Gradient expansion of the exchange energy from second-order density response theory

Svendsen, Per-Sverre; Barth, Ulf von

doi:10.1103/physrevb.54.17402

Cited by 175 publications

(94 citation statements)

References 28 publications

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“…[113] Another important quantity to describe the density behavior is the Laplacian of the density r 2 qðrÞ. This quantity appears in the fourthorder gradient expansions (GE4) of both the exchange [107] and kinetic [114][115][116] energies via the dimensionless reduced Laplacian…”

Section: Input Ingredients Of Meta-gga Functionals 211 | Inhomogementioning

confidence: 99%

“…Using indications from the gradient expansions, [105][106][107][108][109][110] the gradient of the density is generally employed to construct more practical input ingredients. For exchange, the dimensionless reduced gradient [111] s5 jrqj 2k F q 5 jrqj 2ð3p 2 Þ 1=3 q 4=3 ;…”

Section: Input Ingredients Of Meta-gga Functionals 211 | Inhomogementioning

confidence: 99%

“…Another important quantity to describe the density behavior is the Laplacian of the density r 2 qðrÞ. This quantity appears in the fourthorder gradient expansions (GE4) of both the exchange [107] and kinetic [114][115][116] energies via the dimensionless reduced LaplacianThe reduced Laplacian, which is by construction invariant under the uniform density scaling (see Appendix B), is therefore a natural meta-GGA input ingredient. It contains more information than the reduced gradients, being able to distinguish the nuclear region (q !…”

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confidence: 99%

“…[12] Its use was in fact proposed within DFT already in one of the first pioneering articles of Hohenberg and Kohn. [3] The gradient of the density provides the most simple and natural indication on the inhomogeneity of the electron density and it is a basic quantity in different exact properties of the XC energy (see Section 2.2).Using indications from the gradient expansions, [105][106][107][108][109][110] the gradient of the density is generally employed to construct more practical input ingredients. For exchange, the dimensionless reduced gradient [111] s5 jrqj…”

mentioning

confidence: 99%

“…[176] Equations 49 and 51 represent hard tests for any semilocal functional, being related to fractional particle number problems (b 5 1) [72] and self-interaction errors (homogeneous density scaling for b 5 0). [177] 2.2.1.6 | Gradient expansionsThe second-and fourth-order gradient expansions (GE2 and GE4, respectively) of the exchange energy have the following enhancement factors [105,107] F GE2 x 511l GE2 x s 2 ; with l GE2 x 5 10 81 ; with D 5 0 being the best numerical estimate for this coefficient. [65] Note that the F GE2 x and the term 146 2025 q 2 are present in the static HEG (i. e., jellium) linear-response, [178][179][180][181][182] At the meta-GGA level of theory, the second-order gradient expansion can also be applied to a functional with an exchange enhancement factor which depends only on a KS .…”

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Kinetic‐energy‐density dependent semilocal exchange‐correlation functionals

Sala

Fabiano

Constantin

2016

Int J of Quantum Chemistry

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We present the theory of semilocal exchange-correlation (XC) energy functionals which depend on the Kohn-Sham kinetic energy density (KED), including the relevant class of meta-generalized gradient approximation (meta-GGA) functionals. Thanks to the KED ingredient, meta-GGA functionals can satisfy different exact constraints for XC energy and can be made one-electron selfcorrelation free. This leads to a better accuracy over a wider range of properties with respect to GGAs, often reaching the accuracy of hybrid functionals, but at much reduced computational cost.An extensive survey of the relevant literature on existing KED dependent XC functionals is provided, considering nonempirical, semi-empirical, and fully empirical ones. A deeper analysis and a wide benchmark are presented for functionals derived considering only exact constraints and parameters obtained from model and/or atomic systems. K E Y W O R D Sdensity functional theory, kinetic energy, meta-GGA | I N T R O D U C T I O NKohn-Sham (KS)-density functional theory (DFT) [1][2][3][4][5] is nowadays one of the most popular computational approaches in quantum chemistry, solid-state physics, and materials science. [6][7][8][9][10][11] While the ground-state description of a real system of interacting electrons requires a complicate N-electron wavefunction, in KS-DFT this is mapped onto an auxiliary system of noninteracting electrons having the same ground-state density, which can be easily described using N single-particle KS orbitals. [1,2] The correspondence between the many-electron problem and the KS system is in principle exact [1,3] and it is based on the so called exchange-correlation (XC) energy functional, which collects all the quantum electron-electron interaction terms beyond the Hartree one. [12] Within the constrained-search formalism, [13] the XC energy functional is:E xc ½q 5 min W!q hWjT 1V ee jWi2T s ½q 2J½q 5E x ½q 1U c ½q 1T c ½q ;(1) where W is an N-electron ground-state wavefunction yielding the electron density q,T is the kinetic energy (KE) operator,V ee is the electron-electron interaction operator, T s is the KE of the noninteracting system, J is the Coulomb energy, E x is the exact exchange energy, and U c and T c represent, respectively, the potential and the kinetic contributions to the DFT correlation energy E c 5U c 1T c . The important point of Equation1 is that it is an universal functional of the electron density. However, its explicit functional form is not known, and any practical realization of KS-DFT is based on an approximated XC functional. The latter determines in practice the overall accuracy of the KS-DFT calculation.The study of the exact properties of the XC energy functional as well as the development of new and improved approximations have been hot research topics in DFT for more than three decades, and a large number of XC approximations has been proposed. [12,14,15] The different XC functionals are usually classified, according to their complexity, on the so called Jacob's ladder of DFT [14,16] whic...

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Section: Input Ingredients Of Meta-gga Functionals 211 | Inhomogementioning

confidence: 99%

Section: Input Ingredients Of Meta-gga Functionals 211 | Inhomogementioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Kinetic‐energy‐density dependent semilocal exchange‐correlation functionals

Sala

Fabiano

Constantin

2016

Int J of Quantum Chemistry

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From explicit to implicit density functionals

1999

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The concept of orbital-and eigenvalue-dependent exchange-Ž . correlation xc energy functionals is reviewed. We show how such functionals can be derived in a systematic fashion via a perturbation expansion, utilizing the Kohn᎐Sham system as a noninteracting reference system. We demonstrate that the second-order contribution to this expansion of the xc-energy functional includes the leading term of the van der Waals interaction. The optimized-Ž . potential method OPM , which allows the calculation of the multiplicative xc-potential corresponding to an orbital-and eigenvalue-dependent xc-energy functional via an integral equation, is discussed in detail. We examine an approximate analytical solution of the OPM integral equation, pointing out that, for eigenvalue-dependent functionals, the three paths used in the literature for the derivation of this approximation yield different results. Finally, a number of illustrative results, both for the exchange-only limit and for the combination of the exact exchange with various correlation functionals, are given.

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Molecular and solid-state tests of density functional approximations: LSD, GGAs, and meta-GGAs

Kurth

Perdew

Blaha³

1999

Int. J. Quant. Chem.

673

385

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Results from numerical tests of nine approximate exchange–correlation energy functionals are reported for various systems—atoms, molecules, surfaces, and bulk solids. The functional forms can be divided into three categories: (1) the local spin density (LSD) approximation, (2) generalized gradient approximations (GGAs), and (3) meta‐GGAs. In addition to the spin densities and their first gradients, the input to a meta‐GGA includes other semilocal information such as Laplacians of the spin densities or orbital kinetic energy densities. We present a way to visualize meta‐GGA nonlocality which generalizes that for GGA nonlocality, and which stresses the different meta‐GGA descriptions of iso‐orbital and orbital overlap regions of space. While some of the tested approximations were constructed semiempirically with many parameters fitted to chemical data, others were constructed to incorporate key properties of the exact exchange–correlation energy. The latter functionals perform well for both small and extended systems, with the best performance achieved by a meta‐GGA which recovers the correct gradient expansion. While the semiempirical functionals can achieve high accuracy for atoms and molecules, they are typically less accurate for surfaces and solids. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 75: 889–909, 1999

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Gradient expansion of the exchange energy from second-order density response theory

Cited by 175 publications

References 28 publications

Kinetic‐energy‐density dependent semilocal exchange‐correlation functionals

Kinetic‐energy‐density dependent semilocal exchange‐correlation functionals

From explicit to implicit density functionals

Molecular and solid-state tests of density functional approximations: LSD, GGAs, and meta-GGAs

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