Optimized unrestricted Kohn–Sham potentials from <i>ab initio</i> spin densities

Within exact electron density-functional theory, we investigate Kohn-Sham (KS) potentials, orbital energies, and non-interacting kinetic energies of the fractional ions of Li, C and F. We use quantum Monte Carlo densities as input, which are then fitted, interpolated at non-integer electron numbers N , and inverted to produce accurate KS potentials v N s (r). We study the dependence of the KS potential on N , and in particular we numerically reproduce the theoretically predicted spatially constant discontinuity of v N s (r) as N passes through an integer. We further show that, for all the cases considered, the inner orbital energies and the non-interacting kinetic energy are nearly piecewise linear functions of N . This leads us to propose a simple approximation of the KS potential v N s (r) at any fractional electron number N which uses only quantities of the systems with the adjacent integer electron numbers.PACS numbers: 31.15.ae,31.15.E-,31.15.ve Over the past few decades, Kohn-Sham (KS) [1] density-functional theory (DFT) [2] has become one of the most important tools in electronic-structure theory. Given the overwhelming popularity of density-functional approximations (DFAs) (e.g., PBE [3], hybrids [4]), surprisingly few studies have been dedicated to the detailed properties of the exact KS system. This is despite the fact that unusual properties [5] of the fictious noninteracting KS system serve a vitally important role in reproducing the quantum mechanical properties of the interacting system in cases where degeneracies are present in the ground state, or where electrons are added and removed. In this paper we study exact KS DFT properties of difficult, open quantum systems with degenerate ground states -specifically open-shell atoms with noninteger electron numbers. A primary aim is to provide guidance for the construction of new DFAs.In quantum mechanics, open electronic systems with a non-integer average number of electrons naturally arise, for example, as fragments from a molecular dissociation in entangled quantum states. In particular, in DFT the study of systems with fractional electron numbers is of great importance for a better understanding of the theory (for a recent review, see Ref. 6). For such fractional systems, Perdew et al. [7] proved[8] that the energy is a piecewise linear function of the electron number between the adjacent integers. This lead to the theoretical prediction of the discontinuity of the KS potential as the electron number passes through an integer, with many important physical consequences concerning the description of the fundamental gap [9,10], molecular dissociation [7] or charge-transfer excitations [11]. This also lead * t.gould@griffith.edu.au to the explanation that the underestimation of energies obtained with the usual semilocal DFAs for delocalised densities is a consequence of their deviations from the exact piecewise linear behavior of the energy [12][13][14][15][16]. These understandings have guided the design of improved DFT approximations [17][18][19][2...

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“…Unlike other work [26,27], we use a spin-restricted open-shell formalism, and thus avoid dealing with spin dependence.…”

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confidence: 99%

Kohn-Sham potentials in exact density-functional theory at noninteger electron numbers

Gould

Toulouse

2014

Phys. Rev. A

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“…For other methods to circumvent the numerical difficulties due to the nonuniqueness of the optimized effective potential, see Refs. for instance. More recent analysis of the nonuniqueness of the numerical inversion and an approach to deal with it can be found in the work by Jacob (Ref.…”

Section: Introductionmentioning

confidence: 99%

Nonadditive kinetic potentials from inverted Kohn–Sham problem

Banafsheh¹,

Wesołowski²

2017

Int J of Quantum Chemistry

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The nonadditive kinetic potential is a key element in density-dependent embedding methods. The correspondence between the ground-state density and the total effective Kohn-Sham potential provides the basis for various methods to construct the nonadditive kinetic potential for any pair of electron densities. Several research groups used numerical or analytical inversion procedures to explore this strategy which overcomes the failures of known explicit density functional approximations. The numerical inversions, however, apply additional approximations/simplifications. The relations known for the exact quantities cannot be assumed to hold for quantities obtained in numerical inversions. The exact relations are discussed with special emphasis on such issues as: the admissibility of the densities for which the potential is constructed, the choice of densities to be used as independent variables, self-consistency between the potentials and observables calculated using the embedded wavefunction, and so forth. The review focuses on how these issues are treated in practice. The review is supplemented with the analysis of the inverted potentials for weakly overlapping pairs of electron densities-the case not studied previously.density embedding, density functional theory, electron density overlap | I N TR ODU C TI ONThe nonadditive kinetic energy (T nad s ) and potential (v nad t ðrÞ) are key ingredients in numerical simulation methods exploring the density embedding strategy (see section 1.2). Both quantities are defined as functionals of admissible pairs of electron densities: where W s denotes a N-electron single-determinant trial wavefunction. Atomic units are used throughout this work.In practical applications, q A ðrÞ is the density corresponding to the wavefunction W emb associated with the embedded subsystem and q tot ðrÞ is electron density of the total system comprising the part represented by W emb and its "environment." Usually, explicit density functionals approximate the functionals v nad t ½q A ; q tot ðrÞ and/or T nad s ½q A ; q tot . Simulations and studies on model systems reported by several researchers revealed many cases where the currently known explicit density functionals fail (see section 1.2). These failures resulted in recent interest in an alternative -implicit -strategy to construct v nad t ½q A ; q tot ðrÞ (and T nad s ½q A ; q tot ) for arbitrarily chosen pairs of densities q A ðrÞ and q tot ðrÞ. Such constructions are based on the Levy correspondence [1] between the Kohn-Sham potential [2] and the electron density. This correspondence exists for admissible electron Int J Quantum Chem. 2018;118:e25410.

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“…Various options for improving density functionals in this respect exist. One, which we explore currently, is the explicit reconstruction and analysis of the spin‐dependent Kohn–Sham potentials of chemically different systems …”

Section: Discussionmentioning

confidence: 99%

Systematic dependence of transition‐metal coordination energies on density‐functional parametrizations

Weymuth

Reiher

2014

Int J of Quantum Chemistry

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In recent years, the calibration of parameters in approximate exchange‐correlation density functionals was intensified by the proposition of diverse and increasingly unbiased reference datasets. It is, however, not obvious how sensitive the accuracy of a given functional is with respect to a small change of its parameters. Knowledge about this sensitivity would be desirable for the assessment of the general accuracy that can be expected for the calculation of a given observable—especially for notorious cases as found, for instance, in coordination chemistry. At the example of the well‐known BP86 exchange‐correlation functional, we investigate the dependence of the coordination energies in the WCCR10 reference set [Weymuth et al., J. Chem. Theory Comput. 2014, 10, 3092] on the empirical parameters of this density functional. The WCCR10 reactions were found to be a true challenge for contemporary density functionals. Here, we find that the parameter dependence is qualitatively the same for all reactions. This observation is important in view of the fact that the BP86 functional was never parametrized against transition‐metal data. Still, within the parameter intervals investigated, the individual reaction energies vary significantly, which seems to suggest a reoptimization of the empirical parameters. However, it turns out that the overall description of all 10 reactions can be improved by only 9.5 kJ/mol, and therefore, such a reoptimization is not advisable. © 2014 Wiley Periodicals, Inc.

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Optimized unrestricted Kohn–Sham potentials from ab initio spin densities

Cited by 28 publications

References 83 publications

Kohn-Sham potentials in exact density-functional theory at noninteger electron numbers

Kohn-Sham potentials in exact density-functional theory at noninteger electron numbers

Nonadditive kinetic potentials from inverted Kohn–Sham problem

Systematic dependence of transition‐metal coordination energies on density‐functional parametrizations

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