For the first time, the cumulant 2-body reduced density matrix (= 2-matrix) of the spin-unpolarized homogeneous electron gas (HEG) is considered. This
γc 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
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) follows from the 2-body quantity
γc - through a certain contraction procedure, cf Eqs.(2.25)-(2.28). In particular, this
γc 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.
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 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 subject of this study is the exchange-correlation-energy functional of reduced density-matrix functional theory. Approximations of this functional are tested by applying them to the homogeneous electron gas. We find that two approximations recently proposed by Gritsenko et al., [J. Chem. Phys. 122, 204102 (2005)] yield considerably better correlation energies and momentum distributions than previously known functionals. We introduce modifications to these functionals, which, by construction, reproduce the exact correlation energy of the homogeneous electron gas.
Some properties of Grassmann products of one-electron operators and of cumulants and related quantities for two-electron reduced density matrices (2-RDMs) are reported. The results suggest that any physical significance of the Grassmann product plus cumulant decomposition must be based on the physics of the system described, not just on the mathematical structure. The Grassmann product plus cumulant decomposition of the 2-RDM is resolved into components based on unitary invariants. Simple examples using mathematically acceptable but physically unlikely wave functions show that a Grassmann product differing from the usual one may better approximate the 2-RDM, and that the usual Grassmann product, cumulant, and 2-RDM can be mutually proportional.
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)