Reduced density matrices, their spectral resolutions, and the Kimball-Overhauser approach

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 Schr\"odinger 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.Comment: 15 pages, conference contributio

The 2-matrix of the spin-polarized electron gas: contraction sum rules and spectral resolutions

The spin-polarized homogeneous electron gas with densities $\rho_\uparrow$ and $\rho_\downarrow$ for electrons with spin `up' ($\uparrow$) and spin `down' ($\downarrow$), respectively, is systematically analyzed with respect to its lowest-order reduced densities and density matrices and their mutual relations. The three 2-body reduced density matrices $\gamma_{\uparrow\uparrow}$, $\gamma_{\downarrow\downarrow}$, $\gamma_a$ are 4-point functions for electron pairs with spins $\uparrow\uparrow$, $\downarrow\downarrow$, and antiparallel, respectively. From them, three functions $G_{\uparrow\uparrow}(x,y)$, $G_{\downarrow\downarrow}(x,y)$, $G_a(x,y)$, depending on only two variables, are derived. These functions contain not only the pair densities but also the 1-body reduced density matrices. The contraction properties of the 2-body reduced density matrices lead to three sum rules to be obeyed by the three key functions $G_{ss}$, $G_a$. These contraction sum rules contain corresponding normalization sum rules as special cases. The momentum distributions $n_\uparrow(k)$ and $n_\downarrow(k)$, following from $f_\uparrow(r)$ and $f_\downarrow(r)$ by Fourier transform, are correctly normalized through $f_s(0)=1$. In addition to the non-negativity conditions $n_s(k),g_{ss}(r),g_a(r)\geq 0$ [these quantities are probabilities], it holds $n_s(k)\leq 1$ and $g_{ss}(0)=0$ due to the Pauli principle and $g_a(0)\leq 1$ due to the Coulomb repulsion. Recent parametrizations of the pair densities of the spin-unpolarized homogeneous electron gas in terms of 2-body wave functions (geminals) and corresponding occupancies are generalized (i) to the spin-polarized case and (ii) to the 2-body reduced density matrix giving thus its spectral resolutions.Comment: 32 pages, 4 figure

Methods for electronic-structure calculations - an overview from a reduced-density-matrix point of view

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, $N$-representability), and their determining equations (contracted Schr\"odinger 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.Comment: 27 pages, conference contributio

The self-energy of the uniform electron gas in the second order of exchange

The on-shell self-energy of the homogeneous electron gas in second order of exchange, $\Sigma_{2{\rm x}}= {\rm Re} \Sigma_{2{\rm x}}(k_{\rm F},k_{\rm F}^2/2)$, is given by a certain integral. This integral is treated here in a similar way as Onsager, Mittag, and Stephen [Ann. Physik (Leipzig) {\bf 18}, 71 (1966)] have obtained their famous analytical expression $e_{2{\rm x}}={1/6}\ln 2- 3\frac{\zeta(3)}{(2\pi)^2}$ (in atomic units) for the correlation energy in second order of exchange. Here it is shown that the result for the corresponding on-shell self-energy is $\Sigma_{2{\rm x}}=e_{2{\rm x}}$. The off-shell self-energy $\Sigma_{2{\rm x}}(k,\omega)$ correctly yields $2e_{2{\rm x}}$ (the potential component of $e_{2{\rm x}}$) through the Galitskii-Migdal formula. The quantities $e_{2{\rm x}}$ and $\Sigma_{2{\rm x}}$ appear in the high-density limit of the Hugenholtz-van Hove (Luttinger-Ward) theorem.Comment: 12 pages, 2 figure

The on-shell self-energy of the uniform electron gas in its weak-correlation limit

The ring-diagram partial summation (or RPA) for the ground-state energy of the uniform electron gas (with the density parameter $r_s$) in its weak-correlation limit $r_s\to 0$ is revisited. It is studied, which treatment of the self-energy $\Sigma(k,\omega)$ is in agreement with the Hugenholtz-van Hove (Luttinger-Ward) theorem $\mu-\mu_0= \Sigma(k_{\rm F},\mu)$ and which is not. The correlation part of the lhs h as the RPA asymptotics $a\ln r_s +a'+O(r_s)$ [in atomic units]. The use of renormalized RPA diagrams for the rhs yields the similar expression $a\ln r_s+a''+O(r_s)$ with the sum rule $a'= a''$ resulting from three sum rules for the components of $a'$ and $a''$. This includes in the second order of exchange the sum rule $\mu_{2{\rm x}}=\Sigma_{2{\rm x}}$ [P. Ziesche, Ann. Phys. (Leipzig), 2006].Comment: 19 pages, 10 figure

New sum rules relating the 1-body momentum distribution of the homogeneous electron gas to the Overhauser 2-body wave functions of its pair density

The recently derived sum rules for the scattering phase shifts of the Overhauser geminals (being 2-body-wave functions which parametrize the pair density together with an appropriately chosen occupancy) are generalized to integral equations which allow in principle to calculate the momentum distribution, supposed the phase sifts of the Overhauser geminals are known from an effective parity-dependent interaction potential (screened Coulomb repulsion).Comment: 10 page

New measure of electron correlation

We propose to quantify the "correlation" inherent in a many-electron (or many-fermion) wavefunction by comparing it to the unique uncorrelated state that has the same single-particle density operator as it does.Comment: Final version to appear in PR

Screened Exchange Corrections to the Random Phase Approximation from Many-Body Perturbation Theory

The random phase approximation (RPA) systematically overestimates the magnitude of the correlation energy and generally underestimates cohesive energies. This originates in part from the complete lack of exchange terms that would otherwise cancel Pauli exclusion principle violating (EPV) contributions. The uncanceled EPV contributions also manifest themselves in form of an unphysical negative pair density of spin parallel electrons close to electron-electron coalescence. We follow considerations of many-body perturbation theory to propose an exchange correction that corrects the largest set of EPV contributions, while having the lowest possible computational complexity. The proposed method exchanges adjacent particle/hole pairs in the RPA diagrams, considerably improving the pair density of spin-parallel electrons close to coalescence in the uniform electron gas (UEG). The accuracy of the correlation energy is comparable to other variants of second-order screened exchange (SOSEX) corrections although it is slightly more accurate for the spin-polarized UEG. Its computational complexity scales as O(N-5) or O(N-4) in orbital space or real space, respectively. Its memory requirement scales as O(N-2)

Quantum Monte Carlo Algorithm Based on Two-Body Density Functional Theory for Fermionic Many-Body Systems: Application to 3He

We construct a quantum Monte Carlo algorithm for interacting fermions using the two-body density as the fundamental quantity. The central idea is mapping the interacting fermionic system onto an auxiliary system of interacting bosons. The correction term is approximated using correlated wave functions for the interacting system, resulting in an effective potential that represents the nodal surface. We calculate the properties of 3He and find good agreement with experiment and with other theoretical work. In particular, our results for the total energy agree well with other calculations where the same approximations were implemented but the standard quantum Monte Carlo algorithm was usedComment: 4 pages, 3 figures, 1 tabl