Fast multipole method for three-dimensional systems with periodic boundary condition in two directions

Area of Science:

• Computational physics
• Materials science
• Physical chemistry

Background:

• Calculating electrostatic interactions in periodic systems is crucial for molecular simulations.
• Traditional methods like Ewald summation can be computationally intensive, especially for slab geometries.
• Existing methods may struggle with large empty spaces in unit cells.

Purpose of the Study:

• To derive a new, computationally efficient expression for electrostatic interactions in 3D charge-neutral systems under 2D periodic boundary conditions.
• To implement and validate this new expression using the Fast Multipole Method (FMM).
• To demonstrate the method's superiority over conventional techniques for slab geometries.

Main Methods:

• Developed a novel analytical expression for electrostatic interactions incorporating real and reciprocal space terms, plus a self-interaction term.
• Applied the Fast Multipole Method (FMM) to efficiently sum contributions from all image cells.
• Validated the method by calculating electrostatic interactions for a model charge distribution and liquid water from molecular dynamics.

Main Results:

• The new expression accurately calculates electrostatic interactions in slab geometries.
• The FMM implementation achieves an O(N) computational complexity, where N is the number of particles.
• The accuracy is controllable via the FMM expansion degree.

Conclusions:

• The derived FMM-based expression provides a computationally superior alternative for electrostatic interactions in 2D periodic systems.
• This method is more efficient than conventional 2D periodic Ewald and particle mesh Ewald methods for slab geometries.
• The approach offers a scalable and accurate solution for large-scale molecular simulations.