Similarities and differences in the construction of dispersion laws for charge carriers in semiconductor crystals and adiabatic potentials in molecules

TL;DR

This paper compares the construction of dispersion laws for charge carriers in crystals with adiabatic potentials in molecules, highlighting key differences using group theory and invariants, exemplified by a molecule with D3d symmetry.

Contribution

It introduces a detailed comparison of the matrix construction methods for dispersion laws and vibronic potentials, emphasizing their differences in symmetry and physical context.

Findings

Differences in secular matrix construction for crystals and molecules.

Methodology for deriving dispersion laws near the Γ point.

Application to D3d symmetry crystal and molecule examples.

Abstract

Using the group theory and the method of invariants, it is shown how the vibronic potential can be written in a matrix form and the corresponding adiabatic potentials can be found. The molecule having $D_{3d}$ symmetry is considered herein as an example. The symmetries of normal vibrations active in Jahn-Teller's effect were defined. $E-E$ vibronic interaction was considered to obtain vibronic potential energy in a matrix form and thus the adiabatic potential. Significant differences are shown in the construction of a secular matrix $D(\vec{k})$ for defining a dispersion law for charge carriers in the crystals and the matrix of vibronic potential energy, which depends on the normal coordinates of normal vibrations active in Jahn-Teller's effect. Dispersion law of charge carriers in the vicinity of $\Gamma$ point of Brillouin zone of the crystal with $D_{3d}^2$ symmetry was considered as an example.