Splitting of One-Electron Levels in a Tetrahedral Environment

Abstract

The degeneracy of atomic orbitals is lifted when an atom is placed in an environment with no spherical symmetry. A well-studied illustration of this principle is the splitting of d-orbital energy levels in an octahedral environment, where group theory provides a sound method to find the splitting without resorting to explicitly solving the Schrodinger equations. Also common but less discussed cases include f orbitals in a tetrahedral environment. This paper shows that a well-established group theoretical approach relying only on the φ component of a wave function is inadequate to properly predict the splitting of levels in a tetrahedral symmetry, but is the correct way to deduce how degeneracy of atomic orbitals are lifted. A method that takes into account the θ component as well is shown to provide a basic means of obtaining the splitting of energy levels for l = odd orbitals in a tetrahedral symmetry.