# Real-space quadrature: A convenient, efficient representation for multipole expansions

## Abstract

Multipoles are central to the theory and modeling of polarizable and nonpolarizable molecular electrostatics. This has made a representation in terms of point charges a highly sought after goal, since rotation of multipoles is a bottleneck in molecular dynamics implementations. All known point charge representations are orders of magnitude less efficient than spherical harmonics due to either using too many fixed charge locations or due to nonlinear fitting of fewer charge locations. We present the first complete solution to this problem—completely replacing spherical harmonic basis functions by a dramatically simpler set of weights associated to fixed, discrete points on a sphere. This representation is shown to be space optimal. It reduces the spherical harmonic decomposition of Poisson’s operator to pairwise summations over the point set. As a corollary, we also shows exact quadrature-based formulas for contraction over trace-free supersymmetric 3D tensors. Moreover, multiplication of spherical harmonic basis functions translates to a direct product in this representation.

- Authors:

- University of South Florida, 4202 E. Fowler Ave., CHE 205, Tampa, Florida 33620 (United States)

- Publication Date:

- OSTI Identifier:
- 22416134

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Chemical Physics; Journal Volume: 142; Journal Issue: 7; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; COMPUTERIZED SIMULATION; DECOMPOSITION; ELECTROSTATICS; EXPANSION; MATHEMATICAL SOLUTIONS; MOLECULAR DYNAMICS METHOD; MULTIPOLES; NONLINEAR PROBLEMS; POINT CHARGE; QUADRATURES; ROTATION; SPACE; SPHERES; SPHERICAL CONFIGURATION; SPHERICAL HARMONICS; SUPERSYMMETRY; TENSORS

### Citation Formats

```
Rogers, David M., E-mail: davidrogers@usf.edu.
```*Real-space quadrature: A convenient, efficient representation for multipole expansions*. United States: N. p., 2015.
Web. doi:10.1063/1.4907404.

```
Rogers, David M., E-mail: davidrogers@usf.edu.
```*Real-space quadrature: A convenient, efficient representation for multipole expansions*. United States. doi:10.1063/1.4907404.

```
Rogers, David M., E-mail: davidrogers@usf.edu. Sat .
"Real-space quadrature: A convenient, efficient representation for multipole expansions". United States.
doi:10.1063/1.4907404.
```

```
@article{osti_22416134,
```

title = {Real-space quadrature: A convenient, efficient representation for multipole expansions},

author = {Rogers, David M., E-mail: davidrogers@usf.edu},

abstractNote = {Multipoles are central to the theory and modeling of polarizable and nonpolarizable molecular electrostatics. This has made a representation in terms of point charges a highly sought after goal, since rotation of multipoles is a bottleneck in molecular dynamics implementations. All known point charge representations are orders of magnitude less efficient than spherical harmonics due to either using too many fixed charge locations or due to nonlinear fitting of fewer charge locations. We present the first complete solution to this problem—completely replacing spherical harmonic basis functions by a dramatically simpler set of weights associated to fixed, discrete points on a sphere. This representation is shown to be space optimal. It reduces the spherical harmonic decomposition of Poisson’s operator to pairwise summations over the point set. As a corollary, we also shows exact quadrature-based formulas for contraction over trace-free supersymmetric 3D tensors. Moreover, multiplication of spherical harmonic basis functions translates to a direct product in this representation.},

doi = {10.1063/1.4907404},

journal = {Journal of Chemical Physics},

number = 7,

volume = 142,

place = {United States},

year = {Sat Feb 21 00:00:00 EST 2015},

month = {Sat Feb 21 00:00:00 EST 2015}

}