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International Journal of Quantum Chemistry

Correlation measures as benchmarks in reduced density matrix functional theory

Nektarios N. Lathiotakis

Published Mar 27, 2012• 6 citations
Publisher

Abstract

Abstract In reduced density matrix functional theory, approximate total energy expressions in terms of the one‐body reduced density matrix (1RDM) are minimized under the appropriate N‐representability subsidiary conditions. The optimal 1RDM is in general nonidempotent. Measures of nonidempotency are the total charge of weakly occupied states and correlation entropy given in terms of the eigenvalues of the 1RDM. In this work, we calculate these quantities for several systems using different 1RDM functionals and compare with accurate results from a multideterminant method. Our goal is to assess 1RDM functionals with respect to the quality of the obtained 1RDM. © 2012 Wiley Periodicals, Inc.

DOI
10.1002/qua.24069

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Abstract In reduced density matrix functional theory, approximate total energy expressions in terms of the one‐body reduced density matrix (1RDM) are minimized under the appropriate N‐representability subsidiary conditions. The optimal 1RDM is in general nonidempotent. Measures of nonidempotency are the total charge of weakly occupied states and correlation entropy given in terms of the eigenvalues of the 1RDM. In this work, we calculate these quantities for several systems using different 1RDM functionals and compare with accurate results from a multideterminant method. Our goal is to assess 1RDM functionals with respect to the quality of the obtained 1RDM. © 2012 Wiley Periodicals, Inc.
Functionals of the one-body reduced density matrix (1-RDM) are routinely minimized under Coleman’s ensemble N-representability conditions. Recently, the topic of pure-state N-representability conditions, also known as generalized Pauli constraints, received increased attention following the discovery of a systematic way to derive them for any number of electrons and any finite dimensionality of the Hilbert space. The target of this work is to assess the potential impact of the enforcement of the pure-state conditions on the results of reduced density-matrix functional theory calculations. In particular, we examine whether the standard minimization of typical 1-RDM functionals under the ensemble N-representability conditions violates the pure-state conditions for prototype 3-electron systems. We also enforce the pure-state conditions, in addition to the ensemble ones, for the same systems and functionals and compare the correlation energies and optimal occupation numbers with those obtained by the enforcement of the ensemble conditions alone.
Functionals of the one-body reduced density matrix (1-RDM) are routinely minimized under Coleman's ensemble N-representability conditions. Recently, the topic of pure-state N-representability conditions, also known as generalized Pauli constraints, received increased attention following the discovery of a systematic way to derive them for any number of electrons and any finite dimensionality of the Hilbert space. The target of this work is to assess the potential impact of the enforcement of the pure-state conditions on the results of reduced density-matrix functional theory calculations. In particular, we examine whether the standard minimization of typical 1-RDM functionals under the ensemble N-representability conditions violates the pure-state conditions for prototype 3-electron systems. We also enforce the pure-state conditions, in addition to the ensemble ones, for the same systems and functionals and compare the correlation energies and optimal occupation numbers with those obtained by the enforcement of the ensemble conditions alone.
We develop a systematic approach to construct energy functionals of the one-particle reduced density matrix (1RDM) for equilibrium systems at finite temperature. The starting point of our formulation is the grand potential Ω [ G ] regarded as variational functional of the Green’s function G based on diagrammatic many-body perturbation theory and for which we consider either the Klein or Luttinger–Ward form. By restricting the input Green’s function to be one-to-one related to a set on one-particle reduced density matrices (1RDM) this functional becomes a functional of the 1RDM. To establish the one-to-one mapping we use that, at any finite temperature and for a given 1RDM γ in a finite basis, there exists a non-interacting system with a spatially non-local potential v [ γ ] which reproduces the given 1RDM. The corresponding set of non-interacting Green’s functions defines the variational domain of the functional Ω . In the zero temperature limit we obtain an energy functional E [ γ ] which by minimisation yields an approximate ground state 1RDM and energy. As an application of the formalism we use the Klein and Luttinger–Ward functionals in the GW-approximation to compute the binding curve of a model hydrogen molecule using an extended Hubbard Hamiltonian. We compare further to the case in which we evaluate the functionals on a Hartree–Fock and a Kohn–Sham Green’s function. We find that the Luttinger–Ward version of the functionals performs the best and is able to reproduce energies close to the GW energy which corresponds to the stationary point.
One of the major computational bottlenecks in one-body reduced density matrix (1RDM) functional theory is the evaluation of approximate 1RDM functionals and their derivatives. The reason is that more advanced approximate functionals are almost exclusively defined in the natural orbital basis, so a 4-index transformation of the two-electron integrals appears to be unavoidable. I will show that this is not the case and that so-called separable functionals can be evaluated much more efficiently, i.e. only at cubic cost in the basis size. Since most approximate functionals are actually separable, this new algorithm is an important development to make 1RDM functional theory calculations feasible for large electronic systems.
We present an ab initio approach for grand-canonical ensembles in thermal equilibrium (eq) with local or nonlocal external potentials based on the one-reduced density matrix (1RDM). We show that equilibrium properties of a grand-canonical ensemble are determined uniquely by the eq-1RDM and establish a variational principle for the grand potential with respect to its 1RDM. We further prove the existence of a Kohn-Sham system capable of reproducing the 1RDM of an interacting system at finite temperature. Utilizing this Kohn-Sham system as an unperturbed system, we deduce a many-body approach to iteratively construct approximations to the correlation contribution of the grand potential.
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.