This work presents an accurate and efficient approach to the calculation of long-range interactions for molecular modeling and simulation. This method defines a local region for each particle and describes the remaining region as images of the local region statistically distributed in an isotropic and periodic way, which we call isotropic periodic images. Different from lattice sum methods that sum over discrete lattice images generated by periodic boundary conditions, this method sums over the isotropic periodic images to calculate long-range interactions, and is referred to as the isotropic periodic sum (IPS) method. The IPS method is not a lattice sum method and eliminates the need for a reciprocal space sum. Several analytic solutions of IPS for commonly used potentials are presented. It is demonstrated that the IPS method produces results very similar to that of Ewald summation, but with three major advantages, (1) it eliminates unwanted symmetry artifacts raised from periodic boundary conditions, (2) it can be applied to potentials of any functional form and to fully and partially homogenous systems as well as finite systems, and (3) it is more computationally efficient and can be easily parallelized for multiprocessor computers. Therefore, this method provides a general approach to an efficient calculation of long-range interactions for various kinds of molecular systems.

Isotropic periodic sum (IPS) is a method for the calculation of long-range interactions in molecular simulation based on the homogeneity of simulation systems. Three IPS models, 3D IPS, 2D IPS, and 1D IPS have been developed for three common types of homogeneous systems. Based on the fact that 3D IPS can well describe the long-range interactions of a heterogeneous system if a local region larger than the homogeneity scale is used, this work presents a method based on 3D IPS to calculate long-range interactions for all kinds of simulation systems, including homogeneous, heterogeneous, and finite systems. Unlike the original 3D IPS method that uses a local region defined by the cutoff distance, this method uses a local region larger than that defined by the cutoff distance to reach the homogeneity scale. To efficiently calculate interactions within such a large local region, this method split long-range interactions into two parts, a cutoff part and a long-range part. The cutoff part is calculated by summing over atom pairs within a cutoff range (about $10\mathrm{\AA }$ ), and the long-range part is calculated using the discrete fast Fourier transform (DFFT) technique. This method is applied to electrostatic and van der Waals interactions for both periodic and non-periodic systems. Example simulations demonstrate that this method can accurately and efficiently calculate long-range interactions for molecular simulation.

Isotropic periodic sum (IPS) is a method to calculate long-range interactions based on homogeneity of simulation systems. Long-range interactions are represented by interactions with isotropic periodic images of a defined local region and can be reduced to short ranged IPS potentials. The original electrostatic three-dimensional (3D)-IPS potential was derived based on a nonpolar homogeneous approximation and its application is limited to nonpolar or weak polar systems. This work derived a polar electrostatic 3D-IPS potential based on polar interactions. For the convenience of application, polynomial functions with rationalized coefficients are proposed for electrostatic and Lennard-Jones 3D-IPS potentials. Model systems of various polarities and several commonly used solvent systems are simulated to evaluate the 3D-IPS potentials. It is demonstrated that for polar systems the polar electrostatic 3D-IPS potential has much improved accuracy as compared to the nonpolar 3D-IPS potential. For homogeneous systems, the polar electrostatic 3D-IPS potential with a local region radius or cutoff distance of as small as 10 Å can satisfactorily reproduce energetic, structural, and dynamic properties from the particle-meshed-Ewald method. For both homogeneous and heterogeneous systems, the 3D-IPS/discrete fast Fourier transform method using either the nonpolar or the polar electrostatic 3D-IPS potentials results in very similar simulation results.

This work derives a quantitative description of the conformational distribution in self-guided Langevin dynamics (SGLD) simulations. SGLD simulations employ guiding forces calculated from local average momentums to enhance low-frequency motion. This enhancement in low-frequency motion dramatically accelerates conformational search efficiency, but also induces certain perturbations in conformational distribution. Through the local averaging, we separate properties of molecular systems into low-frequency and high-frequency portions. The guiding force effect on the conformational distribution is quantitatively described using these low-frequency and high-frequency properties. This quantitative relation provides a way to convert between a canonical ensemble and a self-guided ensemble. Using example systems, we demonstrated how to utilize the relation to obtain canonical ensemble properties and conformational distributions from SGLD simulations. This development makes SGLD not only an efficient approach for conformational searching, but also an accurate means for conformational sampling.