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arXiv:1108.1912

The two-body reduced density matrix of the high-density electron gas

P. Ziesche

Published Aug 9, 2011• 0 citations
arXivHTMLPDF

Abstract

For the first time, the cumulant 2-body reduced density matrix (= 2-matrix) of the spin-unpolarized homogeneous electron gas (HEG) is considered. This $γ_{\rm c}$ proves to be the common source for both the momentum distribution $n(k)$ and the static structure factor $S(q)$. Within many-body perturbation theory, this $γ_{\rm c}$ is given by only {\it linked} diagrams (with 2 open particle-hole lines as well as with closed loops and interaction lines). Here it is worked out in detail, how the 1-body quantity $n(k)$ follows from the 2-body quantity $γ_{\rm c}$ - through a certain contraction procedure, cf Eqs.(2.25)-(2.28). In particular, this $γ_{\rm c}$ is developed for the high-density HEG. Its correctness is checked by deriving from it $n(k)$ and $S(q)$, known from the random-phase approximation (RPA). This study opens the way to a more sophisticated HEG description in terms of cumulant geminals or/and variational methods. Besides, the cumulant structure factor (CSF) of the exchange in lowest order is explicitly given and sum rules for the CSFs and their small- and large-$q$ behavior (beyond RPA) are systematically summarized within the plasmon sum rule, coalescing theorems, and the inflexion-point trajectory.

DOI
10.48550/arxiv.1108.1912

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For the first time, the cumulant 2-body reduced density matrix (= 2-matrix) of the spin-unpolarized homogeneous electron gas (HEG) is considered. This γcγ_{\rm c}γc​ proves to be the common source for both the momentum distribution n(k)n(k)n(k) and the static structure factor S(q)S(q)S(q). Within many-body perturbation theory, this γcγ_{\rm c}γc​ is given by only {\it linked} diagrams (with 2 open particle-hole lines as well as with closed loops and interaction lines). Here it is worked out in detail, how the 1-body quantity n(k)n(k)n(k) follows from the 2-body quantity γcγ_{\rm c}γc​ - through a certain contraction procedure, cf Eqs.(2.25)-(2.28). In particular, this γcγ_{\rm c}γc​ is developed for the high-density HEG. Its correctness is checked by deriving from it n(k)n(k)n(k) and S(q)S(q)S(q), known from the random-phase approximation (RPA). This study opens the way to a more sophisticated HEG description in terms of cumulant geminals or/and variational methods. Besides, the cumulant structure factor (CSF) of the exchange in lowest order is explicitly given and sum rules for the CSFs and their small- and large-qqq behavior (beyond RPA) are systematically summarized within the plasmon sum rule, coalescing theorems, and the inflexion-point trajectory.
Abstract The cumulant 2‐matrix χ is that part of the two‐body reduced density matrix γ2, which cannot be reduced to products of the one‐body reduced density matrix (1‐matrix) γ. This irreducible part χ is calculated perturbatively for the high‐density electron gas (EG) in its ground state, such that the pair densities and the interaction energy are correctly reproduced in their high‐density limits, which are exactly known and summarized here. From the thus available cumulant 2‐matrix the pair density in momentum space can be derived and used for a fluctuation analysis and compared with the analog analysis in position space, where it is concluded that “correlation suppresses fluctuations”. The perturbatively available cumulant 2‐matrix χ can be used also for the high‐density electron gas to start the iterative solution of the Yasuda integral equation of density matrix functional theory (DMFT), which is a nonlinear functional relation between the cumulant 2‐matrix χ and the 1‐matrix γ recently derived from the contracted Schrödinger equation approach: χY[γ]. From the perturbatively determined χ one can find another functional χ[γ] as an alternative approximation for a DMFT. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002
The methods of quantum chemistry and solid state theory to solve the many-body problem are reviewed. We start with the definitions of reduced density matrices, their properties (contraction sum rules, spectral resolutions, cumulant expansion, NNN-representability), and their determining equations (contracted Schrödinger equations) and we summarize recent extensions and generalizations of the traditional quantum chemical methods, of the density functional theory, and of the quasi-particle theory: from finite to extended systems (incremental method), from density to density matrix (density matrix functional theory), from weak to strong correlation (dynamical mean field theory), from homogeneous (Kimball-Overhauser approach) to inhomogeneous and finite systems. Measures of the correlation strength are discussed. The cumulant two-body reduced density matrix proves to be a key quantity. Its spectral resolution contains geminals, being possibly the solutions of an approximate effective two-body equation, and the idea is sketched of how its contraction sum rule can be used for a variational treatment.
For the ground state of the homogeneous electron gas (jellium), it is shown how the cumulant decomposition of the 2-matrix leads to the cumulant decomposition of the structure factors Sa,p(q) for the antiparallel (a) and parallel (p) spin pairs and how it simultaneously allows one to derive the momentum distribution n(k), which is a one-body quantity [Phys. Rev. A 86, 012508 (2012)]. The small-q and large-q behavior of Sa,p(q), and their normalizations are derived and compared with the results of P. Gori-Giorgi et al. [Physica A 280, 199 (2000) and Phys. Rev. B 61, 7353 (2000)].
Abstract Recently, it has been shown, that the pair density of the homogeneous electron gas can be parametrized in terms of 2‐body wave functions (geminals), which are scattering solutions of an effective 2‐body Schrödinger equation. For the corresponding scattering phase shifts, new sum rules are reported in this paper. These sum rules describe not only the normalization of the pair density (similar to the Friedel sum rule of solid state theory), but also the contraction of the 2‐body reduced density matrix. This allows one to calculate also the momentum distribution, provided that the geminals are known from an appropriate screening of the Coulomb repulsion. An analysis is presented leading from the definitions and (contraction and spectral) properties of reduced density matrices to the Kimball‐Overhauser approach and its generalizations. Thereby cumulants are used. Their size‐extensivity is related to the thermodynamic limit. See erratum Ann. Phys. (Leipzig) 13, 624 (2004)