Reduced-Density-Matrix Theory for Many-electron Correlation

Erik P. Hoy

Conventional low cost computational methods in quantum chemistry such as Hartree-Fock and density functional theory often fail to properly describe electron correlation. Of the two types of electron correlation, dynamic and static, static or multi-reference correlation is particularly difficult to obtain using low cost methods. This type of electron correlation results when multiple Hartree-Fock reference states are needed to describe a system. One class of methods which are able to describe both types effectively are 2-electron reduced density matrix (2-RDM) methods. These methods use the 2-electron density matrix as the primary variable in the electronic Schrodinger equation in place of the wavefunction. By circumventing the wavefunction, one replaces an exponentially scaling problem with a polynomially scaling one, and unlike density functional theory, the energy is a linear functional of the 2-RDM. The lowest cost 2-RDM method currently available is known as the parametric 2-electron reduced density matrix method (p2-RDM). The method has equivalent scaling to configuration interaction with single and double excitations (CISD) and is able to treat both dynamic and static correlation. This work examines both the correlation recovery abilities of the p2-RDM method and potential scaling improvements to the method using tensor factorization. We first investigated several molecular systems including the conversion of hydroxylamine to ammonia oxide, the hydridotrioxygen atmospheric radical, and the benzene and cyclobutadiene diradicals. The purpose of these studies was to further clarify the ability of the parametric method to capture various mixes of both types of correlation. We then investigated using two different tensor factorizations, low rank spectral expansion and tensor hypercontraction, to reduce the scaling of the p2-RDM toward that of Hartree-Fock and density functional theory. Based on these results, we developed a new tensor factorization to address the shortcomings of previous methods and provide a promising, new technique to generate a fourth order p2-RDM method capable of capturing both dynamic and static correlation.

Josef Paldus

Mauricio Rodríguez‐Mayorga, Eloy Ramos‐Cordoba, Ferran Feixas +1 more

The electronic energy of a system of fermions can be obtained from the second-order reduced density matrix through the contracted Schrödinger equation or its anti-Hermitian counterpart. Both energy expressions depend on the third-order reduced density matrix (3-RDM) which is usually approximated from lower-order densities. The accuracy of these methods depends critically on the set of N-representability conditions enforced in the calculation and the quality of the approximate 3-RDM. There are no benchmark studies including most 3-RDM approximations and, thus far, no assessment of the deterioration of the approximations with correlation effects has been performed. In this paper we introduce a series of tests to assess the performance of 3-RDM approximations in a model system with varying electron correlation effects, the three-electron harmonium atom. The results of this work put forward several limitations of the currently most used 3-RDM approximations for systems with important electron correlation effects.

Mauricio Rodríguez‐Mayorga, Klaas J. H. Giesbertz, Lucas Visscher

As a new approach to efficiently describe correlation effects in the relativistic quantum world we propose to consider reduced density matrix functional theory, where the key quantity is the first-order reduced density matrix (1-RDM). In this work, we first introduce the theoretical foundations to extend the applicability of this theory to the relativistic domain. Then, using the so-called no-pair (np) approximation, we arrive at an approximate treatment of the relativistic effects by focusing on electronic wavefunctions and neglecting explicit contributions from positrons. Within the np approximation the theory becomes similar to the nonrelativistic case, with as unknown only the functional that describes the electron-electron interactions in terms of the 1-RDM. This requires the construction of functional approximations, and we therefore also present the relativistic versions of some common RDMFT approximations that are used in the nonrelativistic context and discuss their properties.

Josef Paldus