Symmetry ascent eigenvectors in finite point groups and an expanded matrix element selection rule

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Abstract
A formalism is proposed in which wave-functions quantized in a finite point group may be unambiguously labelled by their relative behaviour under the mapping of individual components from the generic point group R 3. Such a mapping is only possible into highly symmetric finite groups including O h and D 6h . Mapping to lower point groups by conventional symmetry descent creates ambiguities which can be removed by retaining the effect of discriminating virtual operators as parity labels for components. With such labelled wave functions, the formation of unambiguous direct products is possible with the introduction of Symmetry Ascent V Coefficients. By quantizing the wave functions about the desired n-fold axis in complex space, a commutative set of components is obtained. This allows component combination rules similar to those for 3-j symbols to be stated, modified to accommodate the possible mappings in finite groups and to retain the effect of parity. Hamiltonian operators in complex tensor form are treated similarly. The spin-orbit matrix elements for finite groups can thus be written in fully labelled form which when expanded as a scalar product of elements reflects all of the relevant selection rules pertaining to both the representations and components. With the violation of any one such rule, the matrix element vanishes. The electronic symmetry of these systems therefore is higher than that implied by the molecular geometry. The formalism further implies that during descent in symmetry, the number of selection rules for matrix elements can only increase.

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