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arXiv:2209.11663 · Physical Review A

One-body reduced density-matrix functional theory for the canonical ensemble

Sarina Mirjam Sutter, Klaas J. H. Giesbertz

Published Feb 10, 2023• 9 citations
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Abstract

We establish one-body reduced density matrix (1RDM) functional theory for the canonical ensemble in a finite basis set at an elevated temperature. Including temperature guarantees the differentiability of the universal functional by occupying all states and additionally not fully occupying the states in a fermionic system. We use the convexity of the universal functional and invertibility of the potential-to-1RDM map to show that the subgradient contains only one element which is equivalent to differentiability. This allows us to show that all 1RDMs with a purely fractional occupation number spectrum ($0<{n}_{i}<1\phantom{\rule{0.28em}{0ex}}{\ensuremath{\forall}}_{i}$) are uniquely $v$-representable up to a constant.

DOI
10.1103/physreva.107.022210

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We establish one-body reduced density matrix (1RDM) functional theory for the canonical ensemble in a finite basis set at an elevated temperature. Including temperature guarantees the differentiability of the universal functional by occupying all states and additionally not fully occupying the states in a fermionic system. We use the convexity of the universal functional and invertibility of the potential-to-1RDM map to show that the subgradient contains only one element which is equivalent to differentiability. This allows us to show that all 1RDMs with a purely fractional occupation number spectrum (0<ni<1\ensuremath∀i0<{n}_{i}<1\phantom{\rule{0.28em}{0ex}}{\ensuremath{\forall}}_{i}0<ni​<1\ensuremath∀i​) are uniquely vvv-representable up to a constant.
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 The N -representability problem places fundamental constraints on reduced density matrices (RDMs) that originate from physical many-fermion quantum states. Motivated by recent developments in functional theories, we introduce a hierarchy of ensemble one-body N -representability problems that incorporate partial knowledge of the one-body RDMs (1RDMs) within an ensemble of N -fermion states with fixed weights w i . Specifically, we propose a systematic relaxation that reduces the refined problem—where full 1RDMs are fixed for certain ensemble elements—to a more tractable form involving only natural occupation number vectors. Remarkably, we show that this relaxed problem is related to a generalization of Horn’s problem, enabling an explicit solution by combining its constraints with those of the weighted ensemble N -representability conditions. An additional convex relaxation yields a convex polytope that provides physically meaningful restrictions on lattice site occupations in ensemble density functional theory for excited states.
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.
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.
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.