| Summary: | 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.
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