https://doi.org/10.1140/epjp/s13360-026-07642-4
Regular Article
Approximate computation of all eigenvalues using sparse block-low-rank matrices in electronic structure calculations for aggregated molecular systems
1
Research Institute for Value-Added-Information Generation (VAiG), Japan Agency for Marine-Earth Science and Technology (JAMSTEC), 236-0001, Yokohama, Japan
2
Plasma Quantum Processes Unit, National Institute for Fusion Science, 509-5292, Toki, Japan
a
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Received:
10
March
2025
Accepted:
31
March
2026
Published online:
20
April
2026
Abstract
In electronic structure calculations (ESCs) to study various properties of materials, it is necessary to calculate all eigenvalues in such a way that their order is known. Therefore, a tridiagonalization method using the repeated block Householder transformation (HT) is generally employed to solve eigenvalue problems for discretized matrices. Although the matrix can be represented as a sparse matrix with high accuracy, it is treated as a dense matrix from the beginning because the sparse matrix is transformed into a dense matrix in a single block HT. The method based on dense matrix arithmetic requires a memory storage of 
and arithmetic operations of 
. In this study, a construction method of sparse BLR matrices for ESCs and a special block HT for the sparse BLR matrices are proposed to reduce them and provide a pre-screening method to search for promising materials by calculating approximately all the eigenvalues of ESCs. The proposed method realizes efficient arithmetic by determining the block size of sparse BLR matrices on the molecular unit under the assumption of an aggregated molecular system. The validity of the proposed method has been investigated for aggregated fullerene (
) molecules. It is confirmed that the accuracy of the band energy and the band gap calculated from the eigenvalues can be controlled by the tolerance 
for the low-rank approximation in the modified special block HT for sparse BLR matrices. Numerical experiments show that the memory complexity of the proposed method is between 
and depending on the molecular arrangements and the required accuracy. Even when memory is required, it is less than a tenth of that required for the conventional method with dense matrices. Moreover, the computational complexity of the proposed method appears to be 
(
), significantly reduced from (
) of the conventional method.
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© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2026
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
