Automatic Generation of Matrix Element Derivatives for Tight Binding Models

TL;DR

This paper presents a method to automatically generate derivatives of matrix elements in tight binding models, facilitating dynamic simulations without manual derivative calculations.

Contribution

The authors derive general expressions for first and second derivatives of matrix elements and implement them in a program to automate derivative calculations in tight binding models.

Findings

Automated derivative generation reduces manual effort.

General expressions applicable to various tight binding scenarios.

Implementation improves efficiency in dynamic simulations.

Abstract

Tight binding (TB) models are one approach to the quantum mechanical many particle problem. An important role in TB models is played by hopping and overlap matrix elements between the orbitals on two atoms, which of course depend on the relative positions of the atoms involved. This dependence can be expressed with the help of Slater-Koster parameters, which are usually taken from tables. Recently, a way to generate these tables automatically was published. If TB approaches are applied to simulations of the dynamics of a system, also derivatives of matrix elements can appear. In this work we give general expressions for first and second derivatives of such matrix elements. Implemented in a computer program they obviate the need to type all the required derivatives of all occuring matrix elements by hand.