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The Journal of Chemical Physics

Size consistency of explicit functionals of the natural orbitals in reduced density matrix functional theory

Nektarios N. Lathiotakis, Nikitas I. Gidopoulos, N. Helbig

Published Feb 23, 2010• 30 citations
Publisher

Abstract

We report a size-inconsistency problem for several functionals within reduced density matrix functional theory. Being explicit functionals of the natural orbitals and occupation numbers, instead of the one-body reduced density matrix, many of the approximate functionals are not invariant under unitary transformations in the subspace of degenerate occupation numbers. One such transformation mixes the degenerate natural orbitals of identical independent subsystems, delocalizing them. Noninvariance under this transformation results in size inconsistency for some of the approximations while others avoid this pathology by favoring orbital localization.

DOI
10.1063/1.3324699

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We report a size-inconsistency problem for several functionals within reduced density matrix functional theory. Being explicit functionals of the natural orbitals and occupation numbers, instead of the one-body reduced density matrix, many of the approximate functionals are not invariant under unitary transformations in the subspace of degenerate occupation numbers. One such transformation mixes the degenerate natural orbitals of identical independent subsystems, delocalizing them. Noninvariance under this transformation results in size inconsistency for some of the approximations while others avoid this pathology by favoring orbital localization.
In many of the approximate functionals in one-body reduced-density-matrix (1RDM) functional theory, the approximate two-body reduced density matrix (2RDM) in the natural orbital representation only depends on the natural occupation numbers. In Phys. Rev. A 92, 012520 (2015), Wang and Knowles initialized a discussion of to what extent this simplification is valid by introducing two different H4{\mathrm{H}}_{4}H4​ geometries with identical natural occupation numbers but different 2RDMs. Gritsenko has argued that this feature is due symmetry [Phys. Rev. A 97, 026501 (2018)]. This work aims to contribute to the discussion on the following points: (1) one should rather speak of symmetry-restricted variants of the universal functional than saying that the universal functional is symmetry dependent; (2) the unitary invariance of degenerate NOs can lead to large deviations in the 2RDM elements, especially the phase of the NOs; (3) symmetry-restricted functionals are constructed for the H4{\mathrm{H}}_{4}H4​ geometries considered by Wang and Knowles, whose structure could serve as guide in the construction of approximate 1RDM functionals.
In many of the approximate functionals in one-body reduced density matrix (1RDM) functional theory, the approximate two-body reduced density matrix (2RDM) in the natural orbital representation only depends on the natural occupation numbers. In Phys. Rev. A 92, 012520 (2015) Wang and Knowles initialised the discussion to which extend this simplification is valid, by introducing two different H4_44​ geometries with identical natural occupation numbers, but different 2RDMs. Gritsenko has argued that this feature is due symmetry dependence of the exact functional [Phys. Rev. A 97, 026501 (2018)]. This work aims to contribute to the discussion on the following points: 1) the feature that the exact functional can yield different 2RDMs for the same set of natural occupations can be explained without symmetry, so is more general; 2) the exact functional is not symmetry dependent, though it is possible to define symmetry-restricted variants of the exact functionals; 3) symmetry-restricted functionals are derived for the H4_44​ geometries considered by Wang and Knowles, which can serve as guide in the construction of approximate 1RDM functionals.
Recently, an approximate theoretical framework was introduced, called local reduced density matrix functional theory (local-RDMFT), where functionals of the one-body reduced density matrix (1-RDM) are minimized under the additional condition that the optimal orbitals satisfy a single electron Schrödinger equation with a local potential. In the present work, we focus on the character of these optimal orbitals. In particular, we compare orbitals obtained by local-RDMFT with those obtained with the full minimization (without the extra condition) by contrasting them against the exact NOs and orbitals from a density functional calculation using the local density approximation (LDA). We find that the orbitals from local-RMDFT are very close to LDA orbitals, contrary to those of the full minimization that resemble the exact NOs. Since local RDMFT preserves the good quality of the description of strong static correlation, this finding opens the way to a mixed density/density matrix scheme, where Kohn-Sham orbitals obtain fractional occupations from a minimization of the occupation numbers using 1-RDM functionals. This will allow for a description of strong correlation at a cost only minimally higher than a density functional calculation.
Recently, an approximate theoretical framework was introduced, called local reduced density matrix functional theory (local-RDMFT), where functionals of the one-body reduced density matrix (1-RDM) are minimized under the additional condition that the optimal orbitals satisfy a single electron Schrödinger equation with a local potential. In the present work, we focus on the character of these optimal orbitals. In particular, we compare orbitals obtained by local-RDMFT with those obtained with the full minimization (without the extra condition) by contrasting them against the exact NOs and orbitals from a density functional calculation using the local density approximation (LDA). We find that the orbitals from local-RMDFT are very close to LDA orbitals, contrary to those of the full minimization that resemble the exact NOs. Since local RDMFT preserves the good quality of the description of strong static correlation, this finding opens the way to a mixed density/density matrix scheme, where Kohn-Sham orbitals obtain fractional occupations from a minimization of the occupation numbers using 1-RDM functionals. This will allow for a description of strong correlation at a cost only minimally higher than a density functional calculation.
A promising variational approach for determining the ground state energy and its properties is by using the second-order reduced density matrix (2-RDM). However, the leading obstacle with this approach is the N-representability problem. By employing a subset of conditions (typically the P, Q, G, T1 and T2′ conditions) results comparable to those of CCSD(T) can be achieved. However, these conditions do not guarantee size-consistency. In this work, we show that size-consistency can be satisfied if the 2-RDM satisfies the following conditions: (i) the 2-RDM is unitary invariant diagonal N-representable; (ii) the 2-RDM corresponding to each (unspecified) subsystem is the eigenstate of the number of corresponding electrons; and (iii) the 2-RDM satisfies at least one of the P, Q, G, T1 and T2′ conditions. This is the first time that a computationally feasible (though demanding) sufficient condition for the RDM method that guarantees size-consistency in all chemical systems has been published in the literature.
We investigate the suitability of natural orbitals as a basis for describing many-body excitations. We analyze to which extent the natural orbitals describe both bound as well as ionized excited states and show that depending on the specifics of the excited state the ground-state natural orbitals may yield a good approximation. We show that the success of reduced density-matrix functional theory in describing molecular dissociation lies in the flexibility provided by fractional occupation numbers while the role of the natural orbitals is minor.