# Linear-scaling electronic structure theory: Electronic temperature in the Kernel Polynomial Method

## TL;DR

This paper explores the Kernel Polynomial Method for electronic structure calculations, establishing its connection to electron temperature and comparing it with the Fermi operator expansion, enhancing large-scale atomistic simulation efficiency.

## Contribution

It formally connects the Kernel Polynomial Method with electron temperature and compares it to the Fermi operator expansion, providing insights for large-scale simulations.

## Key findings

- Kernel polynomial convolution acts as an effective electron temperature.
- Formal connection established between KPM and Fermi operator expansion.
- Application demonstrated on a tight-binding model.

## Abstract

Linear-scaling electronic structure methods based on the calculation of moments of the underlying electronic Hamiltonian offer a computationally efficient and numerically robust scheme to drive large-scale atomistic simulations, in which the quantum-mechanical nature of the electrons is explicitly taken into account. We compare the kernel polynomial method to the Fermi operator expansion method and establish a formal connection between the two approaches. We show that the convolution of the kernel polynomial method may be understood as an effective electron temperature. The results of a number of possible kernels are formally examined, and then applied to a representative tight-binding model.