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arXiv:2202.04946 · Physical review. B./Physical review. B

Introducing screening in one-body density matrix functionals: Impact on charged excitations of model systems via the extended Koopmans' theorem

S. Di Sabatino, Jaakko Koskelo, J. A. Berger, Pina Romaniello

Published Jun 16, 2022• 19 citations
arXivHTMLPDF

Abstract

In this work we get insight into the impact of reduced density matrix functionals on the quality of removal/addition energies obtained using the extended Koopmans' theorem (EKT). Within the reduced density matrix functional theory (RDMFT) the EKT approach reduces to a matrix diagonalization, whose ingredients are the one- and two-body reduced density matrices. A striking feature of the EKT within RDMFT is that it opens a band gap, although it is too large, in strongly correlated materials, which are a challenge for state-of-the-art methods such as $GW$. Using the one-dimensional Hubbard model and the homogeneous electron gas as test cases, we find that (i) with exact or very accurate density matrices the EKT systematically overestimates the band gap in the Hubbard model and the bandwidth in the homogeneous electron gas and (ii) with approximate density matrices, instead, the EKT can benefit from error cancellation. In particular we test a new approximation which combines random-phase approximation screening with the power functional approximation to the two-body reduced density matrix introduced by Sharma et al. [Phys. Rev. B 78, 201103 (2008)]. An important feature of this approximation is that it reduces the EKT band gap in the studied models; it is hence a promising approximation for correcting the EKT band-gap overestimation in strongly correlated materials.

DOI
10.1103/physrevb.105.235123

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In this work we get insight into the impact of reduced density matrix functionals on the quality of removal/addition energies obtained using the extended Koopmans' theorem (EKT). Within the reduced density matrix functional theory (RDMFT) the EKT approach reduces to a matrix diagonalization, whose ingredients are the one- and two-body reduced density matrices. A striking feature of the EKT within RDMFT is that it opens a band gap, although it is too large, in strongly correlated materials, which are a challenge for state-of-the-art methods such as GWGWGW. Using the one-dimensional Hubbard model and the homogeneous electron gas as test cases, we find that (i) with exact or very accurate density matrices the EKT systematically overestimates the band gap in the Hubbard model and the bandwidth in the homogeneous electron gas and (ii) with approximate density matrices, instead, the EKT can benefit from error cancellation. In particular we test a new approximation which combines random-phase approximation screening with the power functional approximation to the two-body reduced density matrix introduced by Sharma et al. [Phys. Rev. B 78, 201103 (2008)]. An important feature of this approximation is that it reduces the EKT band gap in the studied models; it is hence a promising approximation for correcting the EKT band-gap overestimation in strongly correlated materials.
We explore the extended Koopmans’ theorem (EKT) within the phaseless auxiliary-field quantum Monte Carlo (AFQMC) method. The EKT allows for the direct calculation of electron addition and removal spectral functions using reduced density matrices of the N -particle system and avoids the need for analytic continuation. The lowest level of EKT with AFQMC, called EKT1-AFQMC, is benchmarked using atoms, small molecules, 14-electron and 54-electron uniform electron gas supercells, and a minimal unit cell model of diamond at the Γ-point. Via comparison with numerically exact results (when possible) and coupled-cluster methods, we find that EKT1-AFQMC can reproduce the qualitative features of spectral functions for Koopmans-like charge excitations with errors in peak locations of less than 0.25 eV in a finite basis. We also note the numerical difficulties that arise in the EKT1-AFQMC eigenvalue problem, especially when back-propagated quantities are very noisy. We show how a systematic higher-order EKT approach can correct errors in EKT1-based theories with respect to the satellite region of the spectral function. Our work will be of use for the study of low-energy charge excitations and spectral functions in correlated molecules and solids where AFQMC can be reliably performed for both energy and back propagation.
In this work, we explore the performance of approximations to electron correlation in reduced density-matrix functional theory (RDMFT) and of approximations to the observables calculated within this theory. Our analysis focuses on the calculation of total energies, occupation numbers, removal/addition energies, and spectral functions. We use the exactly solvable Hubbard dimer at 1/4 and 1/2 fillings as test systems. This allows us to analyze the underlying physics and to elucidate the origin of the observed trends. For comparison, we also report the results of the GW approximation, where the self-energy functional is approximated, but no further hypothesis is made concerning the approximations of the observables. In particular, we focus on the atomic limit, where the two sites of the dimer are pulled apart and electrons localize on either site with equal probability, unless a small perturbation is present: this is the regime of strong electron correlation. In this limit, using the Hubbard dimer at 1/2 filling with or without a spin-symmetry-broken ground state allows us to explore how degeneracies and spin-symmetry breaking are treated in RDMFT. We find that, within the used approximations, neither in RDMFT nor in GW, the signature of strong correlation is present, when looking at the removal/addition energies and spectral function from the spin-singlet ground state, whereas both give the exact result for the spin-symmetry broken case. Moreover, we show how the spectroscopic properties change from one spin structure to the other.
Abstract Strongly correlated systems pose a challenge for theoretical methods based on an independent electron approximation. Such methods struggle to predict a nonzero gap in Mott insulators or to capture the correct physics of the insulator-to-metal phase transition in strongly correlated materials. In a recent paper by Shinohara et al (2015 New J. Phys. 17 093038 ) it is shown that strongly correlated materials and correct descriptions of their phase transitions are within the reach of reduced density matrix functional theory (RDMFT) approximations. For a doping-induced phase transition, not only is a satisfactory agreement with experimental spectra found for NiO but it is also shown that the physical picture of the observed Mott transition stays in line with more computationally demanding many-body theories. This is an important step toward providing an RDMFT–based computation tool for studying strongly correlated materials.
From drug design to material science, computational chemistry has become a valuable tool for many fields of applications in the past decades. Quantum chemistry in particular, can provide accurate descriptions of molecules and solids by relying on principles of quantum mechanics with the Schrödinger equation at its core. Unfortunately this equation becomes prohibitively expensive to solve for systems made of more than a few electrons and one has to rely on other approaches. Among the most popular is Density Functional Theory, relatively inexpensive and exact in principle. One has, however, to introduce approximations in practice, deteriorating its accuracy for challenging, so-called strongly correlated, systems. On the contrary, other methods like CASSCF remain reliable for these systems, but although cheaper than the solution of the Schrödinger equation, remain too expensive for extended systems. There is therefore room for methods in between, that could handle large correlated systems with reasonable accuracy. One such method is the One Body Reduced Density Matrix (1-RDM) Functional Theory (RDMFT), based on the fact that an electronic system can, in principle exactly, be described by its 1-RDM only. Although its theory dates back to the 1970s, computational hindrances have prevented this method from shining so far. Indeed, computations in RDMFT rely on a constrained optimization of the eigenvalues and eigenvectors of the 1-RDM, that can be reformulated as an unconstrained optimization by parameterizing the latter but in both cases suffers from convergence problems. In this thesis, the role of this parametrization is analyzed and motivates a now second order optimization algorithm, which improves the convergence speed by two orders of magnitude w.r.t. the state of the art algorithm. This understanding of the problem also inspires an alternative approach with the 1-RDM as the optimization variable. This however requires the development of new approximations and a first investigation in that direction, based on Machine Learning is performed. Lastly, a generalization of the method to excited states is attempted, highlighting some key computational challenges.
Abstract Using the discontinuity of the chemical potential as a function of excess charge, the fundamental gaps for finite systems and the band gaps of extended solids are determined within reduced density matrix functional theory. We also present the necessary and sufficient conditions for the one-body reduced density matrix of a system with fractional charge to be ensemble N-representable. The performance of most modern day reduced density matrix functionals is assessed for the gaps and the correlation energy of finite systems. Our results show that for finite systems the PNOF, BBC3, and power functionals yield very accurate correlation energies while for a correct description of the fundamental gap the removal of self-interaction terms is essential. For extended solids we find that the power functional captures the correct band gap behavior for conventional semiconductors as well as strongly correlated Mott insulators, where a gap is obtained in absence of any magnetic ordering.
Three strict constraints upon the electron-electron repulsion energy functional of the one-electron reduced density matrix (the 1-matrix) are obtained by combining its invariance and stationary properties with the extended Koopmans' theorem. The constraints relate the quantities derived from the functional pertaining to an N-electron system with those of its (N-1)-electron congener. Together with the N-representability requirement for the 1-matrix of the congener, identities involving the electron-electron repulsion energies of the two systems and their derivatives with respect to the 1-matrices seriously narrow down the choices for potential approximate density-matrix functionals. This fact is well illustrated in the case of two-electron systems, where the validity of the new constraints is confirmed and found to originate from a nontrivial cancellation among different terms. Thus, the constraints provide a new tool for the construction and testing of new functionals that complements the previously known conditions such as the reproduction of the homogeneous gas energies and momentum distributions, convexity, and the N-representability of the associated 2-matrices.