# 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:

- Chemical Sciences and Engineering Division, Argonne National Laboratory, Argonne, Illinois 60439 (United States)
- 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}

}