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.
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.
Treatment of two-electron excitations is a fundamental but computationally expensive part of ab initio calculations of many-electron correlation. In this paper we develop a low-rank spectral expansion of two-electron excitations for accelerated electronic-structure calculations. The spectral expansion differs from previous approaches by relying upon both (i) a sum of three expansions to increase the rank reduction of the tensor and (ii) a factorization of the tensor into geminal (rank-two) tensors rather than orbital (rank-one) tensors. We combine three spectral expansions from the three distinct forms of the two-electron reduced density matrix (2-RDM), (i) the two-particle (2)D, (ii) the two-hole (2)Q, and the (iii) particle-hole (2)G matrices, to produce a single spectral expansion with significantly accelerated convergence. While the resulting expansion is applicable to any quantum-chemistry calculation with two-particle excitation amplitudes, it is employed here in the parametric 2-RDM method [D. A. Mazziotti, Phys. Rev. Lett. 101, 253002 (2008)]. The low-rank parametric 2-RDM method scales quartically with the basis-set size, but like its full-rank version it can capture multi-reference correlation effects that are difficult to treat efficiently by traditional single-reference wavefunction methods. Applications are made to computing potential energy curves of HF and triplet OH(+), equilibrium bond distances and frequencies, the HCN-HNC isomerization, and the energies of hydrocarbon chains. Computed 2-RDMs nearly satisfy necessary N-representability conditions. The low-rank spectral expansion has the potential to expand the applicability of the parametric 2-RDM method as well as other ab initio methods to large-scale molecular systems that are often only treatable by mean-field or density functional theories.
Low-rank spectral expansion and tensor hypercontraction are two promising techniques for reducing the size of the two-electron excitation tensor by factorizing it into products of smaller tensors. Both methods can potentially realize an O(r(4)) quantum chemistry method where r is the number of one-electron orbitals. We compare the two factorizations in this paper by applying them to the parametric 2-electron reduced density matrix method with the M functional [D. A. Mazziotti, Phys. Rev. Lett. 101, 253002 (2008)]. We study several inorganic molecules, alkane chains, and potential curves as well as reaction and dissociation energies. The low-rank spectral expansion, we find, is typically more efficient than tensor hypercontraction due to a faster convergence of the energy and a smaller constant prefactor in the energy optimization. Both factorizations are applicable to the acceleration of a wide range of wavefunction and reduced-density-matrix methods.
An improved parametrization of the two-electron reduced density matrix (2-RDM) [D. A. Mazziotti, Phys. Rev. Lett. 101, 253002 (2008)] was recently shown to yield energies and properties that are markedly better than those calculated by traditional ab initio methods of similar computational scaling. In this paper a family of such energy functionals, generalizing the ones obtained previously, is derived through the use of (i) p-particle contraction relations based on the contraction of the cumulant expansions of p-particle RDMs and (ii) Cauchy-Schwarz relations that arise from an important set of N-representability constraints known as the two-positivity conditions. The 2-RDMs are explicitly parameterized in terms of the first-order part of the cumulant 2-RDM and, for the inclusion of single excitations, a second-order part of the 1-RDM. In contrast to earlier formulations based on the coefficients from configuration interaction with single and double excitations (CISD), the cumulant-based parametric 2-RDM methods, from the properties of cumulants, are rigorously size extensive. We also show that writing the energy functionals in terms of correlated 1-RDMs and cumulant 2-RDMs reduces the computational cost of the parametric 2-RDM methods to that of CISD. Applications are made to ground-state energies of several molecules, equilibrium bond distances, and frequencies of HF, F2, and CO, the relative energy of the cis and trans isomers of HO3${}^{\ensuremath{-}}$, and the HCN-HNC isomerization reaction. For bond breaking in hydrogen fluoride the improved and more efficient parametric 2-RDM methods yield energies with similar accuracies at both equilibrium and nonequilibrium geometries in 6-31G** and polarized valence quadruple-$\ensuremath{\zeta}$ basis sets. Computed 2-RDMs very nearly satisfy well-known N-representability conditions.
The electronic Hamiltonian contains only pairwise interactions, allowing the energy of an electronic system to be expressed in terms of the two-electron reduced-density-matrix (2-RDM) in lieu of the many-electron wavefunction. The variable space for the exact N-electron wavefunction scales exponentially with the size of the system, while the 2-RDM is polynomial in scale. By using the 2-RDM as the primary variable in electronic structure calculations, it may be possible to obtain very accurate energies at a much more favorable scaling than wavefunction methods. In this thesis, we will use two existing 2-RDM methods to treat electronic systems. First, we will apply the active-space variational 2-RDM method, which directly minimizes the energy with respect to the 2-RDM, to a cadmium telluride polymer that was recently used to greatly enhance the conductivity of CdTe quantum dots. We find that this polymer is very highly correlated despite a deceptively simple structure. We will then turn to the parametric 2-RDM method (p2-RDM), which parameterizes the 2-RDM in terms of a truncated configuration interaction ansatz, but which includes additional flexibility in order to be size-extensive. We apply p2-RDM to the study of the olympicene molecule, which features both fully aromatic and diradical isomers. The parametric 2-RDM method predicts that all isomers are stable to dissociation, in contrast to coupled cluster methods which do not predict stable diradical states. We then present analytical nuclear gradients for p2-RDM, which greatly decrease the number of calculations required to perform geometry optimizations. We apply these gradients to the study of trans-polyacetylene, for which p2-RDM, unlike many wavefunction methods, is able to predict a bond length alternation (BLA) to within experimental values. Lastly, as a single-reference method, p2-RDM may encounter numerical difficulties when the reference wavefunction is of particularly poor quality. We propose a modification to the parameterization that may render the method more generally robust.
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.
In this thesis we develop the variational two-electron reduced-density-matrix method for extended systems. Extended systems are represented in two ways: i) lattice models describing the dominant valence electronic structure with periodic boundaries to account for their extended nature and ii) a crystalline-orbital basis built from atomic orbitals using the generalization of molecular orbital theory to polymers. The first part of this thesis (Ch. 3-4) examines the performance of the variational 2-RDM method on lattice systems with tunable electron correlation. The first of these systems is the classic Hubbard model with linear and ladder lattice topologies. Because electron correlation functions, such as charge- and spin-ordering, are linear functions of the 2-RDM, the difference in electronic structure between one- and quasi-one-dimensional systems is accurately characterized. The second model contains only two-body interactions and is unique among typical spin models in that it does not have a mean-field reference wave function. The ground state wave functions from all Hamiltonians in the model have the same 1-electron reduced density matrix; consequently, one-electron theories are largely inapplicable. The superconducting eta-pairing ground states make the model a unique tool for demonstrating the necessary N-representability in highly correlated environments. The second part of this thesis (Ch. 5-6) develops a formalism for modeling materials by solving the full Schrodinger equation. Crystalline-orbital Hartree-Fock provides a set of orbitals and integral tensors for the variational 2-RDM method. We demonstrate that time-reversal symmetry, which is implicitly included in position space electronic structure calculations, must be explicitly included as an N-representability constraint on the 2-RDM when using a momentum space basis. The necessity of these equality constraints is demonstrated by the accurate recovery of the binding energy of two polymers and the symmetry of their natural orbital occupations.