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Register a new userGeneralized Pauli constraints in reduced density matrix functional theory
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Added by
Nektarios Lathiotakis
Publication date
2015
fields
Physics
and research's language is
English
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Functionals of the one-body reduced density matrix (1-RDM) are routinely minimized under Colemans 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.
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We investigate the structure of the one-body Reduced Density Matrix (1RDM) of three electron systems, i.e. doublet and quadruplet spin configurations, corresponding to the smallest interacting system with an open-shell ground state. To this end, we use Configuration Interaction (CI) expansions of the exact wave function in Slater determinants built from natural orbitals in a finite dimensional Hilbert space. With the exception of maximally polarized systems, the natural orbitals of spin eigenstates are generally spin dependent, i.e. the spatial parts of the up and down natural orbitals form two different sets. A measure to quantify this spin dependence is introduced and it is shown that it varies by several orders of magnitude depending on the system. We also study the ordering issue of the spin-dependent occupation numbers which has practical implications in Reduced Density Matrix Functional Theory minimization schemes when Generalized Pauli Constraints are imposed and in the form of the CI expansion in terms of the natural orbitals. Finally, we discuss the aforementioned CI expansion when there are GPCs that are almost pinned.
We consider necessary conditions for the one-body-reduced density matrix (1RDM) to correspond to a triplet wave-function of a two electron system. The conditions concern the occupation numbers and are different for the high spin projections, $S_z=pm 1$, and the $S_z=0$ projection. Hence, they can be used to test if an approximate 1RDM functional yields the same energies for both projections. We employ these conditions in reduced density matrix functional theory calculations for the triplet excitations of two-electron systems. In addition, we propose that these conditions can be used in the calculation of triplet states of systems with more than two electrons by restricting the active space. We assess this procedure in calculations for a few atomic and molecular systems. We show that the quality of the optimal 1RDMs improves by applying the conditions in all the cases we studied.
Based on a generalization of Hohenberg-Kohns theorem, we propose a ground state theory for bosonic quantum systems. Since it involves the one-particle reduced density matrix $gamma$ as a natural variable but still recovers quantum correlations in an exact way it is particularly well-suited for the accurate description of Bose-Einstein condensates. As a proof of principle we study the building block of optical lattices. The solution of the underlying $v$-representability problem is found and its peculiar form identifies the constrained search formalism as the ideal starting point for constructing accurate functional approximations: The exact functionals for this $N$-boson Hubbard dimer and general Bogoliubov-approximated systems are determined. The respective gradient forces are found to diverge in the regime of Bose-Einstein condensation, $ abla_{gamma} mathcal{F} propto 1/sqrt{1-N_{mathrm{BEC}}/N}$, providing a natural explanation for the absence of complete BEC in nature.
We present an textit{ab initio} theory for superconductors, based on a unique mapping between the statistical density operator at equilibrium, on the one hand, and the corresponding one-body reduced density matrix $gamma$ and the anomalous density $chi$, on the other. This new formalism for superconductivity yields the existence of a universal functional $mathfrak{F}_beta[gamma,chi]$ for the superconductor ground state, whose unique properties we derive. We then prove the existence of a Kohn-Sham system at finite temperature and derive the corresponding Bogoliubov-de Gennes-like single particle equations. By adapting the decoupling approximation from density functional theory for superconductors we bring these equations into a computationally feasible form. Finally, we use the existence of the Kohn-Sham system to extend the Sham-Schluter connection and derive a first exchange-correlation functional for our theory. This reduced density matrix functional theory for superconductors has the potential of overcoming some of the shortcomings and fundamental limitations of density functional theory of superconductivity.
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 molecule at 1/4 and 1/2 filling 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 are made concerning the approximations of the observables. In particular we focus on the atomic limit, where the two sites of the molecule 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 molecule 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 are present in 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. Our findings can be generalized to other situations, which allows us to make connections to real materials and experiment.
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