Infinite Boundary Terms and Pairwise Interactions: A Unified Framework for Periodic Coulomb Systems
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2025
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| _version_ | 1866917300108525568 |
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| author | Zhao, Yihao Hu, Zhonghan |
| author_facet | Zhao, Yihao Hu, Zhonghan |
| contents | The introduction of the infinite boundary terms and the pairwise interactions [J. Chem. Theory Comput., 10, 5254, (2014)] enables a physically intuitive approach for deriving electrostatic energy and pressure for both neutral and non-neutral systems under the periodic boundary condition (PBC). For a periodic system consisting of $N$ point charges (with charge $q_j$ located at ${\mathbf r}_j$ where $j=1,2,\cdots N$) and one charge distribution of density $ρ({\mathbf r})$ within a primary cell of volume $V$, the derived electrostatic energy can be expressed as, \[ {\mathcal U} = \sum_{i<j}^N q_iq_jν({\mathbf r}_{ij} ) + \sum_{j=1}^N q_j \int_V d{\mathbf r}_0\,ρ({\mathbf r}_0) ν({\mathbf r}_{0j} ) + \frac{1}{2}\int_V d{\mathbf r}_0 \int_V d{\mathbf r}_1\,ρ({\mathbf r}_0)ρ({\mathbf r}_1) ν({\mathbf r}_{01}), \] where ${\mathbf r}_{ij}={\mathbf r}_i - {\mathbf r}_j$ is the relative vector and $ν({\mathbf r})$ represents the effective pairwise interaction under PBC. The charge density $ρ({\mathbf r})$ is free of Delta-function-like divergence throughout the volume but may exhibit discontinuity. This unified formulation directly follows that of the isolated system by replacing the Coulomb interaction $1/\lvert {\mathbf r} \rvert$ or other modified Coulomb interactions with $ν({\mathbf r})$. For a particular system of one-component plasma with a uniform neutralizing background, the implementation of various pairwise formulations clarifies the contribution of the background and subsequently reveals criteria for designing volume-dependent potentials that preserve the simple relation between energy and pressure. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_06023 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Infinite Boundary Terms and Pairwise Interactions: A Unified Framework for Periodic Coulomb Systems Zhao, Yihao Hu, Zhonghan Computational Physics Chemical Physics Plasma Physics The introduction of the infinite boundary terms and the pairwise interactions [J. Chem. Theory Comput., 10, 5254, (2014)] enables a physically intuitive approach for deriving electrostatic energy and pressure for both neutral and non-neutral systems under the periodic boundary condition (PBC). For a periodic system consisting of $N$ point charges (with charge $q_j$ located at ${\mathbf r}_j$ where $j=1,2,\cdots N$) and one charge distribution of density $ρ({\mathbf r})$ within a primary cell of volume $V$, the derived electrostatic energy can be expressed as, \[ {\mathcal U} = \sum_{i<j}^N q_iq_jν({\mathbf r}_{ij} ) + \sum_{j=1}^N q_j \int_V d{\mathbf r}_0\,ρ({\mathbf r}_0) ν({\mathbf r}_{0j} ) + \frac{1}{2}\int_V d{\mathbf r}_0 \int_V d{\mathbf r}_1\,ρ({\mathbf r}_0)ρ({\mathbf r}_1) ν({\mathbf r}_{01}), \] where ${\mathbf r}_{ij}={\mathbf r}_i - {\mathbf r}_j$ is the relative vector and $ν({\mathbf r})$ represents the effective pairwise interaction under PBC. The charge density $ρ({\mathbf r})$ is free of Delta-function-like divergence throughout the volume but may exhibit discontinuity. This unified formulation directly follows that of the isolated system by replacing the Coulomb interaction $1/\lvert {\mathbf r} \rvert$ or other modified Coulomb interactions with $ν({\mathbf r})$. For a particular system of one-component plasma with a uniform neutralizing background, the implementation of various pairwise formulations clarifies the contribution of the background and subsequently reveals criteria for designing volume-dependent potentials that preserve the simple relation between energy and pressure. |
| title | Infinite Boundary Terms and Pairwise Interactions: A Unified Framework for Periodic Coulomb Systems |
| topic | Computational Physics Chemical Physics Plasma Physics |
| url | https://arxiv.org/abs/2504.06023 |