Piecewise Linearity of Approximate Density Functionals Revisited: Implications for Frontier Orbital Energies

Abstract

In the exact Kohn-Sham density-functional theory (DFT), the total energy versus the number of electrons is a series of linear segments between integer points. However, commonly used approximate density functionals produce total energies that do not exhibit this piecewise-linear behavior. As a result, the ionization potential theorem, equating the highest occupied eigenvalue with the ionization potential, is grossly disobeyed. Here, we show that, contrary to conventional wisdom, most of the required piecewise-linearity of an arbitrary approximate density functional can be restored by careful consideration of the ensemble generalization of DFT. Furthermore, the resulting formulation introduces the desired derivative discontinuity to any approximate exchange-correlation functional, even one that is explicitly density-dependent. This opens the door to calculations of the ionization potential and electron affinity even without explicit electron removal or addition. All these advances are achieved while neither introducing empiricism nor changing the underlying functional form. The power of the approach is demonstrated on benchmark systems using the local density approximation as an illustrative example.

Minor correction to Eqs. (10) and (III.7), information added to the Supplementary Material, typos corrected

References in corpus (13)

• A linear response approach to the calculation of the effective interaction parameters in the LDA+U method
• Perspective on density functional theory
• Localization and delocalization errors in density functional theory and implications for band-gap prediction
• Density functional theory in transition-metal chemistry: a self-consistent Hubbard U approach
• Polaronic hole localization and multiple hole binding of acceptors in oxide wide-gap semiconductors
• Quasiparticle spectra from a non-empirical optimally-tuned range-separated hybrid density functional
• Koopmans' condition for density-functional theory
• The discontinuous nature of the exchange-correlation functional -- critical for strongly correlated systems
• Simple iterative construction of the optimized effective potential for orbital functionals, including exact exchange
• Optimized Effective Potential made simple: Orbital functionals, orbital shifts, and the exact Kohn-Sham exchange potential
• Prediction of the derivative discontinuity in density functional theory from an electrostatic description of the exchange and correlation potential
• The flexible nature of exchange, correlation and Hartree physics: resolving "delocalization" errors in a 'correlation free' density functional
• Constraining density functional approximations to yield self-interaction free potentials

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