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Physical Review A

Density-functional derivative discontinuities at the maximum number of bound electrons

Daniel Whitenack, Yu Zhang, Adam Wasserman

Published Apr 5, 2012• 5 citations
Publisher

Abstract

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 (${J}_{\mathrm{max}}$) that can be sustained in the system with an infinite lifetime. At this integer ${J}_{\mathrm{max}}$, 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 ${J}_{\mathrm{max}}$. For integers below ${J}_{\mathrm{max}}$ the jump is purely real because of the real shift in the chemical potential. At ${J}_{\mathrm{max}}$, 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 ${J}_{\mathrm{max}}$ is larger than would have been expected by simply setting the affinity to zero.

DOI
10.1103/physreva.85.042504

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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{J}_{\mathrm{max}}Jmax​) that can be sustained in the system with an infinite lifetime. At this integer Jmax{J}_{\mathrm{max}}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{J}_{\mathrm{max}}Jmax​. For integers below Jmax{J}_{\mathrm{max}}Jmax​ the jump is purely real because of the real shift in the chemical potential. At Jmax{J}_{\mathrm{max}}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{J}_{\mathrm{max}}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 nnn 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][v][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 NNN, 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 III and AAA 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)(I)(I) and electron affinity (A)(A)(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>AI>AI>A for any one-electron system in the absence of magnetic fields.
potential of an open system may jump discontinuously as the particle number crosses an integer, with important physical consequences. Recently, Sagvolden and Perdew (Phys Rev A 2008, 77, 012517) have analyzed the discontinuity of the exchange-correlation potential as the particle number crosses one, with an illustration that uses a model density for the H � ion. In this work, we extend their analysis to the case in which the external potential is the simple harmonic confinement, choosing spring-constant values for which the two-electron hamiltonian has an analytic solution. This way, we can obtain the exact, analytic, exchange and correlation potentials for particle number fluctuating between zero and two, illustrating the discontinuity as the particle number crosses one without introducing any model or approximation. We also discuss exchange and correlation separately. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem 109: 2410–2415, 2009 Key words: density functional theory; electronic correlation; exchange-correlation potentials; harmonic traps
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.
We derive analytical expressions for chemical potentials within the random phase approximation (RPA), equivalently the GWGWGW energy functional evaluated using non interacting Green's functions (GsG_sGs​). 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 GsG_sGs​, both validated with finite difference benchmarks. We show that the functional derivative of the GWGWGW correlation energy$\unicode{x2013}$i.e., the GWGWGW correlation self energy$\unicode{x2013}$exhibits a discontinuity at integer particle numbers with finite jumps. This resolves the apparent inconsistency between accurate GWGWGW quasiparticle energies and the large delocalization errors observed in RPA total energies, as standard GWGWGW 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.
The question of whether the exact exchange-correlation (XC) potential in density functional theory can have discontinuities as a function of the number of electrons is discussed. For a very simple model the exact XC functional is constructed analytically using Levy's constrained-search method and shown to have a discontinuity in its derivative. Exact numerical calculations for the XC potential for a simple finite one-dimensional 'semiconductor' chain are performed. An extrapolation to infinite chain length shows the occurrence of a sizable discontinuity in certain parameter ranges, but the model has a small discontinuity for 'realistic' parameters. The question of how to calculate the value of the discontinuity for a general system is addressed using many-body theory. A formula proposed recently by other researchers is shown to be incorrect.