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Study of the Discontinuity of the Exchange-Correlation Potential in an Exactly Soluble Case

Paola Gori‐Giorgi, Andreas Savin B

Published Jan 1, 2008• 29 citations
OpenAlex

Abstract

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

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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
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.
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.
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.
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.
The derivative discontinuity of the exchange-correlation (xc) energy at an integer particle number is a property of the exact, unknown xc functional of density functional theory (DFT) which is absent in many popular local and semilocal approximations. In lattice DFT, approximations exist which exhibit a discontinuity in the xc potential at half-filling. However, due to convergence problems of the Kohn-Sham (KS) self-consistency cycle, the use of these functionals is mostly restricted to situations where the local density is away from half-filling. Here a numerical scheme for the self-consistent solution of the lattice KS Hamiltonian with a local xc potential with rapid (or quasidiscontinuous) density dependence is suggested. The problem is formulated in terms of finite-temperature DFT where the discontinuity in the xc potential emerges naturally in the limit of zero temperature. A simple parametrization is suggested for the xc potential of the uniform one-dimensional (1D) Hubbard model at finite temperature which is obtained from the solution of the thermodynamic Bethe ansatz. The feasibility of the numerical scheme is demonstrated by application to a model of fermionic atoms in a harmonic trap. The corresponding density profile exhibits a plateau of integer occupation at low temperatures which melts away for higher temperatures.
We generalize the exact strong-interaction limit of the exchange-correlation energy of Kohn-Sham density functional theory to open systems with fluctuating particle numbers. When used in the self-consistent Kohn-Sham procedure on strongly interacting systems, this functional yields exact features crucial for important applications such as quantum transport. In particular, the steplike structure of the highest-occupied Kohn-Sham eigenvalue is very well captured, with accurate quantitative agreement with exact many-body chemical potentials. While it can be shown that a sharp derivative discontinuity is present only in the infinitely strongly correlated limit, at finite correlation regimes we observe a slightly smoothened discontinuity, with qualitative and quantitative features that improve with increasing correlation. From the fundamental point of view, our results obtain the derivative discontinuity without making the assumptions used in its standard derivation, offering independent support for its existence.