A boundary integral method for computing electron repulsion integrals in heterogeneous dielectric environments
Abstract
We present a boundary integral method for evaluating electron repulsion integrals (ERIs) in dielectric environments of piecewise constant permittivity. This generalizes conventional bare-Coulomb ERIs by incorporating screening from structured dielectric surroundings, such as the substrates and/or encapsulating layers present in modern two-dimensional-material devices. We formulate the problem as a second-kind boundary integral equation for the induced surface charge on the dielectric interfaces, and discretize it with a Nyström method on adaptively refined panels. The incident potential generated by each smooth orbital source density is evaluated by a Fourier truncated-kernel method in or near to its source box, and by fast multipole method (FMM) accelerated quadrature farther away. The linear system is solved by a Krylov iteration also accelerated by the FMM. We focus on cuboidal device geometries, and retain efficiency at high aspect ratios via close-to-touching panel quadratures. We further develop a distributed pipeline that computes the full ERI tensor, comprising thousands of orbital pairs, embedded in a single dielectric geometry. We demonstrate the method on an effective three-layer graphene slab over a Si/SiO2 heterojunction, focusing on the central graphene layer represented by $198$ pz-like orbitals, retaining the 26132 non-negligible entries of the 1984-sized ERI tensor. The model is resolved on $1.3$ million surface unknowns, achieving a relative error of 0.004%, at a cost of $22$ node-hours.
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BibTeX
@misc{gao2026boundary,
title = {{A boundary integral method for computing electron repulsion integrals in heterogeneous dielectric environments}},
author = {Xuanzhao Gao and Malte Rösner and Jason Kaye and Alex Barnett},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26745},
eprint = {2610.26745},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26745},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A boundary integral method for computing electron repulsion integrals in heterogeneous dielectric environments
AU - Xuanzhao Gao
AU - Malte Rösner
AU - Jason Kaye
AU - Alex Barnett
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26745
UR - https://arxiv.org/abs/2610.26745
AB - We present a boundary integral method for evaluating electron repulsion integrals (ERIs) in dielectric environments of piecewise constant permittivity. This generalizes conventional bare-Coulomb ERIs by incorporating screening from structured dielectric surroundings, such as the substrates and/or encapsulating layers present in modern two-dimensional-material devices. We formulate the problem as a second-kind boundary integral equation for the induced surface charge on the dielectric interfaces, and discretize it with a Nyström method on adaptively refined panels. The incident potential generated by each smooth orbital source density is evaluated by a Fourier truncated-kernel method in or near to its source box, and by fast multipole method (FMM) accelerated quadrature farther away. The linear system is solved by a Krylov iteration also accelerated by the FMM. We focus on cuboidal device geometries, and retain efficiency at high aspect ratios via close-to-touching panel quadratures. We further develop a distributed pipeline that computes the full ERI tensor, comprising thousands of orbital pairs, embedded in a single dielectric geometry. We demonstrate the method on an effective three-layer graphene slab over a Si/SiO_2 heterojunction, focusing on the central graphene layer represented by $198$ p_z-like orbitals, retaining the 2613^2 non-negligible entries of the 198^4-sized ERI tensor. The model is resolved on $1.3$ million surface unknowns, achieving a relative error of 0.004%, at a cost of $22$ node-hours.
ER -