How to Convert SPME to P3M: Influence Functions and Error Estimates

Area of Science:

• Computational physics
• Molecular dynamics
• Electrostatics

Background:

• Particle mesh Ewald methods are crucial for simulating long-range interactions in large systems.
• Smooth particle mesh Ewald (SPME) and particle-particle particle mesh (P3M) are widely used but distinct implementations.
• Accurate error estimation is vital for efficient parameter selection in these methods.

Purpose of the Study:

• To demonstrate the mathematical equivalence between SPME and P3M methods.
• To derive a unified error estimation framework applicable to both methods.
• To provide practical guidance for optimizing simulation parameters.

Main Methods:

• Mathematical transformation between SPME and P3M formulations.
• Derivation of an error estimate for SPME based on P3M's energy-conserving scheme.
• Analytical subtraction of self-forces within particle mesh calculations.
• Approximation of the P3M lattice Green function with a closed-form expression.

Main Results:

• Explicit mathematical transformation linking SPME and P3M is presented.
• A novel error estimate for SPME, valid for all parameter values, is derived.
• Plots illustrating SPME force accuracy across various parameters are generated.
• A simplified, on-the-fly computable approximation for the P3M lattice Green function is introduced.

Conclusions:

• SPME and P3M methods are fundamentally interchangeable, simplifying error analysis.
• The derived error estimate facilitates efficient parameter space exploration for optimal accuracy and computational cost.
• Code conversion between SPME and P3M is straightforward, enhancing methodological flexibility.