3. Electrostatic interactions

In the filtered Hamiltonian hPF formalism [Bore and Cascella, 2020] the particles are intrinsically smeared, filtered density distributions. The electrostatic interactions between such smeared densities takes on the form of the long-range part of ordinary particle–mesh Ewald.

The charge density is projected onto the computational grid by use of the CIC window function, and subsequently filtered

\[\tilde\rho =\int\mathrm{d}\mathbf{x}\,H(\mathbf{r}-\mathbf{x})\sum_{i=1}^Nq_i P(\mathbf{r}-\mathbf{r}_i),\]

with \(q_i\) being the charge of particle \(i\) and \(H\) is the filtering function (see Filtering). The value of the charge grid at vertex \((i,j,k)\) is found by

\[\tilde\rho_{ijk}=\text{FFT}^{-1}[\text{FFT}(\rho)\text{FFT}(H)]\]

and the electrostatic potential \(\Psi_{ijk}\)

\[\Psi_{ijk}=\text{FFT}^{-1}\left[\frac{k_\text{e}}{|\mathbf{k}^2|}\text{FFT}(\rho)\text{FFT}(H)\right],\]

where \(k_\text{e}\) is the Coulomb constant \(1/4\pi\varepsilon_0\). The electric field is obtained by differentiation of the electrostatic potential in Fourier space,

\[\mathbf{E}_{ijk}=\text{FFT}^{-1}[-i\mathbf{k}\text{FFT}(\Psi)]\]

from which the forces are interpolated back to the particle positions.

3.1. Specifying electrostatics

In HyMD, electrostatics are specified by the coulombtype and dielectric_const keywords in the configuration file (see Configuration file) and the /charge dataset in the HDF5 format structure/topology input file (see Topology and structure file). In addition, the helical propensity peptide dihedral type induces topological reconstruction of peptide dipoles which adds electrostatic interactions.