Unitary Group Approach to Configuration Interaction Calculations of the Electronic Structure of Atoms and Molecules

Shavitt, Isaiah

doi:10.1007/978-1-4684-6363-7_11

Cited by 55 publications

(34 citation statements)

References 25 publications

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“…The differences in these numbers relative to full-CI single-headed Shavitt graphs always includes one term of Z [Eq. (8)] and sometimes additional boundary value terms. Z(a, c) is a measure of the available full and empty orbitals that are the necessary sources of additional singly occupied orbitals.…”

Section: Discussionmentioning

confidence: 99%

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Spin–orbit interaction with nonlinear wave functions

Brozell

Shepard

Zhang

2007

Int J of Quantum Chemistry

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ABSTRACT:The computation of the spin-orbit interaction is discussed for electronic wave functions expressed in the new nonlinear expansion form. This form is based on spin eigenfunctions using the graphical unitary group approach (GUGA). The nodes of a Shavitt graph in GUGA are connected by arcs, and a Configuration State Function (CSF) is represented as a walk along arcs from the vacuum node to a head node. The wave function is a linear combination of product functions each of which is a linear combination of all CSFs, wherein each CSF coefficient is a product of nonlinear arc factors. When the spin-orbit interaction is included the Shavitt graph is a union of single-headed Shavitt graphs each with the same total number of electrons and orbitals. Thus spin-orbit Shavitt graphs are multiheaded. For full-CI multiheaded Shavitt graphs, analytic expressions are presented for the number of walks, the number of nodes, the number of arcs, and the number of node pairs in the associated auxiliary pair graph.

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Section: Discussionmentioning

confidence: 99%

“…This form is based on the graphical unitary group approach (GUGA) of Shavitt [4][5][6][7][8]. The new methodology is intended for MCSCF [9,10] and CI [6,11] calculations, and it is being developed within the COLUMBUS Program System [11][12][13].…”

Section: Introduction Imentioning

confidence: 99%

Spin–orbit interaction with nonlinear wave functions

Brozell

Shepard

Zhang

2007

Int J of Quantum Chemistry

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“…I n the graphically contracted function (GCF) method [1][2][3][4], the wave function is represented using the graphical unitary group approach [5][6][7] in which the expansion configuration state functions (CSF) of the unitary group approach [8][9][10] are represented graphically. The wave function is expanded as a linear combination of GCFs…”

Section: Introductionmentioning

confidence: 99%

An efficient recursive algorithm to compute wave function optimization gradients for the graphically contracted function method

Shepard

Gidofalvi

Hovland

2010

Int J of Quantum Chemistry

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An efficient recursive algorithm is presented to compute orbital-level Hamiltonian matrices for wave functions expanded in a basis of graphically contracted functions (GCF). Each GCF depends on a nonlinear set of parameters called arc factors. The orbital-level Hamiltonian matrices characterize the dependence of the energy on the wave function changes associated with a subset of these nonlinear parameters corresponding to an individual molecular orbital. From these Hamiltonian matrices, gradients with respect to the nonlinear arc factor parameters may be computed and other arc factor optimization algorithms may be used. The recursive algorithm allows the orbital-level Hamiltonian matrices to be computed with O(N 2 GCF xn 4 ) total effort, where N GCF is the dimension of the GCF basis, n is the dimension of the orbital basis, and where the scale factor x depends on the number of electrons N and ranges from O(N 0 ) to O(N 2 ) depending on the complexity of the underlying Shavitt graph. This effort is about two to five times that required to compute an energy expectation value for a given set of arc factors; thus the energy and gradient have the same scaling behavior with increasing molecule size, N GCF dimension, and orbital basis size. Timings are given for wave functions that correspond to configuration state function expansions over 10 73 in length, many orders of magnitude larger than can be considered using traditional electronic structure methods.

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“…The rows of the Paldus array may be conveniently organized by employing the Graphical Unitary Group Approach (GUGA) developed by Shavitt. [4][5][6][7][8][9][10] Within GUGA the set of Paldus arrays is represented by a Shavitt graph where individual orbitals correspond to vertical levels of the graph and each level contains a set of nodes that correspond to distinct rows of the Paldus array. Nodes at a particular level of the Shavitt graph are connected to nodes at adjacent levels by four different types of arcs with step numbers d k .…”

Section: Introductionmentioning

confidence: 99%

Computation of determinant expansion coefficients within the graphically contracted function method

Gidofalvi¹,

Shepard²

2009

J Comput Chem

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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 depends 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.

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Unitary Group Approach to Configuration Interaction Calculations of the Electronic Structure of Atoms and Molecules

Cited by 55 publications

References 25 publications

Spin–orbit interaction with nonlinear wave functions

Spin–orbit interaction with nonlinear wave functions

An efficient recursive algorithm to compute wave function optimization gradients for the graphically contracted function method

Computation of determinant expansion coefficients within the graphically contracted function method

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