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Parallel Two-Electron Reduced Density Matrix Based Electronic Structure Software for Highly Correlated Molecules and Materials

Eugene DePrince

Published Apr 7, 2015• 0 citations
OpenAlex

Abstract

Abstract : Variational two-electron reduced-density-matrix (2-RDM) methods provide a reference independent description of the electronic structure ofmany-electron systems; these methods also naturally capture multireference correlation effects. However, existing implementations of thevariational 2-RDM (v-2RDM) method exhibit a time-to-solution that is often prohibitively large for the general description of largemolecules or materials. It is the goal of this STTR to develop an efficient, parallel, and commercializable 2-RDM-based electronic structure software suite. In Phase I of this STTR, we have: (i) developed a prototype commercializable software product to perform electronicstructure computations using the variational 2-RDM method, (ii) developed two semidefinite programming solvers and identified which ismore likely to be of use for solving the variational 2-RDM problem for general systems, (iii) interfaced this solver with the libraries of theQ-Chem electronic structure package, (iv) implemented algorithms that use either loop-based or libtensor-based tensor manipulation andidentified the libtensor algorithm as the more promising for massively parallel variational 2-RDM computations, (v) developed a pilotdistributed-memory parallel version of the solver through an interface between libtensor and the Cyclops Tensor Framework (CTF), (vi)enabled the automatic detection and use of graphics processing units (GPUs) for certain portions of the algorithm, and (vii) extended someof our algorithms for the description of open-shell systems and to include partial three-particle N-representability conditions.

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Abstract : Variational two-electron reduced-density-matrix (2-RDM) methods provide a reference independent description of the electronic structure ofmany-electron systems; these methods also naturally capture multireference correlation effects. However, existing implementations of thevariational 2-RDM (v-2RDM) method exhibit a time-to-solution that is often prohibitively large for the general description of largemolecules or materials. It is the goal of this STTR to develop an efficient, parallel, and commercializable 2-RDM-based electronic structure software suite. In Phase I of this STTR, we have: (i) developed a prototype commercializable software product to perform electronicstructure computations using the variational 2-RDM method, (ii) developed two semidefinite programming solvers and identified which ismore likely to be of use for solving the variational 2-RDM problem for general systems, (iii) interfaced this solver with the libraries of theQ-Chem electronic structure package, (iv) implemented algorithms that use either loop-based or libtensor-based tensor manipulation andidentified the libtensor algorithm as the more promising for massively parallel variational 2-RDM computations, (v) developed a pilotdistributed-memory parallel version of the solver through an interface between libtensor and the Cyclops Tensor Framework (CTF), (vi)enabled the automatic detection and use of graphics processing units (GPUs) for certain portions of the algorithm, and (vii) extended someof our algorithms for the description of open-shell systems and to include partial three-particle N-representability conditions.
Variational two-electron reduced-density-matrix (2RDM) theory is applied to computing energy spectra and properties of few-electron quantum dots. The model system is a two-dimensional electron gas with a central confinement potential. For each orbital angular momentum JJJ, the energy and 2RDM are computed by the variational 2RDM method in which the energy is minimized as a functional of the 2RDM. In the minimization, which is performed by semidefinite programming, the 2RDM is constrained to represent a NNN-electron wave function with angular momentum JJJ by NNN- and JJJ-representability conditions [D. A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)]. Advantages of the variational 2RDM method include (i) lower bounds on the energies for all JJJ values, (ii) calculation of approximate 2RDMs in polynomial time without many-electron wave functions, (iii) exploitation of angular symmetry in the sparse block-diagonal structure of the 2RDM, (iv) accurate description of multireference correlation (entanglement) effects, and (v) direct calculation of one- and two-electron properties from the 2RDM. With the 2RDM we directly compute pair-correlation functions, radial charge densities, and average radial electron displacements in the quantum dot. Energies and properties are compared to those from solving the NNN-electron Schr\"odinger equation by large-scale, exact diagonalization. It is found that the accuracy of the variational 2RDM approach is sensitive to the total orbital angular momentum and the symmetry of the final wave function. For quantum dots of high symmetry, the variational algorithm isolates a highly accurate solution that recovers the correlation energy within a few percent.
Abstract The two‐electron reduced density matrix (2RDM) carries enough information to evaluate the electronic energy of a many‐electron system. The variational 2RDM (v2RDM) approach seeks to determine the 2RDM directly, without knowledge of the wave function, by minimizing this energy with respect to variations in the elements of the 2RDM, while also enforcing known N ‐representability conditions. In this tutorial review, we provide an overview of the theoretical underpinnings of the v2RDM approach and the N ‐representability constraints that are typically applied to the 2RDM. We also discuss the semidefinite programming (SDP) techniques used in v2RDM computations and provide enough Python code to develop a working v2RDM code that interfaces to the libSDP library of SDP solvers. This article is categorized under: Electronic Structure Theory > Ab Initio Electronic Structure Methods Software > Quantum Chemistry
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.
The ground-state energy and properties of any many-electron atom or molecule may be rigorously computed by variationally computing the two-electron reduced density matrix rather than the many-electron wavefunction. While early attempts fifty years ago to compute the ground-state 2-RDM directly were stymied because the 2-RDM must be constrained to represent an N-electron wavefunction, recent advances in theory and optimization have made direct computation of the 2-RDM possible. The constraints in the variational calculation of the 2-RDM require a special optimization known as a semidefinite programming. Development of first-order semidefinite programming for the 2-RDM method has reduced the computational costs of the calculation by orders of magnitude [Mazziotti, Phys. Rev. Lett. 93 (2004) 213001]. The variational 2-RDM approach is effective at capturing multi-reference correlation effects that are especially important at non-equilibrium molecular geometries. Recent work on 2-RDM methods will be reviewed and illustrated with particular emphasis on the importance of advances in large-scale semidefinite programming.
Variational two-electron reduced-density-matrix (2-RDM) methods can provide a reference-independent description of the electronic structure of strongly correlated molecules and materials. These methods represent one of few ways of performing large active-space-based computations that are beyond the scope of conventional configuration-interaction-based approaches. However, implementations of the method are quite rare, making it difficult for the quantum chemistry community to assess the utility of the approach. Here, we discuss an open-shell implementation of the variational 2-RDM method and explore its ability to describe different spin states in several model systems, including linear hydrogen chains and linear acenes. The largest calculations considered are comparable to complete-active-space computations with 50 electrons in 50 orbitals.
Molecular two-electron reduced density matrices (2-RDMs) are computed variationally without the many-electron wave function by constraining the 2-RDM with a set of three-particle NNN-representability conditions known as three-positivity conditions. These constraints restrict four distinct three-particle probability distributions, which can be defined for any NNN-particle system, to be nonnegative. The variational calculation of the 2-RDM with full three-positivity conditions is implemented with the first-order semidefinite programming algorithm [D.A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)]. We also derive and implement a generalization of the T2{T}_{2}T2​ representability condition, which is a subset of the three-positivity conditions. Energies and 2-RDMs are computed for several molecules as well as the nitrogen molecule at both equilibrium and nonequilibrium geometries. The ground-state energies for nitrogen are within 0.0015a.u.0.0015\phantom{\rule{0.3em}{0ex}}\mathrm{a.u.}0.0015a.u. of the values from full configuration interaction at all internuclear distances.