Revisiting infinite lattice sums with the periodic fast multipole method

Area of Science:

• Computational physics
• Mathematical methods

Background:

• The periodic fast multipole method (PFM) is crucial for simulating large-scale periodic systems.
• Accurate and efficient computation of lattice sums is essential for PFM performance.
• Existing methods for lattice sums can be computationally intensive or lack sufficient accuracy.

Purpose of the Study:

• To reinvestigate the evaluation of lattice sums and stress lattice sums within the PFM framework.
• To derive simple, accurate, and efficient recurrence expressions for these sums.
• To provide practical guidance on the computation of lattice sums, including convergence and accuracy considerations.

Main Methods:

• The renormalization method is adapted to derive new recurrence relations for lattice sums.
• Lattice sums and stress lattice sums are evaluated using the derived recurrence expressions.
• Numerical computation of the first few nonzero lattice sum terms in a 3D cubic lattice.

Main Results:

• Novel recurrence expressions for lattice sums and stress lattice sums are presented.
• The derived expressions offer improved simplicity, accuracy, and efficiency.
• The first few nonzero lattice sum terms for a 3D cubic lattice are computed and tabulated.

Conclusions:

• The developed recurrence expressions provide a more efficient and accurate approach to lattice sum evaluation in PFM.
• The findings facilitate improved performance and reliability of simulations using the periodic fast multipole method.
• Practical considerations discussed aid researchers in optimizing lattice sum computations.