Structure factor in Particle Mesh Ewald algorithm

1 September, 2021

This document aims to explain the ‘‘normalisation’’ of the structure factor $S_f$ in the Particle Mesh Ewald Algorithm. For background, one can consult this JCTC paper. According to Coulomb’s law, the exact electrostatic energy of a charge distribution $\rho(\bf r)$, periodic under translations $\bf n$, is $$ E = \frac{1}{2} \iint \rho(\bf r) \sum_{\bf n} \frac{\rho ( \bf r’ + \bf n)}{|\bf r - \bf r^\prime|} \mathrm{d^3}{\bf r’} \mathrm{d^3}{\bf r} $$ Albeit its simple form, \ref{eq:Coulomb} is hardly practical for those who seek to capture long-ranged effects in organic macromolecules, where the integration is made unaffordable by infinite periodic images of a discontinuous distribution. ...