Electron Correlations and Materials Properties 2 pp 307-323 | Cite as

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

## Abstract

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 (4*πr* _{ s } ^{3} /3)*a* _{B} ^{3} = 1/*ρ* and the Fermi wave number *k* _{F} from *k* _{F} ^{3} = 3*π* ^{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* _{1} − *r* _{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}.

## Keywords

Momentum Distribution Coulomb Repulsion Virial Theorem Pair Density Reduce Density Matrice## Preview

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## References

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