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.
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.
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.
Variational minimization of the ground-state energy as a function of the two-electron reduced density matrix (2-RDM), constrained by necessary N-representability conditions, provides a polynomial-scaling approach to studying strongly correlated molecules without computing the many-electron wave function. Here we introduce a route to enhancing necessary conditions for N representability through rank restriction of the 2-RDM. Rather than adding computationally more expensive N-representability conditions, we directly enhance the accuracy of two-particle (2-positivity) conditions through rank restriction, which removes degrees of freedom in the 2-RDM that are not sufficiently constrained. We select the rank of the particle-hole 2-RDM by deriving the ranks associated with model wave functions, including both mean-field and antisymmetrized geminal power (AGP) wave functions. Because the 2-positivity conditions are exact for quantum systems with AGP ground states, the rank of the particle-hole 2-RDM from the AGP ansatz provides a minimum for its value in variational 2-RDM calculations of general quantum systems. To implement the rank-restricted conditions, we extend a first-order algorithm for large-scale semidefinite programming. The rank-restricted conditions significantly improve the accuracy of the energies; for example, the percentages of correlation energies recovered for HF, CO, and N2 improve from 115.2%, 121.7%, and 121.5% without rank restriction to 97.8%, 101.1%, and 100.0% with rank restriction. Similar results are found at both equilibrium and nonequilibrium geometries. While more accurate, the rank-restricted N-representability conditions are less expensive computationally than the full-rank conditions.
Molecular systems in chemistry often have wave functions with substantial contributions from two-or-more electronic configurations. Because traditional complete-active-space self-consistent-field (CASSCF) methods scale exponentially with the number N of active electrons, their applicability is limited to small active spaces. In this paper we develop an active-space variational two-electron reduced-density-matrix (2-RDM) method in which the expensive diagonalization is replaced by a variational 2-RDM calculation where the 2-RDM is constrained by approximate N-representability conditions. Optimization of the constrained 2-RDM is accomplished by large-scale semidefinite programming [Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)]. Because the computational cost of the active-space 2-RDM method scales polynomially as r(a)(6) where r(a) is the number of active orbitals, the method can be applied to treat active spaces that are too large for conventional CASSCF. The active-space 2-RDM method performs two steps: (i) variational calculation of the 2-RDM in the active space and (ii) optimization of the active orbitals by Jacobi rotations. For large basis sets this two-step 2-RDM method is more efficient than the one-step, low-rank variational 2-RDM method [Gidofalvi and Mazziotti, J. Chem. Phys. 127, 244105 (2007)]. Applications are made to HF, H(2)O, and N(2) as well as n-acene chains for n=2-8. When n>4, the acenes cannot be treated by conventional CASSCF methods; for example, when n=8, CASSCF requires optimization over approximately 1.47x10(17) configuration state functions. The natural occupation numbers of the n-acenes show the emergence of bi- and polyradical character with increasing chain length.
The variational two-electron reduced-density-matrix (2-RDM) method allows for the computation of accurate ground-state energies and 2-RDMs of atoms and molecules without the explicit construction of an N-electron wave function. While previous work on variational 2-RDM theory has focused on calculating full configuration-interaction energies, this work presents the first application toward approximating multiconfiguration self-consistent-field (MCSCF) energies via low-rank restrictions on the 1- and 2-RDMs. The 2-RDM method with two- or three-particle N-representability conditions reduces the exponential active-space scaling of MCSCF methods to a polynomial scaling. Because the first-order algorithm [Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)] represents each form of the 1- and 2-RDMs by a matrix factorization, the RDMs are readily defined to have a low rank rather than a full rank by setting the matrix factors to be rectangular rather than square. Results for the potential energy surfaces of hydrogen fluoride, water, and the nitrogen molecule show that the low-rank 2-RDM method yields accurate approximations to the MCSCF energies. We also compute the energies along the symmetric stretch of a 20-atom hydrogen chain where traditional MCSCF calculations, requiring more than 17x10(9) determinants in the active space, could not be performed.