Efficient Computation of the Hartree–Fock Exchange in Real-Space with Projection Operators

Boffi, Nicholas M.; Jain, Manish; Natan, Amir

doi:10.1021/acs.jctc.6b00376

Cited by 25 publications

(26 citation statements)

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“…For performing hybrid functional calculations, we implement the Fock operator in DARSEC using the self-consistent scheme described in Ref. [63]. We note that efficient projection approximations for the Fock exchange operator are given Refs.…”

Section: B Exact Remainder Potentialsmentioning

confidence: 99%

Exact Generalized Kohn-Sham Theory for Hybrid Functionals

et al. 2020

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Hybrid functionals have proven to be of immense practical value in density-functional-theory calculations. While they are often thought to be a heuristic construct, it has been established that this is in fact not the case. Here, we present a rigorous and formally exact analysis of generalized Kohn-Sham (GKS) density-functional theory of hybrid functionals, in which exact remainder exchange-correlation potentials combine with a fraction of Fock exchange to produce the correct ground-state density. First, we extend formal GKS theory by proving a generalized adiabatic connection theorem. We then use this extension to derive two different definitions for a rigorous distinction between multiplicative exchange and correlation components-one new and one previously postulated. We examine their density-scaling behavior and discuss their similarities and differences. We then present a new algorithm for obtaining exact GKS potentials by inversion of accurate reference electron densities and employ this algorithm to obtain exact potentials for simple atoms and ions. We establish that an equivalent description of the many-electron problem is indeed obtained with any arbitrary global fraction of Fock exchange, and we rationalize the Fock-fraction dependence of the computed remainder exchange-correlation potentials in terms of the new formal theory. Finally, we use the exact theoretical framework and numerical results to shed light on the exchange-correlation potential used in approximate hybrid functional calculations and to assess the consequences of different choices of fractional exchange.

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Section: B Exact Remainder Potentialsmentioning

confidence: 99%

Exact Generalized Kohn-Sham Theory for Hybrid Functionals

et al. 2020

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“…For performing hybrid functional calculations, we have implemented the Fock operator in DARSEC using the self-consistent scheme described in ref. [60]. We note that efficient projection approximations for the Fock exchange operator have been given [60][61][62][63], but for the small systems studied here use of the full (unprojected) Fock operator was found to be sufficient.…”

Section: B Exact Remainder Potentialsmentioning

confidence: 87%

Exact Generalized Kohn-Sham Theory for Hybrid Functionals

Garrick¹,

Natan²,

Gould³

et al. 2020

Preprint

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Hybrid functionals have proven to be of immense practical value in density functional theory calculations. While they are often thought to be a heuristic construct, it has been established that this is in fact not the case. Here, we present a rigorous and formally exact analysis of generalized Kohn-Sham (GKS) density functional theory of hybrid functionals, in which exact remainder exchange-correlation potentials combine with a fraction of Fock exchange to produce the correct ground state density. First, we extend formal GKS theory by proving a generalized adiabatic connection theorem. We then use this extension to derive two different definitions for a rigorous distinction between multiplicative exchange and correlation components - one new and one previously postulated. We examine their density-scaling behavior and discuss their similarities and differences. We then present a new algorithm for obtaining exact GKS potentials by inversion of accurate reference electron densities and employ this algorithm to obtain exact potentials for simple atoms and ions. We establish that an equivalent description of the many-electron problem is indeed obtained with any arbitrary global fraction of Fock exchange and we rationalize the Fock-fraction dependence of the computed remainder exchange-correlation potentials in terms of the new formal theory. Finally, we use the exact theoretical framework and numerical results to shed light on the exchange-correlation potential used in approximate hybrid functional calculations and to assess the consequences of different choices of fractional exchange.<br><br>

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“…In contrast, the exact HF exchange interaction is nonlocal; however, it can be truncated to finite range in large systems, especially if exact exchange is only included within a short range, as in the popular Heyd‐Scuseria‐Ernzerhof functional . Alternatively, the evaluation of exact exchange can be sped up with projection operators; or the Krieger‐Li‐Iafrate (KLI) approximation can be used to build a fully local exchange potential, also allowing exact‐exchange calculations on large systems . Further necessary steps in the implementation of large‐scale real‐space approaches involve replacing the full matrix diagonalization in the conventional Roothaan orbital update of Equation with an iterative approach such as the Davidson method that is used to solve only for the new occupied subspace, or avoiding diagonalization altogether by using, for example, Green's function methods to solve for the new occupied orbitals, as in the Helmholtz kernel method …”

Section: Applicationsmentioning

confidence: 99%

A review on non‐relativistic, fully numerical electronic structure calculations on atoms and diatomic molecules

Lehtola

2019

Int J of Quantum Chemistry

102

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The need for accurate calculations on atoms and diatomic molecules is motivated by the opportunities and challenges of such studies. The most commonly used approach for all-electron electronic structure calculations in general-the linear combination of atomic orbitals (LCAO) method-is discussed in combination with Gaussian, Slater a.k.a. exponential, and numerical radial functions. Even though LCAO calculations have major benefits, their shortcomings motivate the need for fully numerical approaches based on, for example, finite differences, finite elements, or the discrete variable representation, which are also briefly introduced. Applications of fully numerical approaches for general molecules are briefly reviewed, and their challenges are discussed. It is pointed out that the high level of symmetry present in atoms and diatomic molecules can be exploited to fashion more efficient fully numerical approaches for these special cases, after which it is possible to routinely perform allelectron Hartree-Fock and density functional calculations directly at the basis set limit on such systems. Applications of fully numerical approaches to calculations on atoms as well as diatomic molecules are reviewed. Finally, a summary and outlook is given.atomic calculations, density functional theory, diatomic calculations, fully numerical electronic structure theory, Hartree-Fock | INTRODUCTIONThanks to decades of development in approximate density functionals and numerical algorithms, density functional theory [1,2] (DFT) is now one of the cornerstones of computational chemistry and solid-state physics. [3][4][5] Since atoms are the basic building block of molecules, a number of popular density functionals [6][7][8][9][10] have been parametrized using ab initio data on noble gas atoms; these functionals have been used in turn as the starting point for the development of other functionals. A comparison of the results of density-functional calculations to accurate ab initio data on atoms has, however, recently sparked controversy on the accuracy of a number of recently published, commonly used density functionals. [11][12][13][14][15][16][17][18][19][20][21] This underlines that there is still room for improvement in DFT even for the relatively simple case of atomic studies.The development and characterization of new density functionals require the ability to perform accurate atomic calculations. Because atoms are the simplest chemical systems, atomic calculations may seem simple; however, they are in fact sometimes surprisingly challenging. For instance, a long-standing discrepancy between the theoretical and experimental electron affinity and ionization potential of gold has been resolved only very recently with high-level ab initio calculations. [22] Atomic calculations can also be used to formulate efficient starting guesses for molecular electronic structure calculations via either the atomic density [23,24] or the atomic potential as discussed in Ref. [25].

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Efficient Computation of the Hartree–Fock Exchange in Real-Space with Projection Operators

Cited by 25 publications

References 43 publications

Exact Generalized Kohn-Sham Theory for Hybrid Functionals

Exact Generalized Kohn-Sham Theory for Hybrid Functionals

Exact Generalized Kohn-Sham Theory for Hybrid Functionals

A review on non‐relativistic, fully numerical electronic structure calculations on atoms and diatomic molecules

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