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Title: Momentum density and spatial form of correlated density matrix in model two-electron atoms with harmonic confinement

Abstract

The detailed nature of the correlated first-order density matrix for the model atoms in the title for arbitrary interparticle interaction u(r{sub 12}) is studied. One representation with contracted information is first explored by constructing the momentum density {rho}(p) in terms of the wave function of the relative motion, say {psi}{sub R}(r{sub 12}), which naturally depends on the choice of u(r{sub 12}). For u(r{sub 12})=e{sup 2}/r{sub 12}, the so-called Hookean atom, and for the inverse square law u(r{sub 12})={lambda}/r{sub 12}{sup 2}, plots are presented of the above density {rho}(p) in momentum space. The correlated kinetic energy is recovered from averaging p{sup 2}/2m, m denoting the electron mass, with respect to {rho}(p). The second method developed is in coordinate space and expands the density matrix {gamma}(r{sub 1},r{sub 2}) in Legendre polynomials, using relative coordinate r{sub 1}-r{sub 2}, center-of-mass coordinate (r{sub 1}+r{sub 2})/2 and the angle, {theta} say, between these two vectors. For the Moshinsky atom in which u(r{sub 12})=(1/2)kr{sub 12}{sup 2} only the s term (l=0) contributes to the Legendre polynomial expansion. The specific example we present of the inverse square law model is shown to be characterized by the low-order terms (s+d) of the Legendre expansion. The Wigner function is finallymore » calculated analytically for both Moshinsky and inverse square law models.« less

Authors:
;  [1];  [1]
  1. Departamento de Fisica de Materiales, Facultad de Ciencias Quimicas, Universidad del Pais Vasco, Centro Mixto UPV-CSIC, and Donostia International Physics Center (DIPC), 20018 San Sebastian (Spain)
Publication Date:
OSTI Identifier:
21015970
Resource Type:
Journal Article
Journal Name:
Physical Review. A
Additional Journal Information:
Journal Volume: 76; Journal Issue: 3; Other Information: DOI: 10.1103/PhysRevA.76.032510; (c) 2007 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1050-2947
Country of Publication:
United States
Language:
English
Subject:
74 ATOMIC AND MOLECULAR PHYSICS; ATOMS; CENTER-OF-MASS SYSTEM; CONFINEMENT; DENSITY MATRIX; ELECTRONS; EXPANSION; HELIUM; INTERACTIONS; KINETIC ENERGY; LEGENDRE POLYNOMIALS; MASS; WAVE FUNCTIONS; WIGNER DISTRIBUTION

Citation Formats

Akbari, Ali, Rubio, Angel, European Theoretical Spectroscopy Facility, March, Norman H, Department of Physics, University of Antwerp, Antwerp, and Oxford University, Oxford. Momentum density and spatial form of correlated density matrix in model two-electron atoms with harmonic confinement. United States: N. p., 2007. Web. doi:10.1103/PHYSREVA.76.032510.
Akbari, Ali, Rubio, Angel, European Theoretical Spectroscopy Facility, March, Norman H, Department of Physics, University of Antwerp, Antwerp, & Oxford University, Oxford. Momentum density and spatial form of correlated density matrix in model two-electron atoms with harmonic confinement. United States. https://doi.org/10.1103/PHYSREVA.76.032510
Akbari, Ali, Rubio, Angel, European Theoretical Spectroscopy Facility, March, Norman H, Department of Physics, University of Antwerp, Antwerp, and Oxford University, Oxford. 2007. "Momentum density and spatial form of correlated density matrix in model two-electron atoms with harmonic confinement". United States. https://doi.org/10.1103/PHYSREVA.76.032510.
@article{osti_21015970,
title = {Momentum density and spatial form of correlated density matrix in model two-electron atoms with harmonic confinement},
author = {Akbari, Ali and Rubio, Angel and European Theoretical Spectroscopy Facility and March, Norman H and Department of Physics, University of Antwerp, Antwerp and Oxford University, Oxford},
abstractNote = {The detailed nature of the correlated first-order density matrix for the model atoms in the title for arbitrary interparticle interaction u(r{sub 12}) is studied. One representation with contracted information is first explored by constructing the momentum density {rho}(p) in terms of the wave function of the relative motion, say {psi}{sub R}(r{sub 12}), which naturally depends on the choice of u(r{sub 12}). For u(r{sub 12})=e{sup 2}/r{sub 12}, the so-called Hookean atom, and for the inverse square law u(r{sub 12})={lambda}/r{sub 12}{sup 2}, plots are presented of the above density {rho}(p) in momentum space. The correlated kinetic energy is recovered from averaging p{sup 2}/2m, m denoting the electron mass, with respect to {rho}(p). The second method developed is in coordinate space and expands the density matrix {gamma}(r{sub 1},r{sub 2}) in Legendre polynomials, using relative coordinate r{sub 1}-r{sub 2}, center-of-mass coordinate (r{sub 1}+r{sub 2})/2 and the angle, {theta} say, between these two vectors. For the Moshinsky atom in which u(r{sub 12})=(1/2)kr{sub 12}{sup 2} only the s term (l=0) contributes to the Legendre polynomial expansion. The specific example we present of the inverse square law model is shown to be characterized by the low-order terms (s+d) of the Legendre expansion. The Wigner function is finally calculated analytically for both Moshinsky and inverse square law models.},
doi = {10.1103/PHYSREVA.76.032510},
url = {https://www.osti.gov/biblio/21015970}, journal = {Physical Review. A},
issn = {1050-2947},
number = 3,
volume = 76,
place = {United States},
year = {Sat Sep 15 00:00:00 EDT 2007},
month = {Sat Sep 15 00:00:00 EDT 2007}
}