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arXiv:2210.14475 · Chinese Journal of Chemical Physics

Copula approach to exchange-correlation hole in many-electron systems with strong correlations

JingChun Wang, Su Yu, Haoyang Cheng, Yao Wang, Rui‐Xue Xu

Published Feb 21, 2023• 3 citations
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Abstract

Electronic correlation is a fundamental topic in many-electron systems. To characterize this correlation, one may introduce the concept of exchange-correlation hole. In this work, we first briefly revisit its definition and relation to electron and geminal densities, followed by their intimate relations to copula functions in probability theory and statistics. We then propose a copula-based approach to estimate the exchange-correlation hole from the electron density. It is anticipated that the proposed scheme will become a promising ingredient towards the future development of strongly correlated electronic structure calculations.

DOI
10.1063/1674-0068/cjcp2210154

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Electronic correlation is a fundamental topic in many-electron systems. To characterize this correlation, one may introduce the concept of exchange-correlation hole. In this work, we first briefly revisit its definition and relation to electron and geminal densities, followed by their intimate relations to copula functions in probability theory and statistics. We then propose a copula-based approach to estimate the exchange-correlation hole from the electron density. It is anticipated that the proposed scheme will become a promising ingredient towards the future development of strongly correlated electronic structure calculations.
The one-particle Green function of a many-electron system is traditionally formulated within the self-energy picture. A different formalism was recently proposed, in which the self-energy is replaced by a dynamical exchange-correlation field, which acts on the Green function locally in both space and time. It was found that there exists a fundamental quantity, referred to as the dynamical exchange-correlation hole, which can be interpreted as effective density fluctuations induced in a many-electron system when a hole or an electron is introduced into the system, as in photoemission and inverse photoemission experiments. The dynamical exchange-correlation potential is simply the Coulomb potential of this exchange-correlation hole, which fulfils a sum rule and an exact constraint, identical to those satisfied by the static exchange-correlation hole in density-functional theory. The proposed formalism has been applied to a number of model systems such as the half-filled one-dimensional Hubbard model, the one-dimensional antiferromagnetic Heisenberg model, and the single-impurity Anderson model. The dynamical exchange-correlation hole and field of the homogeneous electron gas have also been studied with the view of constructing a density-functional approximation such as the local-density approximation. The availability of simple but accurate approximations for the exchange-correlation potential would circumvent costly computations of the traditional self-energy. The formalism may also provide new perspectives and insights into the many-body problem.
The Hubbard I decoupling approximation [J. Hubbard, Proc. R. Soc. London Ser. A 276, 238 (1963)] is extended by introducing a k-dependent self-energy which describes a mobile correlation hole of opposite spin propagating along with the electron in its way around the lattice. The theory, despite its simplicity, (1) becomes exact in the strong-coupling limit; and (2) reproduces with great accuracy the ground-state energy, the double occupancy, the on-site correlation functions, and the effective hopping for all the values of the Hubbard Coulomb repulsion U. It also gives reasonable spectral functions leading to (3) a single-particle Green function showing the well-known spectral-weight transfer between the Hubbard bands as soon as the electron density deviates from half-filling, and (4) a momentum distribution as well as magnetic and charge structure factors in qualitative agreement with quantum Monte Carlo simulations, slave-boson calculations, and conserving approximations like fluctuation exchange and parquet. On the other hand, as in most decoupling schemes, liftime effects and the near-Fermi-level additional structure characteristic of strongly correlated electron systems are missing.
We present exchange-correlation energy densities exc,{e}_{\mathrm{xc}},exc​, total energies Exc,{E}_{\mathrm{xc}},Exc​, and holes, calculated for strongly inhomogeneous electron gases using the nonlocal weighted density approximation. The results closely resemble variational Monte Carlo simulations performed recently [Phys. Rev. Lett. 87, 036401 (2001)], demonstrating the effectiveness of a nonlocal density functional description.
We propose a new approach toward approximating the density-to-pair-density map based on copula theory from statistics. We extend the copula theory to multi-dimensional marginals and deduce that one can describe any (exact or approximate) pair density by the single-particle density and a copula. We present analytical formulas for the exact copula in scaling limits, numerically compute the copula for dissociating systems with two to four particles in one dimension, and propose accurate approximations of the copula between equilibrium and dissociation for two-particle systems.
The discovery of high-Tc superconductivity in cuprates in 1986 moved strongly correlated systems from exotic worlds interesting only for pure theorists to the focus of solid-state research. In recent decades, the majority of hot topics in condensed matter physics (high-Tc superconductivity, colossal magnetoresistance, multiferroicity, ferromagnetism in diluted magnetic semiconductors, etc.) have been related to strongly correlated transition metal compounds. The highly successful electronic structure calculations based on density functional theory lose their predictive power when applied to such compounds. It is necessary to go beyond the mean field approximation and use the many-body theory. The methods and models that were developed for the description of strongly correlated systems are reviewed together with the examples of response function calculations that are needed for the interpretation of experimental information (inelastic neutron scattering, optical conductivity, resonant inelastic X-ray scattering, electron energy loss spectroscopy, angle-resolved photoemission, electron spin resonance, and magnetic and magnetoelectric properties). The peculiarities of (quasi-) 0-, 1-, 2-, and 3- dimensional systems are discussed.
We use the exact strong-interaction limit of the Hohenberg-Kohn energy density functional to approximate the exchange-correlation energy of the restricted Kohn-Sham scheme. Our approximation corresponds to a highly nonlocal density functional whose functional derivative can be easily constructed, thus transforming exactly, in a physically transparent way, an important part of the electron-electron interaction into an effective local one-body potential. We test our approach on quasi-one-dimensional systems, showing that it captures essential features of strong correlation that restricted Kohn-Sham calculations using the currently available approximations cannot describe.