Proposed definitions of the correlation energy density from a Hartree-Fock starting point: The two-electron Moshinsky model atom as an exactly solvable model

Norman Henry March, A. Cabo, Francisco H. Claro +1 more

arXiv:0803.2594Physical Review AApr 4, 200816 citations

Physics Letters A

Exact correlated kinetic energy related to the electron density for two-electron model atoms with harmonic confinement

Norman Henry March, Ali Akbari, Ángel Rubio

Published Jun 6, 2007• 1 citation

Publisher

DOI: 10.1016/j.physleta.2007.05.097

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Correlated kinetic energy density functional of ground states of harmonically confined two-electron atoms for arbitrary interparticle interaction

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Utilizing the earlier work of Holas et al (2003 Phys. Lett. A 310 451) and the more recent contribution of Akbari et al (2009 Phys. Rev. A 80 032509), we construct an integral equation for the relative motion (RM) contribution tRM(r) to the correlated kinetic energy density for modelling two-electron atoms with harmonic confinement but arbitrary interparticle interaction. It is stressed that tRM = tRM[f(G)], where f(G) is the atomic scattering factor: the Fourier transform of the density ρ(r). As a simple illustrative example of this functional relation for the correlated kinetic energy density, the harmonic Moshinsky case is investigated, the scattering factor then having a Gaussian form.Utilizing the earlier work of Holas et al (2003 Phys. Lett. A 310 451) and the more recent contribution of Akbari et al (2009 Phys. Rev. A 80 032509), we construct an integral equation for the relative motion (RM) contribution tRM(r) to the correlated kinetic energy density for modelling two-electron atoms with harmonic confinement but arbitrary interparticle interaction. It is stressed that tRM = tRM[f(G)], where f(G) is the atomic scattering factor: the Fourier transform of the density ρ(r). As a simple illustrative example of this functional relation for the correlated kinetic energy density, the harmonic Moshinsky case is investigated, the scattering factor then having a Gaussian form.

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Recent Progress in Constructing Nonlocal Energy Density Functionals

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Both model results and some exact relations concerning kinetic, exchange, and correlation energy density functionals will be presented. As to single-particle kinetic energy, a model of harmonic confinement, motivated by recent experimental activity on Fermion vapors, of an arbitrary number of Fermions will be used to construct exact nonlocal kinetic energy density functionals in both one and three dimensions. Then, some further exact results concerning nuclear cusp conditions for exchange only, and for exchange plus correlation, will be discussed. Finally, in Appendices, the one-shell harmonic confinement is generalized to include Coulomb repulsion in the two-electron Hookean atom and some results on the Pauli potentials for many-Fermion systems are obtained. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003Both model results and some exact relations concerning kinetic, exchange, and correlation energy density functionals will be presented. As to single-particle kinetic energy, a model of harmonic confinement, motivated by recent experimental activity on Fermion vapors, of an arbitrary number of Fermions will be used to construct exact nonlocal kinetic energy density functionals in both one and three dimensions. Then, some further exact results concerning nuclear cusp conditions for exchange only, and for exchange plus correlation, will be discussed. Finally, in Appendices, the one-shell harmonic confinement is generalized to include Coulomb repulsion in the two-electron Hookean atom and some results on the Pauli potentials for many-Fermion systems are obtained. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003

World Scientific series in 20th century physicsSep 1, 20090 citations

Proposed definitions of the correlation energy density from a Hartree-Fock starting point: The two-electron Moshinsky model atom as an exactly solvable model

Norman Henry March, A. Cabo, Francisco H. Claro +1 more

In both molecular physics and condensed matter theory, deeper understanding of the correlation energy density ${ϵ}_{c}(\mathbf{r})$ remains a high priority. By adopting L\"owdin's definition of correlation energy as the difference between the exact and the Hartree-Fock values, here we propose two alternative routes to define this. One of these involves both exact and Hartree-Fock (HF) wave functions, while the second requires a coupling constant integration. As an exact analytical example of the first route, we treat the two-electron model atom of Moshinsky, for which both confinement potential and interactions are harmonic. Though the correlation energy density ${ϵ}_{c}(\mathbf{r})$ is known analytically, we also investigate numerically its relation to the exact ground-state density in this example.In both molecular physics and condensed matter theory, deeper understanding of the correlation energy density ${ϵ}_{c}(\mathbf{r})$ remains a high priority. By adopting L\"owdin's definition of correlation energy as the difference between the exact and the Hartree-Fock values, here we propose two alternative routes to define this. One of these involves both exact and Hartree-Fock (HF) wave functions, while the second requires a coupling constant integration. As an exact analytical example of the first route, we treat the two-electron model atom of Moshinsky, for which both confinement potential and interactions are harmonic. Though the correlation energy density ${ϵ}_{c}(\mathbf{r})$ is known analytically, we also investigate numerically its relation to the exact ground-state density in this example.

arXiv:0803.2594Physical Review AApr 4, 200816 citations

In both molecular physics and condensed matter theory, deeper understanding of the correlation energy density

{ϵ}_{c}(\mathbf{r})

remains a high priority. By adopting L\"owdin's definition of correlation energy as the difference between the exact and the Hartree-Fock values, here we propose two alternative routes to define this. One of these involves both exact and Hartree-Fock (HF) wave functions, while the second requires a coupling constant integration. As an exact analytical example of the first route, we treat the two-electron model atom of Moshinsky, for which both confinement potential and interactions are harmonic. Though the correlation energy density

{ϵ}_{c}(\mathbf{r})

is known analytically, we also investigate numerically its relation to the exact ground-state density in this example.