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arXiv:1901.06942 · Journal of Physics Condensed Matter

Natural orbital functional for spin-polarized periodic systems

Raúl Quintero-Monsebaiz, Ion Mitxelena, Mauricio Rodríguez‐Mayorga, Alberto Vela, Mario Piris

Published Jan 23, 2019• 8 citations
arXivHTMLPDF

Abstract

Natural orbital functional theory is considered for systems with one or more unpaired electrons. An extension of the Piris natural orbital functional (PNOF) based on electron pairing approach is presented, specifically, we extend the independent pair model, PNOF5, and the interactive pair model PNOF7 to describe spin-uncompensated systems. An explicit form for the two-electron cumulant of high-spin cases is only taken into account, so that singly occupied orbitals with the same spin are solely considered. The rest of the electron pairs with opposite spins remain paired. The reconstructed two-particle reduced density matrix fulfills certain N-representability necessary conditions, as well as guarantees the conservation of the total spin. The theory is applied to model systems with strong non-dynamic (static) electron correlation, namely, the one-dimensional Hubbard model with periodic boundary conditions and hydrogen rings. For the latter, PNOF7 compares well with exact diagonalization results so the model presented here is able to provide a correct description of the strong-correlation effects.

DOI
10.1088/1361-648x/ab0170

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Natural orbital functional theory is considered for systems with one or more unpaired electrons. An extension of the Piris natural orbital functional (PNOF) based on electron pairing approach is presented, specifically, we extend the independent pair model, PNOF5, and the interactive pair model PNOF7 to describe spin-uncompensated systems. An explicit form for the two-electron cumulant of high-spin cases is only taken into account, so that singly occupied orbitals with the same spin are solely considered. The rest of the electron pairs with opposite spins remain paired. The reconstructed two-particle reduced density matrix fulfills certain N-representability necessary conditions, as well as guarantees the conservation of the total spin. The theory is applied to model systems with strong non-dynamic (static) electron correlation, namely, the one-dimensional Hubbard model with periodic boundary conditions and hydrogen rings. For the latter, PNOF7 compares well with exact diagonalization results so the model presented here is able to provide a correct description of the strong-correlation effects.
The one-particle reduced density matrix functional theory in its natural orbital functional (NOF) version is used to study strongly correlated electrons. We show the ability of the Piris NOF 7 (PNOF7) to describe non-dynamic correlation effects in one-dimensional (1D) systems. An extensive study of 1D systems that includes Hydrogen (H) chains and the 1D Hubbard model with periodic boundary conditions is provided. Different filling situations and large sizes with up to 122 electrons are considered. Compared to quasi-exact results, PNOF7 is accurate in different correlation regimes for the 1D Hubbard model even away from the half-filling, and maintains its accuracy when the system size increases. The symmetric and asymmetric dissociations of the linear H chain composed of 50 atoms are described to remark the importance of long-range interactions in presence of strong correlation effects. Our results compare remarkably well with those obtained at the density-matrix renormalization group level of theory.
Strongly correlated materials are now under intense development, and natural orbital functional (NOF) methods seem to be able to capture the physics of these systems. We present a benchmark based on the Hubbard model for a class of commonly used NOF approximations (also known as reduced density matrix functional approximations). Our findings highlight the importance of imposing ensemble N-representability conditions in order to obtain consistent results in systems with either weak or strong electronic correlation, such as the Hubbard system with a varying two-particle interaction parameter. Based on the accuracy of the results obtained using PNOF7, which retrieves a large amount of the total strong nondynamic correlation, the Hubbard model points out that N-representability gives solid foundations for NOF development.
This work deals with the problem of strongly correlated electrons in two-dimensions. We give a reduced density matrix (RDM) based tool through which the ground-state energy is given as a functional of the natural orbitals and their occupation numbers. Specifically, the Piris Natural Orbital Functional 7 (PNOF7) is used for studying the 2D Hubbard model and hydrogen square lattices. The singlet ground-state is studied, as well as the doublet mixed quantum state obtained by extracting an electron from the system. Our method satisfies two-index necessary N-representability conditions of the two-particle RDM (2RDM) and guarantees the conservation of the total spin. We show the ability of PNOF7 to describe strong correlation effects in two-dimensional (2D) systems by comparing our results with the exact diagonalization, density matrix renormalization group (DMRG), and auxiliary-field quantum Monte Carlo calculations. PNOF7 overcomes variational 2RDM methods with two- and three-index positivity N-representability conditions, reducing computational cost to mean-field scaling. Consistent results are obtained for small and large systems up to 144 electrons, weak and strong correlation regimes, and many filling situations. Unlike other methods, there is no dependence on dimensionality in the results obtained with PNOF7 and no particular difficulties have been observed to converge PNOF7 away from half-filling. Smooth double occupancy of sites is obtained, regardless of the filling. Symmetric dissociation of 2D hydrogen lattices shows that long-range nondynamic correlation dramatically affects electron detachment energies. PNOF7 compares well with DMRG along the dissociation curve.
The natural orbital functional (NOF) theory is briefly reviewed. The meaning of the top‐down and bottom‐up approaches for the construction of a NOF is analyzed. A particular reconstruction of the two‐particle reduced density matrix (2‐RDM) based on the cumulant expansion is discussed. The cumulant is expressed by two auxiliary matrices, which are constrained to certain bounds due to the N‐representabilty conditions of the 2‐RDM. Appropriate forms of these matrices lead to different implementations known in the literature as PNOFi (i = 1–5). The strengths and weaknesses of PNOF5 are assessed. Its main strength is its ability to deal with the intrapair electron correlation at a reasonable computational cost. Its main limitation is the absence of the interpair electron correlation. The inclusion of the missing correlation via a multiconfigurational perturbation theory is shortly described. The growing interest in methods based on NOF theory points to a promising future in this field. © 2014 Wiley Periodicals, Inc.
Accurate modeling of bond breaking remains a central challenge for reduced density matrix functional theory (RDMFT). Although some modern functionals can yield reasonably accurate dissociation energies, they often fail to reproduce key properties of the dissociated fragments, such as a vanishing fragment population covariance (also known as the delocalization index) and the correct total spin angular momentum of each fragment (local spin). In this work, we revisit the construction of natural orbital functionals by correcting the cumulant contribution produced by the PNOF5 functional. Our method enforces known contributions of the cumulant to local spin fragments and the delocalization index at the dissociation limit. We obtain the closest cumulant consistent with these physically motivated constraints and subsequently purify the corresponding one- and two-electron reduced density matrices by imposing the standard P,Q,and GP, Q, \text{and } GP,Q,and G NNN-representability conditions. The resulting functional yields improved behavior in strongly correlated regimes. Benchmarking on the dissociation of the singlet states of \ce{N2}, \ce{NO+}, \ce{O2}, \ce{S2}, and \ce{CO} shows that in the dissociation regime the energies computed from the updated cumulant exactly reproduce the complete active space self-consistent field (CASSCF) energies. We further analyze the limitations of the approach and identify scenarios in which the current approach performs poorly. This work provides a pathway for systematically improving natural orbital functionals to achieve reliable bond-breaking calculations within RDMFT.
Wave functions constructed from electron-pair states can accurately model strong electron correlation effects and are promising approaches especially for larger many-body systems. In this article, we analyze the nature and the type of electron correlation effects that can be captured by wave functions restricted to electron-pair states. We focus on the pair-coupled-cluster doubles (pCCD) ansatz also called the antisymmetric product of the 1-reference orbital geminal (AP1roG) method, combined with an orbital optimization protocol presented in Boguslawski et al. [Phys. Rev. B 89, 201106(R) (2014)], whose performance is assessed against electronic structures obtained form density-matrix renormalization-group reference data. Our numerical analysis covers model systems for strong correlation: the one-dimensional Hubbard model with a periodic boundary condition as well as metallic and molecular hydrogen rings. Specifically, the accuracy of pCCD/AP1roG is benchmarked using the single-orbital entropy, the orbital-pair mutual information, as well as the eigenvalue spectrum of the one-orbital and two-orbital reduced density matrices. Our study indicates that contributions from singly occupied states become important in the strong correlation regime which highlights the limitations of the pCCD/AP1roG method. Furthermore, we examine the effect of orbital rotations within the pCCD/AP1roG model on correlations between orbital pairs.