Density-Matrix Functional Theory and the High-Density Electron Gas

  • P. Ziesche


Often the ground state (GS) of the unpolarized uniform electron gas (EG) serves as a good model to describe the influence of the interelectron Coulomb repulsion on the energies and reduced densities 1. Essential kinematic features of this quantum-many-body system are described by the electron density ρ, the momentum distribution n(k), and the dimensionless pair distribution functions, or pair densities (PD), g and g pairs with parallel and antiparallel spins, respectively. n(k) satisfies 0 ≤ n(k) ≤ 1 (n(k) is non-idempotent), and the g’s are probabilities, thus g , g ≥ 0. ρ fixes the dimensionless density parameter r s from (4πr s 3 /3)a B 3 = 1/ρ and the Fermi wave number k F from k F 3 = 3π 2 ρ. The PD’s g and g are each referred to as either exchange or Fermi hole, and correlation or Coulomb hole, respectively. They are functions of the interelectron distance r 12 = |r 1r 2|. All the quantities of the EG depend parametrically on the density parameter r s . The Coulomb repulsion between each electron pair causes the phenomenon denoted electron correlation 2–9, which shows up in the correlation ‘tails’ of n(k), which means n(k < 1) ⪅ 1 for holes and n(k > 1) ⪆ 0 for particles, where k is measured in units of k F. Naturally related to these tails is z F < 1, the quasiparticle weight or reduced jump discontinuity of n(k) at k = 1, which causes the oscillatory long-range behavior of g and g for r 12 → ∞. Short-range (or dynamical) correlations show up in the curvature of g for zero-interelectron distance. We refer to this situation as the electrons being “in contact” with or “on-top” of each other. They also appear in the on-top value of g , which determines simultaneously both the on-top slope of g from the coalescing cusp theorem 10, and the asymptotic behavior of n(k) for k → ∞. The spin-traced PD g = (g + g )/2 and its parallel and antiparallel ‘components’ have been repeatedly studied over the years 11–35 This is especially true for the on-top value of the Coulomb hole g 25–27, 34, 35 and for the on-top curvature of the Fermi hole g 17, 28, 34, 35 which are short-range correlation properties. Examples of studies of the Fermi and Coulomb holes in molecules are given in Ref. 36.


Momentum Distribution Coulomb Repulsion Virial Theorem Pair Density Reduce Density Matrice 
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© Springer Science+Business Media New York 2002

Authors and Affiliations

  • P. Ziesche
    • 1
  1. 1.Max Planck Institute for the Physics of Complex SystemsDresdenGermany

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