the magnitude of the derivative discontinuity at
Jmax is larger than would have been expected by simply setting the affinity to zero.
Based on the Runge-Gross theorem for ensembles we investigate the influence of particle-number-changes on the exchange-correlation potential in time-dependent density-functional theory. We show that the potential changes discontinuously when the particle number crosses an integer value. Real-time simulations of an atomic ionization process demonstrate that this discontinuity appears naturally in the theory of the time-dependent optimized effective potential. The importance of such a discontinuity for physical processes, even such ones in which the particle number is a constant, is discussed.
Zahariev and Wang [Phys. Rev. A, 70, 042503 (2004)] discuss the density functional theory of noninteger average particle numbers. Among their many results is one we dispute: that the exact exchange-correlation potential (more precisely, the exact functional derivative of the exchange-correlation energy with respect to the density) cannot have a discontinuity as the particle number crosses an integer, in contradiction to works by two of us (J.P.P. and M.L.) in collaboration with co-workers [Phys. Rev. Lett. 49, 1691 (1982); Phys. Rev. Lett.51, 1884 (1983)] and by Sham and Schl\"uter [Phys. Rev. Lett. 51, 1888 (1983)]. We point to a counterexample to Zahariev and Wang's claim, which two of us (E.S. and J.P.P.) have presented in a separate paper: A rigorous proof that, in the absence of external magnetic fields, the exchange-correlation potential jumps by the difference between the ionization potential
(I) and electron affinity
(A) when the particle number crosses 1, given that $I\ensuremath{\geqslant}A$. We point out that Zahariev and Wang's derivation neglects an order-of-limits problem. We also prove that
I>A for any one-electron system in the absence of magnetic fields.
This is the first of a series of papers dealing with many-particle systems from a unified, nonperturbative point of view. It contains derivations and discussions of various field-theoretical techniques which will be applied in subsequent papers. In a short introduction the general method of approach is summarized, and its relationship to other field-theoretic problems indicated. In the second section the macroscopic properties of the spectra of many-particle systems are described. Asymptotic evaluations are performed which characterize these macroscopic features in terms of intensive parameters, and the relationship of these parameters to thermodynamics is discussed. The special characteristics of the ground state are shown to follow as a limiting case of the asymptotic evaluations. The third section is devoted to the time-dependent field correlation functions, or Green's functions, which describe the microscopic behavior of a multiparticle system. These functions are defined, and related to intensive macroscopic variables when the energy and number of particles are large. Spectral representations and other properties of various one-particle Green's functions are derived. In the fourth section the treatment of non-equilibrium processes is considered. As a particular example, the electromagnetic properties of a system are expressed in terms of the special two-particle Green's function which describes current correlation. The discussion yields specifically a fluctuation-dissipation theorem, a sum rule for conductivity, and certain dispersion relations. The fifth section deals with the differential equations which determine the Green's functions. The boundary conditions that characterize the Green's function equations are exhibited without reference to adiabatic decoupling. A method for solving the equations approximately, by treating the correlations among successively larger numbers of particles, is considered. The first approximation in this sequence is shown to yield a generalized Hartree-like equation. A related, but rigorous, identity for the single-particle Green's function is then derived. A second approximation, which takes certain two-particle correlations into account, is shown to produce various additional effects: The interaction between particles is altered in a manner characterized by the intensive macroscopic parameters, and the modification and spread of the energy-momentum relation come into play. In the final section compact formal expressions for the Green's functions and other physical quantities are derived. Alternative equations and systematic approximations for the Green's functions are obtained.
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, ${\ensuremath{\delta}{F}^{\ensuremath{\lambda}}[\ensuremath{\rho}]∕\ensuremath{\delta}\ensuremath{\rho}(\mathbit{r})\ensuremath{\mid}}_{\ensuremath{\rho}={\ensuremath{\rho}}_{0}}$, at ground-state densities with arbitrary normalizations $(⟨{\ensuremath{\rho}}_{0}(\mathbit{r})⟩=n∊{\mathcal{R}}_{+})$ and an electron-electron interaction strength $\ensuremath{\lambda}$, 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 ${\ensuremath{\rho}}_{0,n}^{v,\ensuremath{\lambda}}(\mathbit{r})$ of an external potential $v(\mathbit{r})$, we show that ${\ensuremath{\delta}{F}^{\ensuremath{\lambda}}[\ensuremath{\rho}]∕\ensuremath{\delta}\ensuremath{\rho}(\mathbit{r})\ensuremath{\mid}}_{\ensuremath{\rho}={\ensuremath{\rho}}_{0,n}^{v,\ensuremath{\lambda}}}={\ensuremath{\mu}}_{\mathrm{SM}}^{n}\ensuremath{-}v(\mathbit{r})$, where the constant ${\ensuremath{\mu}}_{\mathrm{SM}}^{n}$ is given by the following chain of dependences: ${\ensuremath{\rho}}_{0,n}^{v,\ensuremath{\lambda}}(\mathbit{r})\ensuremath{\mapsto}[v]\ensuremath{\mapsto}{E}_{0}^{v,\ensuremath{\lambda}}(n)\ensuremath{\mapsto}{\ensuremath{\mu}}_{\mathrm{SM}}^{n}={\ensuremath{\partial}{E}_{0}^{v,\ensuremath{\lambda}}(k)∕\ensuremath{\partial}k\ensuremath{\mid}}_{k=n}$. Here
[v] is the class of the external potential $v(\mathbit{r})$ up to a real constant, and ${\ensuremath{\mu}}_{\mathrm{SM}}^{n}$ 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 ${\ensuremath{\mu}}_{\mathrm{SM}}^{N}$, whose exact value is generally not the popular preference of the negative of Mulliken's electronegativity, $\ensuremath{-}\frac{1}{2}(I+A)$, 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 ${\ensuremath{\mu}}_{\mathrm{SM}}^{N}=\ensuremath{-}I$. Finally, the equality ${\ensuremath{\mu}}_{\mathrm{DFT}}={\ensuremath{\mu}}_{\mathrm{SM}}^{n}$ is rigorously derived via an alternative route, where ${\ensuremath{\mu}}_{\mathrm{DFT}}$ is the Lagrangian multiplier used to constrain the normalization of the density in the traditional DFT approach.