Semiclassical Origins of Density Functionals

Elliott, Peter; Lee, Dong‐Hyung; Cangi, Attila; Burke, Kieron

doi:10.1103/physrevlett.100.256406

Cited by 96 publications

(168 citation statements)

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“…8 Adding and subtracting NE F , eqs A1 and A2 can be written in the more convenient form where, in order to make contact with the work of Elliot et al, we restrict ourselves to the 1D case. Taking again the functional derivative with respect to δV(y), we obtain The particular model studied by Elliot et al 8 consists of a one-dimensional box with potential V(x), 0 e x e L, and E F > V(x) everywhere. Hard-wall boundary conditions were imposed at x ) 0 and L. The corresponding G s semi [V](x,ε) is given by…”

Section: Appendixmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Three-dimensional generalizations have also been formulated. 7 On the basis of the work of Kohn and Sham, highly accurate potential functionals in 1D have recently been developed by Elliot et al 8 We The basic idea of PFT is to find good approximations for T [V] and W [V] so that the total energy of a given system, characterized by the external potential V 0 (r), is obtained directly by plugging V 0 (r) in the functional (eq 4).Most potential functionals known to date have been obtained by semiclassical considerations. [3][4][5][6] Highly accurate approximations have recently been constructed 8 for the kinetic energy and the density of noninteracting particles in one spatial dimension (1D).…”

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Adiabatic Connection and the Kohn−Sham Variety of Potential−Functional Theory

Gross

Proetto

2009

J. Chem. Theory Comput.

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Abstract:In potential-functional theory the total electronic energy is expressed as a functional of the external potential. We discuss how approximations, T s app [v], of the noninteracting kinetic energy functional can be exploited for interacting systems. Two possibilities are discussed: (a) Via an adiabatic connection formula, T s app [v 0 ] can be used directly with the external potential v 0 of the interacting system, and (b) by employing the variational principle of density functional theory, the kinetic energy functional T s app [v s ] is evaluated at the Kohn-Sham potential v s , which, in turn, is determined by an iterative procedure. Advantages and disadvantages of the two approaches are discussed.The Kohn-Sham equations of density functional theory (DFT) are the method of choice to calculate medium to large electronic systems of up to 10000-100000 electrons. The basic strategy of the Kohn-Sham method is to map the interacting electronic system of interest onto a system of noninteracting particles such that the latter has the same ground-state density as the interacting system. Solving the Kohn-Sham single-particle Schrödinger equations rather than the interacting many-body Schrödinger equation makes the problem numerically tractable. However, for larger systems, even the solution of the Kohn-Sham equations becomes too costly. Here, orbital-free DFT, 1 that is, the representation of the total energy as an explicit functional of the density, is the ultimate method. Alternatively, one may express the total energy as a functional of the external potential. 2 This alternative approach, called potentialfunctional theory (PFT), will be addressed in this communication. The approach has its roots in semiclassical Wigner-Kirkwood-type expansions. [3][4][5] The design of more refined semiclassical approximations was outlined in the 1960s by Kohn and Sham 6 in one-dimensional systems. Three-dimensional generalizations have also been formulated. 7 On the basis of the work of Kohn and Sham, highly accurate potential functionals in 1D have recently been developed by Elliot et al. 8 We The basic idea of PFT is to find good approximations for T [V] and W [V] so that the total energy of a given system, characterized by the external potential V 0 (r), is obtained directly by plugging V 0 (r) in the functional (eq 4).Most potential functionals known to date have been obtained by semiclassical considerations. [3][4][5][6] Highly accurate approximations have recently been constructed 8 for the kinetic energy and the density of noninteracting particles in one spatial dimension (1D). To use these approximations as part of the total energy functional of interacting systems, it appears desirable to have a coupling constant integration formula (or adiabatic connection) in PFT. We will deduce such a formula in the following. Consider the λ-dependent Hamiltonian where λ with 0 e λ e 1 allows us to switch off the electron-electron interaction. The V(r) is an external potential which, in contrast to the adiabatic connection of...

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Section: Appendixmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Adiabatic Connection and the Kohn−Sham Variety of Potential−Functional Theory

Gross

Proetto

2009

J. Chem. Theory Comput.

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“…Moreover, we recall that while in real atoms the electrons far from the nucleus experience a screened nuclear charge so that the corresponding orbitals differ from the hydrogenic ones, for large atoms or very positive ions, this screening effect becomes vanishingly small, and the simple model of hydrogenic orbitals becomes exact [69]. This model system has been largely used in DFT [68,[70][71][72], is very important for semiclassical physics [70,73,74] and has been used as a main reference system in recent GGA functionals [20,55].…”

Section: Kinetic and Exchange Energy Densities At The Nuclear Cusp Inmentioning

confidence: 99%

Kinetic and Exchange Energy Densities near the Nucleus

2016

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Abstract:We investigate the behavior of the kinetic and the exchange energy densities near the nuclear cusp of atomic systems. Considering hydrogenic orbitals, we derive analytical expressions near the nucleus, for single shells, as well as in the semiclassical limit of large non-relativistic neutral atoms. We show that a model based on the helium iso-electronic series is very accurate, as also confirmed by numerical calculations on real atoms up to two thousands electrons. Based on this model, we propose non-local density-dependent ingredients that are suitable for the description of the kinetic and exchange energy densities in the region close to the nucleus. These non-local ingredients are invariant under the uniform scaling of the density, and they can be used in the construction of non-local exchange-correlation and kinetic functionals.

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“…Equation 56 is based on the ThomasFermi scaling (Appendix B) and it has been used in the construction of GGA [95,189] and meta-GGA [152,153] functionals. Moreover, modified GE2 [73,104] and modified GE4 [189] (MGE2 and MGE4, respectively) have been derived from the semiclassical theory of atoms. The MGE2 and MGE4 exchange enhancement factors are:…”

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confidence: 99%

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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Semiclassical Origins of Density Functionals

Cited by 96 publications

References 22 publications

Adiabatic Connection and the Kohn−Sham Variety of Potential−Functional Theory

Adiabatic Connection and the Kohn−Sham Variety of Potential−Functional Theory

Kinetic and Exchange Energy Densities near the Nucleus

Kinetic‐energy‐density dependent semilocal exchange‐correlation functionals

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