Abstract
We propose an approach for approximating electrostatic charge distributions with a small number of point charges to optimally represent the original charge distribution. By construction, the proposed optimal point charge approximation (OPCA) retains many of the useful properties of point multipole expansion, including the same far-field asymptotic behavior of the approximate potential. A general framework for numerically computing OPCA, for any given number of approximating charges, is described. We then derive a 2-charge practical point charge approximation, PPCA, which approximates the 2-charge OPCA via closed form analytical expressions, and test the PPCA on a set of charge distributions relevant to biomolecular modeling. We measure the accuracy of the new approximations as the RMS error in the electrostatic potential relative to that produced by the original charge distribution, at a distance the extent of the charge distribution–the mid-field. The error for the 2-charge PPCA is found to be on average 23% smaller than that of optimally placed point dipole approximation, and comparable to that of the point quadrupole approximation. The standard deviation in RMS error for the 2-charge PPCA is 53% lower than that of the optimal point dipole approximation, and comparable to that of the point quadrupole approximation. We also calculate the 3-charge OPCA for representing the gas phase quantum mechanical charge distribution of a water molecule. The electrostatic potential calculated by the 3-charge OPCA for water, in the mid-field (2.8 Å from the oxygen atom), is on average 33.3% more accurate than the potential due to the point multipole expansion up to the octupole order. Compared to a 3 point charge approximation in which the charges are placed on the atom centers, the 3-charge OPCA is seven times more accurate, by RMS error. The maximum error at the oxygen-Na distance (2.23 Å ) is half that of the point multipole expansion up to the octupole order.
Highlights
Point multipole expansions are widely used to gain physical insight by providing a simplified expression for a complex distribution of sources of potential fields, such as electrostatic potential due to a charge distribution
For the charge distributions described in the Results section below, we found that setting DDr1{r2DD~R0=4 to determine the value of q in Eq (18), instead of the much more complex Eq (22), results in the electrostatic potential that is on average within 4% of the optimal K~2 optimal point charge approximation (OPCA) solution (figure 2(b))
We have introduced an alternative to the point multipole expansion– the optimal point charge approximation (OPCA)
Summary
Point multipole expansions are widely used to gain physical insight by providing a simplified expression for a complex distribution of sources of potential fields, such as electrostatic potential due to a charge distribution. Since each successive term in the multipole expansion decays more rapidly with distance than the previous term, the impact of high order terms becomes small in the far-field, i.e. at distances R such that R&R0, where R0 is the distance of the furthest charge from the expansion center. This property has allowed the point multipole expansion to simplify many practical calculations.
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