Edge counts for the auxiliary pair graph within the graphical unitary group approach

Abstract

Closed-form expressions are presented for the numbers of edges in the auxiliary pair graphs (APGs) associated with non spin–orbit and spin–orbit Shavitt graphs for full configuration interaction expansions. A Shavitt graph is a visual representation of a configuration state function expansion space constructed via the graphical unitary group approach (GUGA). An APG is an organisational aid and a programmatic tool generated from a Shavitt graph. The number of edges in an APG determines bounds on the computational scaling as a function of the total numbers of electrons, orbitals, and spin multiplicities. The edge counts extend a suite of Shavitt graph statistics based on these functional parameters. The derivation and the presentation of the formulas for the edge counts has been assisted by the bra–ket interchange symmetry and the particle-hole interchange symmetry in the GUGA formalism. These symmetry operators produce one-to-one correspondences between various sets of edges, and this yields identities among some edge count formulas. There are 208 possible edge types. Of these, some do not contribute to two-electron operators, some are related by bra–ket interchange symmetry, and some are related by particle-hole interchange symmetry. For the remaining unique edge types, explicit expressions are derived for the numbers of edges.

GRAPHICAL ABSTRACT

Keywords:

• Shavitt graph statistics
• graphical unitary group approach
• full configuration interaction
• particle-hole symmetry
• bra–ket symmetry

Acknowledgments

SRB acknowledges the use of computational facilities at the Ohio Supercomputer Center [52] and The Ohio State University.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the US Department of Energy (DOE), Office of Science, Basic Energy Sciences, Chemical Sciences, Geosciences, and Biosciences Division, Gas Phase Chemical Physics program through Argonne National Laboratory under Contract DE-AC02-06CH11357.