skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: The multifacet graphically contracted function method. I. Formulation and implementation

Abstract

The basic formulation for the multifacet generalization of the graphically contracted function (MFGCF) electronic structure method is presented. The analysis includes the discussion of linear dependency and redundancy of the arc factor parameters, the computation of reduced density matrices, Hamiltonian matrix construction, spin-density matrix construction, the computation of optimization gradients for single-state and state-averaged calculations, graphical wave function analysis, and the efficient computation of configuration state function and Slater determinant expansion coefficients. Timings are given for Hamiltonian matrix element and analytic optimization gradient computations for a range of model problems for full-CI Shavitt graphs, and it is observed that both the energy and the gradient computation scale as O(N{sup 2}n{sup 4}) for N electrons and n orbitals. The important arithmetic operations are within dense matrix-matrix product computational kernels, resulting in a computationally efficient procedure. An initial implementation of the method is used to present applications to several challenging chemical systems, including N{sub 2} dissociation, cubic H{sub 8} dissociation, the symmetric dissociation of H{sub 2}O, and the insertion of Be into H{sub 2}. The results are compared to the exact full-CI values and also to those of the previous single-facet GCF expansion form.

Authors:
;  [1];  [2]
  1. Chemical Sciences and Engineering Division, Argonne National Laboratory, Argonne, Illinois 60439 (United States)
  2. Department of Chemistry and Biochemistry, Gonzaga University, 502 E. Boone Ave., Spokane, Washington 99258-0102 (United States)
Publication Date:
OSTI Identifier:
22420011
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Chemical Physics; Journal Volume: 141; Journal Issue: 6; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
37 INORGANIC, ORGANIC, PHYSICAL AND ANALYTICAL CHEMISTRY; DENSITY MATRIX; DISSOCIATION; ELECTRONIC STRUCTURE; ELECTRONS; HAMILTONIANS; KERNELS; SLATER METHOD; SPIN

Citation Formats

Shepard, Ron, Brozell, Scott R., and Gidofalvi, Gergely. The multifacet graphically contracted function method. I. Formulation and implementation. United States: N. p., 2014. Web. doi:10.1063/1.4890734.
Shepard, Ron, Brozell, Scott R., & Gidofalvi, Gergely. The multifacet graphically contracted function method. I. Formulation and implementation. United States. doi:10.1063/1.4890734.
Shepard, Ron, Brozell, Scott R., and Gidofalvi, Gergely. Thu . "The multifacet graphically contracted function method. I. Formulation and implementation". United States. doi:10.1063/1.4890734.
@article{osti_22420011,
title = {The multifacet graphically contracted function method. I. Formulation and implementation},
author = {Shepard, Ron and Brozell, Scott R. and Gidofalvi, Gergely},
abstractNote = {The basic formulation for the multifacet generalization of the graphically contracted function (MFGCF) electronic structure method is presented. The analysis includes the discussion of linear dependency and redundancy of the arc factor parameters, the computation of reduced density matrices, Hamiltonian matrix construction, spin-density matrix construction, the computation of optimization gradients for single-state and state-averaged calculations, graphical wave function analysis, and the efficient computation of configuration state function and Slater determinant expansion coefficients. Timings are given for Hamiltonian matrix element and analytic optimization gradient computations for a range of model problems for full-CI Shavitt graphs, and it is observed that both the energy and the gradient computation scale as O(N{sup 2}n{sup 4}) for N electrons and n orbitals. The important arithmetic operations are within dense matrix-matrix product computational kernels, resulting in a computationally efficient procedure. An initial implementation of the method is used to present applications to several challenging chemical systems, including N{sub 2} dissociation, cubic H{sub 8} dissociation, the symmetric dissociation of H{sub 2}O, and the insertion of Be into H{sub 2}. The results are compared to the exact full-CI values and also to those of the previous single-facet GCF expansion form.},
doi = {10.1063/1.4890734},
journal = {Journal of Chemical Physics},
number = 6,
volume = 141,
place = {United States},
year = {Thu Aug 14 00:00:00 EDT 2014},
month = {Thu Aug 14 00:00:00 EDT 2014}
}
  • Practical algorithms are presented for the parameterization of orthogonal matrices Q ∈ R {sup m×n} in terms of the minimal number of essential parameters (φ). Both square n = m and rectangular n < m situations are examined. Two separate kinds of parameterizations are considered, one in which the individual columns of Q are distinct, and the other in which only Span(Q) is significant. The latter is relevant to chemical applications such as the representation of the arc factors in the multifacet graphically contracted function method and the representation of orbital coefficients in SCF and DFT methods. The parameterizations aremore » represented formally using products of elementary Householder reflector matrices. Standard mathematical libraries, such as LAPACK, may be used to perform the basic low-level factorization, reduction, and other algebraic operations. Some care must be taken with the choice of phase factors in order to ensure stability and continuity. The transformation of gradient arrays between the Q and (φ) parameterizations is also considered. Operation counts for all factorizations and transformations are determined. Numerical results are presented which demonstrate the robustness, stability, and accuracy of these algorithms.« less
  • The graphically contracted function (GCF) method is extended to include an effective one-electron spin-orbit (SO) operator in the Hamiltonian matrix construction. Our initial implementation is based on a multiheaded Shavitt graph approach that allows for the efficient simultaneous computation of entire blocks of Hamiltonian matrix elements. Two algorithms are implemented. The SO-GCF method expands the spin-orbit wave function in the basis of GCFs and results in a Hamiltonian matrix of dimension N{sub dim} = N{sub a} ((S{sub max} + 1){sup 2} ? S{sub min}{sup 2}). N{sub a} is the number of sets of nonlinear arc factor parameters, and S{sub min}more » and S{sub max} are respectively the minimum and maximum values of an allowed spin range in the wave function expansion. The SO-SCGCF (SO spin contracted GCF) method expands the wave function in a basis of spin contracted functions and results in a Hamiltonian matrix of dimension N{sub dim} = N{sub a}. For a given N{sub a} and spin range, the number of parameters defining the wave function is the same in the two methods after accounting for normalization. The full Hamiltonian matrix construction with both approaches scales formally as O(N{sub a}{sup 2}{omega}n{sup 4}) for n molecular orbitals. The {omega} factor depends on the complexity of the Shavitt graph and includes factors such as the number of electrons, N, and the number of interacting spin states. Timings are given for Hamiltonian matrix construction for both algorithms for a range of wave functions with up to N = n = 128 and that correspond to an underlying linear full-CI CSF expansion dimension of over 10{sup 75} CSFs, many orders of magnitude larger than can be considered using traditional CSF-based spin-orbit CI approaches. For Hamiltonian matrix construction, the SO-SCGCF method is slightly faster than the SO-GCF method for a given N{sub a} and spin range. The SO-GCF method may be more suitable for describing multiple states, whereas the SO-SCGCF method may be more suitable for describing single states.« less
  • An efficient algorithm is presented to compute spin-density matrices from wave functions expanded in a basis of graphically contracted functions (GCF). The GCFs are based on the graphical unitary group approach (GUGA), which is a 'spin-free' formulation of the electronic wave function. The spin-density matrix elements are computed from one-particle and two-particle charge-density matrix elements. The recursive algorithm allows the spin-density matrix to be computed with O(N{sub 2}{sup GCF}{omega}n{sup 2}) total effort where N{sub GCF} is the dimension of the GCF basis and n is the dimension of the orbital basis. The scale factor {omega} depends on the number ofmore » electrons N and ranges from O(N{sup 0}) to O(N{sup 2}) depending on the complexity of the underlying Shavitt graph. Because the 'spin-free' GCF formulation eliminates the need to expand the wave function in a spin-dependent Slater determinant basis, it is possible to treat wave functions with large numbers of electrons and orbitals. Timings are given for wave functions that correspond to determinantal expansions over 10{sup 200} in length. The implementation is applicable to arbitrary spin states and to both ground and excited electronic states.« less
  • Most electronic structure methods express the wavefunction as an expansion of N-electron basis functions that are chosen to be either Slater determinants or configuration state functions. Although the expansion coefficient of a single determinant may be readily computed from configuration state function coefficients for small wavefunction expansions, traditional algorithms are impractical for systems with a large number of electrons and spatial orbitals. In this work, we describe an efficient algorithm for the evaluation of a single determinant expansion coefficient for wavefunctions expanded as a linear combination of graphically contracted functions. Each graphically contracted function has significant multiconfigurational character and dependsmore » on a relatively small number of variational parameters called arc factors. Because the graphically contracted function approach expresses the configuration state function coefficients as products of arc factors, a determinant expansion coefficient may be computed recursively more efficiently than with traditional configuration interaction methods. Although the cost of computing determinant coefficients scales exponentially with the number of spatial orbitals for traditional methods, the algorithm presented here exploits two levels of recursion and scales polynomially with system size. Hence, as demonstrated through applications to systems with hundreds of electrons and orbitals, it may readily be applied to very large systems.« less
  • No abstract prepared.