Ensemble averages are an approximation technique for connecting macroscopic and microscopic properties of a system. For systems open with respect to exchange of particles with a bath, the microscopic states are those with integer numbers of particles. When a property of the open system is represented as an ensemble average over these microscopic states, self-consistency dictates several implications for the properties of both for open-system energy density functionals and the integer-state functionals describing the microscopic states. The first is that each integer-state energy density functional is a functional of the original type discovered by Levy. Another is that the dependence of the open-system functional on the ensemble density is linear whereas the dependence of the integer-state functional is decidedly nonlinear. Finally, the derivative discontinuity behavior with respect to particle number of some open-system density functionals appears to be connected to the long-range behavior of the effective external potentials of the integer-state functionals governing the interactions among subsystems in the ensemble.
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.
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.
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.
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.
We derive analytical expressions for chemical potentials within the random phase approximation (RPA), equivalently the
GW energy functional evaluated using non interacting Green's functions (
Gs). The chemical potential is obtained using two formally equivalent approaches: a direct derivative of the total energy with respect to particle number, and a functional derivative via the chain rule through
Gs, both validated with finite difference benchmarks. We show that the functional derivative of the
GW correlation energy$\unicode{x2013}$i.e., the
GW correlation self energy$\unicode{x2013}$exhibits a discontinuity at integer particle numbers with finite jumps. This resolves the apparent inconsistency between accurate
GW quasiparticle energies and the large delocalization errors observed in RPA total energies, as standard
GW self energies neglect this nonanalytic behavior. Our results suggest that derivative discontinuities are a fundamental feature of correlation energy functionals, analogous to the known discontinuity in the exact exchange correlation energy.