Cubic-scaling iterative solution of the Bethe-Salpeter equation for finite systems

TL;DR

This paper introduces a cubic-scaling iterative method for solving the Bethe-Salpeter equation in finite systems, enabling efficient and accurate calculation of neutral excitations in molecules with large basis sets.

Contribution

The authors develop a local basis set formulation and an iterative Haydock recursion scheme for the BSE, reducing computational complexity and allowing beyond Tamm-Dancoff approximation calculations.

Findings

Computational cost scales as O(N^3) with the number of atoms.

The method converges with a low number of recursion coefficients.

Runtime is dominated by O(N^2) Coulomb kernel operations for large systems.

Abstract

The Bethe-Salpeter equation (BSE) is currently the state of the art in the description of neutral electron excitations in both solids and large finite systems. It is capable of accurately treating charge-transfer excitations that present difficulties for simpler approaches. We present a local basis set formulation of the BSE for molecules where the optical spectrum is computed with the iterative Haydock recursion scheme, leading to a low computational complexity and memory footprint. Using a variant of the algorithm we can go beyond the Tamm-Dancoff approximation (TDA). We rederive the recursion relations for general matrix elements of a resolvent, show how they translate into continued fractions, and study the convergence of the method with the number of recursion coefficients and the role of different terminators. Due to the locality of the basis functions the computational cost of each iteration scales asymptotically as $O(N^3)$ with the number of atoms, while the number of iterations is typically much lower than the size of the underlying electron-hole basis. In practice we see that , even for systems with thousands of orbitals, the runtime will be dominated by the $O(N^2)$ operation of applying the Coulomb kernel in the atomic orbital representation