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Physical review. B, Condensed matter

Normalization sum rules for the two-body scattering phase shifts of the electron-gas pair densities

P. Ziesche

Published Jun 20, 2003• 15 citations
Publisher

Abstract

It is shown how the normalization sum rules of the spin-parallel and spin-antiparallel (or equivalently singlet and triplet) pair densities of the homogeneous electron gas can be rewritten equivalently as sum rules for the scattering phase shifts of those two-body wave functions (geminals), with which the pair densities have been parametrized recently. These sum rules relate two-body quantities to a one-body quantity: namely, the asymptotics of the pair-density geminals to the momentum distribution.

DOI
10.1103/physrevb.67.233102

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Physical review. B, Condensed matter•Jun 15, 1978•38 citations
It is shown how the normalization sum rules of the spin-parallel and spin-antiparallel (or equivalently singlet and triplet) pair densities of the homogeneous electron gas can be rewritten equivalently as sum rules for the scattering phase shifts of those two-body wave functions (geminals), with which the pair densities have been parametrized recently. These sum rules relate two-body quantities to a one-body quantity: namely, the asymptotics of the pair-density geminals to the momentum distribution.
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)
We present a calculation of the spin-averaged and spin-resolved pair distribution functions for a homogeneous gas of electrons moving in a plane with e2/r interactions at coupling strength rs = 10. The calculation is based on the solution of a two-electron scattering problem for both parallelspin- and antiparallel-spin-pairs interacting via effective spin-dependent many-body potentials. The scattering potentials are modeled within the approach proposed by Kukkonen and Overhauser to treat exchange and correlations under close constraints imposed by sum rules. We find very good agreement with quantum MonteCarlo data for the spin-averaged pair density. We also find that short-range pairing between parallel-spin electrons is beginning to emerge in the paramagnetic fluid at this coupling strength, as a precursor of a transition to a fully spin-polarized fluid state occurring at stronger coupling.
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)].
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.
The effects of short-range correlations on the properties of the electron gas are studied using an approach phrased in the polarization potential language of Pines and Nozi\`eres. It was argued by Lowy and Brown that two-body terms make up the dominant contribution to the effective interaction for large momentum transfers. In the region of small-momentum transfers it is known that the random-phase approximation provides an adequate picture of the electron gas. With these two regions of large- and small-momentum transfers in mind, a polarization propagator is constructed to interpolate between them. A number of sum rules, among them the fff-sum rule, involving the imaginary part of the polarization propagator, were found to be satisfied. The failure of our theory to satisfy the compressibility sum rule is discussed in detail. We present here the results for the local ttt-matrix, the dynamic and static form factors, and the pair correlation function. That our pair-correlation function remains positive and is essentially the same as in the paper of Lowy and Brown indicates that the physics and not the formalism is responsible for the results we obtain.