Normalized irreducible tensorial matrices and the Wigner–Eckart theorem for unitary groups: A superposition hamiltonian constructed from octahedral NITM

Abstract

The concepts of normalized irreducible tensorial matrices (NITM) are extended to all finite and compact unitary groups by a development that clarifies their relationship to group theory and matrix algebra. NITM for a unitary group G are shown to be elements of a basis obtained by symmetry adapting to G the matrix basis of a matrix space M ( a , X a*). a z ) is said to be simple. A compound NITM basis of a matrix space results when the space is partitioned into two or more subspaces, each spanned by a simple NITM basis. NITM determined from Griffiths Vcoefficients for the octahedral group are tabulated and used to construct a six-coordinate superposition Hamiltonian.

Using Group Theory to Obtain Eigenvalues of Nonsymmetric Systems by Symmetry Averaging

“…On a symmetry-adapted basis, the elements of an operator, H sym , that commutes with G satisfy the relation [9]:…”

Symmetry-Adapted Bases of Matrix Spaces Applied to Quantum Chemistry

“…Then B(ω) can be written where For G a finite group, the original basis B(ω) can be symmetry-adapted to B(ω) by means of matrix basis elements e rs R of the Frobenius algebra A(G). 2,12 An element X∈A(G) is expressed on the matric basis according to where the matric basis elements multiply according to…”

“…2. When both representations Γ ω 1 and Γ ω ˆ2 are completely reduced, the NITM basis is a compound basis 2 Elements of the NITM[n r FR ] ω 1 ;F 1 R 1 ,ω 2 ;F 2 R 2 are zero except in the block identified by…”

“…1 The Hamiltonian may contain rectangular blocksssubmatricessinvariant to the symmetry group. 2 The set of matrices with m rows and n columns of complex numbers is an mn-dimensional Hilbert space.…”

“…If the system exhibits symmetry, the Hamiltonian matrix must commute with the corresponding group of unitary operations . The Hamiltonian may contain rectangular blockssubmatricesinvariant to the symmetry group . The set of matrices with m rows and n columns of complex numbers is an mn -dimensional Hilbert space.…”