Koopmans-compliant functionals and their performance against reference molecular data

Abstract

Koopmans-compliant functionals emerge naturally from extending the constraint of piecewise linearity of the total energy as a function of the number of electrons to each fractional orbital occupation. When applied to approximate density-functional theory, these corrections give rise to orbital-density-dependent functionals and potentials. We show that the simplest implementations of Koopmans' compliance provide accurate estimates for the quasiparticle excitations and leave the total energy functional almost or exactly intact, i.e., they describe correctly electron removals or additions, but do not necessarily alter the electronic charge density distribution within the system. Additional functionals can then be constructed that modify the potential energy surface, including e.g. Perdew-Zunger corrections. These functionals become exactly one-electron self-interaction free and, as all Koopmans-compliant functionals, are approximately many-electron self-interaction free. We discuss in detail these different formulations, and provide extensive benchmarks for the 55 molecules in the reference G2-1 set, using Koopmans-compliant functionals constructed from local-density or generalized-gradient approximations. In all cases we find excellent performance in the electronic properties, comparable or improved with respect to that of many-body perturbation theories, such as G0W0 and self-consistent GW, at a fraction of the cost and in a variational framework that also delivers energy derivatives. Structural properties and atomization energies preserve or slightly improve the accuracy of the underlying density-functional approximations (Note: Supplemental Material is included in the source).

First-Principles Photoemission Spectroscopy of DNA and RNA Nucleobases from Koopmans-Compliant Functionals

“…35 The condition of Koopmans' compliance is naturally akin to that of enforcing a correct description of charged excitations 28,36 , and can therefore lead to orbital energies that are comparable to the quasiparticle excitation energies of photoemission experiments. In a previous work 36 , we showed the remarkable performance of KC functionals in predicting ultraviolet photoemission spectra (UPS) and orbital tomography momentum maps for photovoltaic molecules, showing an agreement with experiments for frontier orbital energies [ionization potentials (IPs) and electron affinities (EAs)] that is comparable (in some cases even slightly superior) to state-of-the-art methods in many-body perturbation theory, while preserving a moderate computational cost and scaling 32 , and the quality of potential energy surfaces of the underlying DFT functionals 32 .…”

“…(8) are the result of numerical simulations involving an approximate DFT functional with Koopmans' corrections. As shown by Dabo et al 28 and Borghi et al 32 , a constant screening coefficient is sufficient to accurately predict IPs and EAs of a variety of atomic and molecular systems, even though an orbital-dependent α might be more accurate in the case of large or extended systems, or in systems composed of sub-systems having very different orbital relaxation properties. (v) and α (c) computed on top of the initial structures, PBE (@PBE), or on top of the structure resulting from scf-KIPZ optimization (@KIPZ) of different DNA/RNA nucleobases.…”

“…As mentioned, KC functionals [26][27][28][29][30][31][32] can be seen as a generalization of DFT+U aimed at explicitly enforcing PWL to an entire electronic manifold. These functionals are obtained by removing, orbital-by-orbital, the nonlinear (Slater) contribution to the total energy and by replacing it by a linear (Koopmans) term.…”

“…KI functionals, described in detail in Ref. 32, are obtained from an approximate functional E app as…”

“…Recently, Koopmans-compliant (KC) functionals were introduced [26][27][28][29][30][31][32] to enforce a generalized criterion of piecewise linearity (PWL) in the energy of approximate DFT functionals with respect to the fractional removal or addition of an electron from any orbital of the system. This PWL condition is a generalization to the entire manifold of the molecular DFT+U approach 33,34 , stemming from similar linearization ideas in the solid state.…”