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The role of exchange for the spin-parallel structure factor of jellium

P. Ziesche, Felix Hummel

Published May 27, 2016• 0 citations
OpenAlex

Abstract

The random phase approximation (RPA) provides a qualitatively good approximation of electronic correlation in many systems. It has been successfully applied to the spin-unpolarized homogeneous electron gas (HEG) for decades, especially in the studies of its many properties such as correlation energy, momentum distribution and the spin resolved pair correlation function and structure factor. However, the RPA neglects important contributions to the spin-parallel structure factor originating from exchange effects. We derive an approximation for the spin-parallel structure factor including the lowest order exchange process beyond the RPA analytically by means of contour integration and evaluate it numerically by means of a Monte-Carlo integration. A generalized Hellmann-Feynman theorem with functional derivatives is used.

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How screened exchange improves on violations of the Pauli principle in the random phase approximation of jellium

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Within many-body perturbation theory (MBPT) of the spin-unpolarized jellium the second order exchange-term is explicitly renormalized using the Random Phase Approximation (RPA). Regarding total energies this has been studied under the name Second Order Screened Exchange (SOSEX). We study the pair correlation function (PCF) and the structure factor for parallel spins to investigate to what extend the exchange terms correct for violations of the Pauli principle introduced by the RPA. The PCF is directly evaluated for the thermodynamic limit by means of a Monte-Carlo integration of the imaginary frequency propagators of the homogeneous electron gas.

arXiv:1605.08643?context=cond-mat.str-el•May 27, 2016•0 citations

Convergence behavior of the random phase approximation renormalized correlation energy

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Based on the random phase approximation (RPA), RPA renormalization [J. E. Bates and F. Furche, J. Chem. Phys. 139, 171103 (2013)] is a robust many-body perturbation theory that works for molecules and materials because it does not diverge as the Kohn-Sham gap approaches zero. Additionally, RPA renormalization enables the simultaneous calculation of RPA and beyond-RPA correlation energies since the total correlation energy is the sum of a series of independent contributions. The first-order approximation (RPAr1) yields the dominant beyond-RPA contribution to the correlation energy for a given exchange-correlation kernel, but systematically underestimates the total beyond-RPA correction. For both the homogeneous electron gas model and real systems, we demonstrate numerically that RPA renormalization beyond first order converges monotonically to the infinite-order beyond-RPA correlation energy for several model exchange-correlation kernels and that the rate of convergence is principally determined by the choice of the kernel and spin polarization of the ground state. The monotonic convergence is rationalized from an analysis of the RPA renormalized correlation energy corrections, assuming the exchange-correlation kernel and response functions satisfy some reasonable conditions. For spin-unpolarized atoms, molecules, and bulk solids, we find that RPA renormalization is typically converged to 1 meV error or less by fourth order regardless of the band gap or dimensionality. Most spin-polarized systems converge at a slightly slower rate, with errors on the order of 10 meV at fourth order and typically requiring up to sixth order to reach 1 meV error or less. Slowest to converge, however, open-shell atoms present the most challenging case and require many higher orders to converge.

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Inhomogeneous random-phase approximation and many-electron trial wave functions

R. Gaudoin, Maziar Nekovee, W. M. C. Foulkes +2 more

The long-range electronic correlations in a uniform electron gas may be deduced from the random-phase approximation (RPA) of Bohm and Pines [Phys. Rev. 92, 609 (1953)]. Here we generalize the RPA to nonuniform systems and use it to derive many-electron Slater-Jastrow trial wave functions for quantum Monte Carlo simulations. The RPA theory fixes the long-range behavior of the inhomogeneous two-body terms in the Jastrow factor and provides an accurate analytic expression for the one-body terms. It also explains the success of Slater-Jastrow trial functions containing determinants of Hartree-Fock or density-functional orbitals, even though these theories do not include Jastrow factors. After adjusting the RPA Jastrow factor to incorporate the known short-range behavior, we test it using variational Monte Carlo simulations. In the small inhomogeneous electron gas system we consider, the analytic RPA-based Jastrow factor slightly outperforms the standard numerically optimized form. The inhomogeneous RPA theory therefore enables us to reduce or even avoid the costly numerical optimization process.

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F. Bechstedt

The correlation energy of an inhomogeneous electron gas is ruled by the density correlation function $L$. It obeys a Bethe-Salpeter equation (BSE), whose kernel is the functional derivative of the particle self-energy. It is solved on equal footing for two-particle and single-particle Green's functions within Hedin's GW approximation. According to the Hartree and exchange-correlation contributions to the kernel, the correlation expression decays into random-phase-approximation (RPA) -like and screened-exchange-like parts. The actual treatment of the screened potential $W$ in the BSE for $L$ leads to different second-order screened exchange corrections (SOSEX). Including correlation in the kernel even yields approximations beyond SOSEX. The progress beyond RPA is illustrated for spinless and spin-polarized homogeneous electron gases, indicating that the screening treatment is substantial.

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Francesco Pederiva, E. Lipparini, K Takayanagi

We have evaluated the density-density response of the two-dimensional electron gas at zero temperature by solving the Dyson equation for the particle-hole Green's function, including exchange Coulomb matrix elements and short-range contributions in the ladder approximation. We study the effect of these correlations on the total energy, compressibility per particle, local field factor G ( q ), static structure factor and pair-correlation function. Results are compared with the normal random-phase approximation, local field theories and quantum Monte Carlo calculations.

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

P. Ziesche

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.

arXiv:1108.1912•Aug 9, 2011•0 citations
The random phase approximation (RPA) provides a qualitatively good approximation of electronic correlation in many systems. It has been successfully applied to the spin-unpolarized homogeneous electron gas (HEG) for decades, especially in the studies of its many properties such as correlation energy, momentum distribution and the spin resolved pair correlation function and structure factor. However, the RPA neglects important contributions to the spin-parallel structure factor originating from exchange effects. We derive an approximation for the spin-parallel structure factor including the lowest order exchange process beyond the RPA analytically by means of contour integration and evaluate it numerically by means of a Monte-Carlo integration. A generalized Hellmann-Feynman theorem with functional derivatives is used.
Within many-body perturbation theory (MBPT) of the spin-unpolarized jellium the second order exchange-term is explicitly renormalized using the Random Phase Approximation (RPA). Regarding total energies this has been studied under the name Second Order Screened Exchange (SOSEX). We study the pair correlation function (PCF) and the structure factor for parallel spins to investigate to what extend the exchange terms correct for violations of the Pauli principle introduced by the RPA. The PCF is directly evaluated for the thermodynamic limit by means of a Monte-Carlo integration of the imaginary frequency propagators of the homogeneous electron gas.
Based on the random phase approximation (RPA), RPA renormalization [J. E. Bates and F. Furche, J. Chem. Phys. 139, 171103 (2013)] is a robust many-body perturbation theory that works for molecules and materials because it does not diverge as the Kohn-Sham gap approaches zero. Additionally, RPA renormalization enables the simultaneous calculation of RPA and beyond-RPA correlation energies since the total correlation energy is the sum of a series of independent contributions. The first-order approximation (RPAr1) yields the dominant beyond-RPA contribution to the correlation energy for a given exchange-correlation kernel, but systematically underestimates the total beyond-RPA correction. For both the homogeneous electron gas model and real systems, we demonstrate numerically that RPA renormalization beyond first order converges monotonically to the infinite-order beyond-RPA correlation energy for several model exchange-correlation kernels and that the rate of convergence is principally determined by the choice of the kernel and spin polarization of the ground state. The monotonic convergence is rationalized from an analysis of the RPA renormalized correlation energy corrections, assuming the exchange-correlation kernel and response functions satisfy some reasonable conditions. For spin-unpolarized atoms, molecules, and bulk solids, we find that RPA renormalization is typically converged to 1 meV error or less by fourth order regardless of the band gap or dimensionality. Most spin-polarized systems converge at a slightly slower rate, with errors on the order of 10 meV at fourth order and typically requiring up to sixth order to reach 1 meV error or less. Slowest to converge, however, open-shell atoms present the most challenging case and require many higher orders to converge.
The long-range electronic correlations in a uniform electron gas may be deduced from the random-phase approximation (RPA) of Bohm and Pines [Phys. Rev. 92, 609 (1953)]. Here we generalize the RPA to nonuniform systems and use it to derive many-electron Slater-Jastrow trial wave functions for quantum Monte Carlo simulations. The RPA theory fixes the long-range behavior of the inhomogeneous two-body terms in the Jastrow factor and provides an accurate analytic expression for the one-body terms. It also explains the success of Slater-Jastrow trial functions containing determinants of Hartree-Fock or density-functional orbitals, even though these theories do not include Jastrow factors. After adjusting the RPA Jastrow factor to incorporate the known short-range behavior, we test it using variational Monte Carlo simulations. In the small inhomogeneous electron gas system we consider, the analytic RPA-based Jastrow factor slightly outperforms the standard numerically optimized form. The inhomogeneous RPA theory therefore enables us to reduce or even avoid the costly numerical optimization process.
The correlation energy of an inhomogeneous electron gas is ruled by the density correlation function LLL. It obeys a Bethe-Salpeter equation (BSE), whose kernel is the functional derivative of the particle self-energy. It is solved on equal footing for two-particle and single-particle Green's functions within Hedin's GW approximation. According to the Hartree and exchange-correlation contributions to the kernel, the correlation expression decays into random-phase-approximation (RPA) -like and screened-exchange-like parts. The actual treatment of the screened potential WWW in the BSE for LLL leads to different second-order screened exchange corrections (SOSEX). Including correlation in the kernel even yields approximations beyond SOSEX. The progress beyond RPA is illustrated for spinless and spin-polarized homogeneous electron gases, indicating that the screening treatment is substantial.
We have evaluated the density-density response of the two-dimensional electron gas at zero temperature by solving the Dyson equation for the particle-hole Green's function, including exchange Coulomb matrix elements and short-range contributions in the ladder approximation. We study the effect of these correlations on the total energy, compressibility per particle, local field factor G ( q ), static structure factor and pair-correlation function. Results are compared with the normal random-phase approximation, local field theories and quantum Monte Carlo calculations.
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.