Two electrons in an external oscillator potential: Particular analytic solutions of a Coulomb correlation problem

Taut, M.

doi:10.1103/physreva.48.3561

Cited by 417 publications

(459 citation statements)

References 9 publications

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“…Its separability, in center-of-mass and relative coordinates, was pointed out by Kestner and Sinanoglu [20], and the fact that for certain values of the force constant it is analytically solvable was noted by Kais et al [21] and, more systematically, by Taut [22]. Several studies for Hooke's atom have been performed, aimed at deriving exact density functionals or Kohn-Sham type exchange-correlation potentials [23][24][25], most notably by Burke et al [26][27][28][29][30] and Ludena et al [31].…”

Section: Introductionmentioning

confidence: 99%

Nonuniversality of commonly used correlation-energy density functionals

Katriel

Roy

Springborg

2006

The Journal of Chemical Physics

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The correlation energies of the helium isoelectronic sequence and of Hooke's atom isoelectronic sequence have been evaluated using an assortment of local, gradient and meta-gradient density functionals. The results are compared with the exact correlation energies, showing that while several of the more recent density functionals reproduce the exact correlation energies of the helium isoelectronic sequence rather closely, none is satisfactory for Hooke's atom isoelectronic sequence. It is argued that the uniformly acceptable results for the helium sequence can be explained through simple scaling arguments that do not hold for Hooke's atom sequence, so that the latter system provides a more sensitive testing ground for approximate density functionals. This state of affairs calls for further effort towards formulating correlation-energy density functionals that would be truly universal at least for spherically-symmetric two-fermion systems.

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Section: Introductionmentioning

confidence: 99%

Nonuniversality of commonly used correlation-energy density functionals

Katriel

Roy

Springborg

2006

The Journal of Chemical Physics

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“…The basic HA is two electrons interacting by the Coulomb potential but bound to a nucleus by a harmonic potential that mimics a nuclear-electron attraction. For certain values of the confinement strength, there exist exact solutions for the HA ground state [1,2]. When the HA is placed in a perpendicular magnetic field, Taut provided analytical solutions for a two-dimensional (2D) HA at particular values of the magnetic field [3].…”

Section: Introductionmentioning

confidence: 99%

Dynamical screening of the Coulomb interaction for two confined electrons in a magnetic field

Simonović

Nazmitdinov

2008

Phys. Rev. A

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We show that a difference in time scales of vertical and lateral dynamics permits one to analyze the problem of interacting electrons confined in an axially symmetric three-dimensional potential with a lateral oscillator confinement by means of the effective two-dimensional Hamiltonian with a screened Coulomb interaction. Using an adiabatic approximation based on action-angle variables, we present solutions for the effective charge of the Coulomb interaction (screening) for a vertical confinement potential simulated by parabolic, square, and triangular wells. While for the parabolic potential the solution for the effective charge is given in a closed anlytical form, for the other cases similar solutions can be easily calculated numerically.

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“…[416][417][418][419][420][421][422] The exact ground-state solutions of the Hooke's atom are known for several values of x, see Refs. [423] and [421]. The inset of Figure 10 for different frequencies, where we have used the classical electron distance.…”

Section: Hooke's Atommentioning

confidence: 99%

“…The inset of Figure 10 for different frequencies, where we have used the classical electron distance. [423] r 0 5ðx 2 =2Þ 21=3 :…”

Section: Hooke's Atommentioning

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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Two electrons in an external oscillator potential: Particular analytic solutions of a Coulomb correlation problem

Cited by 417 publications

References 9 publications

Nonuniversality of commonly used correlation-energy density functionals

Nonuniversality of commonly used correlation-energy density functionals

Dynamical screening of the Coulomb interaction for two confined electrons in a magnetic field

Kinetic‐energy‐density dependent semilocal exchange‐correlation functionals

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