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The Finite Coulomb Lattice Sum: A Resolution of Conditional Convergence through Exact Shape and Size
Yang He1,2, Zhonghan Hu1,2
1Key Laboratory of Laser & Infrared System of Ministry of Education, Shandong University, Qingdao 266237, P. R. China.
The Journal of Physical Chemistry Letters
|June 24, 2026
Summary
This study decomposes Coulomb lattice sums into bulk, boundary, and finite-size terms. This provides a rigorous framework for analyzing electrostatic interactions in finite systems.
Area of Science:
- Condensed matter physics
- Computational physics
- Electrostatics
Background:
- Conditionally convergent series, such as Coulomb lattice sums, pose challenges in periodic boundary condition calculations.
- Accurate calculation of electrostatic interactions is crucial for understanding material properties.
Purpose of the Study:
- To rigorously decompose conditionally convergent Coulomb lattice sums.
- To analyze the components of a finite lattice sum under periodic boundary conditions.
- To compare the new formulation with existing methods in the literature.
Main Methods:
- Decomposition of the Coulomb lattice sum into three distinct terms: periodic bulk, shape-dependent nonperiodic boundary, and finite-size correction.
- Parametrization of finite lattice geometry by its exact shape and size.
- Analysis and comparison with existing derivations of lattice sums.
Main Results:
- The finite lattice sum cleanly decomposes into periodic bulk (νpbc), shape-dependent nonperiodic boundary (νb), and finite-size correction (νcorr) terms.
- The formulation provides an effective pairwise form that explicitly parametrizes lattice geometry.
- Detailed analysis and comparison with literature derivations are presented.
Conclusions:
- The developed finite lattice sum offers a rigorous and effective method for calculating electrostatic interactions in finite systems.
- This framework facilitates analytical developments for arbitrarily shaped crystals.
- It paves the way for numerical mesh-type algorithms in condensed matter simulations.
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