Exact matrix elements for general two-body central-force interactions, expressed as sums of products

ABSTRACT

In a manner similar to but distinct from concurrent tensor efforts in electronic structure, it is shown that the Laplace transform can serve as a generator for a sum-of-products (SOP) form that allows one to write essentially any function of distance between two particles (i.e. any central force potential) as an exact two-body matrix. In particular, exact expressions for the Coulomb, Yukawa and long-range Ewald two-body operators are evaluated in a band-limited (Sinc function) basis. The resultant exact, full-basis, SOP representations for these interaction potentials – acting in conjunction with an external harmonic confining field – are validated via comparison with energy eigenstate solutions obtained via an independent calculation based on separation of variables. The new two-body matrix representations may have substantial impact in any of the many disciplines in which pair-wise central force interactions are relevant – especially, electronic structure and dynamics.

GRAPHICAL ABSTRACT

KEYWORDS:

• Sum of products
• Laplace transform
• Coulomb
• Yukawa
• long-range Ewald
• Hookean system

Acknowledgments

All authors acknowledge the High Performance Computing Center (HPCC) at Texas Tech University for providing HPC resources that have contributed to the research results reported within this paper [URL: http://cmsdev.ttu.edu/hpcc]. Poirier would also like to acknowledge Dieter Cremer, and the wonderful discussions they have had over the years.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Jonathan Jerke http://orcid.org/0000-0002-2449-4848

Jacek Karwowski http://orcid.org/0000-0003-1508-2929

Additional information

Funding

One author (Poirier) acknowledges support from The Robert A. Welch Foundation (D-1523).