Length-Gauge Optical Matrix Elements in WIEN2k

Oleg Rubel,Peter Blaha

doi:10.3390/computation10020022

Abstract

Hybrid exchange-correlation functionals provide superior electronic structure and optical properties of semiconductors or insulators as compared to semilocal exchange-correlation potentials due to admixing a portion of the non-local exact exchange potential from a Hartree–Fock theory. Since the non-local potential does not commute with the position operator, the momentum matrix elements do not fully capture the oscillator strength, while the length-gauge velocity matrix elements do. So far, length-gauge velocity matrix elements were not accessible in the all-electron full-potential WIEN2k package. We demonstrate the feasibility of computing length-gauge matrix elements in WIEN2k for a hybrid exchange-correlation functional based on a finite difference approach. To illustrate the implementation we determined matrix elements for optical transitions between the conduction and valence bands in GaAs, GaN, (CH3NH3)PbI3 and a monolayer MoS2. The non-locality of the Hartree–Fock exact exchange potential leads to a strong enhancement of the oscillator strength as noticed recently in calculations employing pseudopotentials (Laurien and Rubel: arXiv:2111.14772 (2021)). We obtained an analytical expression for the enhancement factor for the difference in eigenvalues not captured by the kinetic energy. It is expected that these results can also be extended to other non-local potentials, e.g., a many-body GW approximation.

Highlights

Calculations of linear optical properties of solids require matrix elements for electric dipole transitions
The local potential is selected here since both the lengthand velocity-gauge should lead to identical results under these circumstances
It is important to get a feeling for the step size q at which the finite difference approximation converges to the accurate result given by the momentum matrix element

Summary

Introduction

Calculations of linear optical properties of solids require matrix elements for electric dipole transitions. Momentum matrix elements pmn (k) = hm, k| − i ∇r |n, ki Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Are widely used in full-potential codes with periodic boundary conditions [1] when optical properties are computed with local potentials (e.g., LDA (see end of the paper for the full list of abbreviations) or GGA XC functionals) and referred to in the literature as a velocity gauges. Starace [2] emphasised the limitations of Equation (1) when representing matrix elements for electric dipole transitions. The more general velocity matrix elements should be used vmn (k) = hm, k|i [ Ĥ, r]|n, ki (2).

Methods

Results

Discussion

Conclusion