# Efficient algorithm based on Liechtenstein method for computing exchange coupling constants using localized basis set

## TL;DR

This paper introduces an efficient algorithm for calculating exchange coupling constants using a localized basis set and a simplified energy integration method, significantly reducing computational cost while maintaining accuracy.

## Contribution

The authors adapt the Liechtenstein formula with a finite pole approximation for localized orbitals, improving computational efficiency for large-scale magnetic system simulations.

## Key findings

- Number of poles needed for 0.05 meV accuracy is ~60
- Compared distance dependence of J_ij with KKR formalism
- Algorithm reduces computational cost by an order of magnitude

## Abstract

For large-scale computation of the exchange coupling constants $J_{ij}$, we reconstruct the Liechtenstein formula for localized orbital representation and simplify the energy integrations by adopting the finite pole approximation of the Fermi function proposed by Ozaki [Phys. Rev. B 75, 035123 (2007)]. We calculate the exchange coupling constant $J_{\mathrm{1NN}}$ of the first-nearest-neighbor sites in body-centered-cubic Fe systems of various sizes to estimate the optimal computational parameters that yield appropriate values at the lowest computational cost. It is shown that the number of poles needed for a computational accuracy of 0.05 meV is determined as $\sim$ 60, whereas the number of necessary Matsubara poles needed to obtain similar accuracy, which was determined in previous studies, is on the order of 1000. Finally, we show $J_{ij}$ as a function of atomic distance, and compared it with one derived from Korringa-Kohn-Rostoker Green's function formalism. The distance profile of $J_{ij}$ derived by KKR formalism agrees well with that derived by our study, and this agreement supports the reliability of our newly derived formalism.