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The Journal of Chemical Physics

Model derived reduced density matrix restrictions for correlated fermions

Joel H. Sebold, Jerome K. Percus

Published May 1, 1996• 6 citations
Publisher

Abstract

We illustrate the importance of restrictions in improving ground state energy lower bounds of a model of correlated electrons on a lattice. A reduced density matrix (RDM) formalism is employed. The restrictions are derived from closely related and exactly solved models. Such conditions raise the estimates without resorting to increasing the size of the physical space, thus improving computational efficiency. Our main motivation for this work is the problematic picture of Hohenberg–Kohn–Sham density functional theory for strongly correlated fermions. We find that using small cluster representations, errors can be reduced by more than 50% depending on the nature of the model and parameter regime studied. We obtain results for one- and two-dimensional lattices at half filling in the thermodynamic limit, although the method could be easily adapted to finite molecular structures as well.

DOI
10.1063/1.471379

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The Journal of Chemical Physics•Aug 22, 2017•37 citations
We illustrate the importance of restrictions in improving ground state energy lower bounds of a model of correlated electrons on a lattice. A reduced density matrix (RDM) formalism is employed. The restrictions are derived from closely related and exactly solved models. Such conditions raise the estimates without resorting to increasing the size of the physical space, thus improving computational efficiency. Our main motivation for this work is the problematic picture of Hohenberg–Kohn–Sham density functional theory for strongly correlated fermions. We find that using small cluster representations, errors can be reduced by more than 50% depending on the nature of the model and parameter regime studied. We obtain results for one- and two-dimensional lattices at half filling in the thermodynamic limit, although the method could be easily adapted to finite molecular structures as well.
We derive necessary conditions for any two-fermion reduced density matrix (2-RDM) to be representable by a pure NNN-fermion density matrix $\mathrm{\ensuremath{\Psi}}{\mathrm{\ensuremath{\Psi}}}^{*}$ where $\mathrm{\ensuremath{\Psi}}$ is the wave function. These pure NNN-representability conditions of the 2-RDM are important because they provide stringent constraints beyond those from the Pauli and the generalized Pauli constraints on the structure of many-fermion 2-RDMs and their wave functions. The pure 2-RDM conditions are derived as generalized Pauli constraints on effective one-fermion reduced density matrices (1-RDMs) generated by the removal or addition of a fermion from the wave function. Computationally, we show for four-electron molecules that the derived pure NNN-representability conditions are nontrivially active for exact ground-state 2-RDMs and that they provide significant restrictions beyond the D, Q, and G ensemble NNN-representability conditions. Constraints on higher-order ppp-RDMs where p>2p>2p>2 are derived in a similar fashion. The constraints have potentially significant applications to computing strongly correlated many-fermion states with enhanced accuracy and decreased computational complexity.
A density-functional theory (DFT) of lattice fermion models is presented, which uses the single-particle density matrix ${\ensuremath{\gamma}}_{\mathrm{ij}}$ as basic variable. A simple, explicit approximation to the interaction-energy functional $W[\ensuremath{\gamma}]$ of the Hubbard model is derived from exact dimer results, scaling properties of $W[\ensuremath{\gamma}]$ and known limits. Systematic tests on the one-dimensional chain show a remarkable agreement with the Bethe-ansatz exact solution for all interaction regimes and band fillings. New results are obtained for the ground-state energy and charge-excitation gap in two dimensions. A successful description of strong electron correlations within DFT is achieved.
The one-particle reduced density-matrix (1-RDM) functional theory is a promising alternative to density-functional theory (DFT) that uses the 1-RDM rather than the electronic density as a basic variable. However, long-standing challenges such as the lack of the Kohn-Sham scheme and the complexity of the pure N-representability conditions are still impeding its wild utilization. Fortunately, ensemble N-representability conditions derived in the natural orbital basis are known and trivial such that almost every functional of the 1-RDM is actually natural orbital functional, which does not perform well for all the correlation regimes. In this work, we propose a variational minimization scheme in the ensemble N-representable domain that is not restricted to the natural orbital representation of the 1-RDM. We show that splitting the minimization into the diagonal and off-diagonal parts of the 1-RDM can open the way toward the development of functionals of the orbital occupations, which remains a challenge for the generalization of site-occupation functional theory in chemistry. Our approach is tested on the uniform Hubbard model using the Müller and the Töws-Pastor functionals, as well as on the dihydrogen molecule using the Müller functional.
Reduced density-matrix functional theory (RDMFT) has become an appealing alternative to density-functional theory to describe electronic properties of strongly correlated systems. Here we derive exact conditions for the suitability of RDMFT to describe the two-dimensional homogeneous electron gas, which is the base system for semiconductor quantum dots and quantum Hall devices, for example. Following the method of Cioslowski and Pernal [J. Chem. Phys. 111, 3396 (1999)] we focus on the properties of power functionals of the form $f(n,{n}^{\ensuremath{'}})={(n{n}^{\ensuremath{'}})}^{\ensuremath{\alpha}}$ for the scaling function in the exchange-correlation energy. We show that in order to have stable and analytic solutions, and for fff to satisfy the homogeneous scaling constraint, the power is restricted to $1/4\ensuremath{\leqslant}\ensuremath{\alpha}\ensuremath{\leqslant}3/4$. Applying a reasonable ansatz for the momentum distribution and the lower bound for the exchange-correlation energy tightens the physical regime further to $0.64\ensuremath{\lesssim}\ensuremath{\alpha}\ensuremath{\leqslant}0.75$.
The formalism of Kohn and Sham uses a specific (model) hamiltonian which highly simplifies the many-electron problem to that of noninteracting fermions. The theorem of Hohenberg and Kohn tells us that, for a given ground state density, this hamiltonian is unique. In principle, this density can be chosen as that of the real, interacting system. To obtain the energy, or other properties of the real system, approximations are needed. Working with non interacting fermions is an important simplification, but it may be easier to produce approximations with different choices of the model hamiltonian. The feature that the exact density is (ideally) reproduced can be kept in the newly defined fictitious systems. Using model hamiltonians having the same form as the physical one, that is, being built of one- and two-body operators, allows to approach the physical hamiltonian arbitrarily close, and thus a systematic reduction of the approximations.
Full configuration interaction (FCI) restricted to a pairing space yields size-extensive correlation energies but its cost scales exponentially with molecular size. Restricting the variational two-electron reduced-density-matrix (2-RDM) method to represent the same pairing space yields an accurate lower bound to the pair FCI energy at a mean-field-like computational scaling of O(r3) where r is the number of orbitals. In this paper, we show that localized molecular orbitals can be employed to generate an efficient, approximately size-extensive pair 2-RDM method. The use of localized orbitals eliminates the substantial cost of optimizing iteratively the orbitals defining the pairing space without compromising accuracy. In contrast to the localized orbitals, the use of canonical Hartree-Fock molecular orbitals is shown to be both inaccurate and non-size-extensive. The pair 2-RDM has the flexibility to describe the spectra of one-electron RDM occupation numbers from all quantum states that are invariant to time-reversal symmetry. Applications are made to hydrogen chains and their dissociation, n-acene from naphthalene through octacene, and cadmium telluride 2-, 3-, and 4-unit polymers. For the hydrogen chains, the pair 2-RDM method recovers the majority of the energy obtained from similar calculations that iteratively optimize the orbitals. The localized-orbital pair 2-RDM method with its mean-field-like computational scaling and its ability to describe multi-reference correlation has important applications to a range of strongly correlated phenomena in chemistry and physics.