Theory of Symmetry Projections in Applied Quantum Mechanics

Publisher Website
Google Scholar
Abstract
The Wigner projection operator used in group theory for finding symmetry-adapted states in quantum mechanics is represented as a matrix in a given basis of trial state vectors. By diagonalizing this projection matrix, the redundancies occurring when the projection operator is applied directly are removed automatically. A straightforward method is given for this diagonalization procedure using only general properties of projection matrices. This method has proved very powerful in numerical applications using electronic computers. Particularly for the ligand-field treatment of a part of a crystal or in the case of a molecule of general point-group symmetry, the projection matrix to be used in the MO-LCAO method is constructed and simplified. This is done also for the nonsymmorphic space groups of crystals in the tight-binding approximation of band theory. The whole procedure of constructing symmetry-adapted states in this case has been programmed for an electronic computer.