Wigner /?-function symmetry properties

Bouckaert, L. P., Smoluchowski, R. and Wigner E. (1936) Theory of Brillouin zones and symmetry properties of wave functions in crystals. Phys. Rev. 50, 58-67. 

An important use of vector coupling coefficients lies in the calculation of matrix elements of the operators in the vibronic Hamiltonian. Knowing the symmetry properties of the basis functions and of the operators, the ratio of the matrix elements can be deduced by inspection of the vector coupling coefficients. Without resorting to complicated formulae, a restricted use of the Wigner Eckart theorem may be illustrated as follows. First let us reduce Table 1 to those columns involving only the decomposition products of E symmetry (Table 2). 

The symmetry properties of the spatial wave functions of a Wigner super-multiplet cannot be simply described in the many-body case, because the function may be symmetric for spatial exchange of some pairs of particles and antisymmetric for exchange of other pairs. The supermultiplet may be characterised by four numbers with Xi = A the mass number, representing... 

Bohm, D., Pines, D. (1951). A collective description of electron interactions. 1. magnetic interactions. Phys. Rev. 82, 625 ibid. (1953). A collective description of electron interactions III. Coulomb interactions in a degenerate electron gas. 92, 609. Bouckaert, L. P, Smoluchowski, R., Wigner, E. (1936). Theory of briUouin zones and symmetry properties of wave functions in crystals. Phys. Rev. 50, 58-67 Callaway, J. (1961). Energy bands in lithium. Phys. Rev. 124,1824-1827. 

The Wigner d-functions are real and have the following symmetry properties [58] ... 

Wigner rotation/adiabatic-to-diabatic transformation matrices, 92 Electronic structure theory, electron nuclear dynamics (END) structure and properties, 326-327 theoretical background, 324-325 time-dependent variational principle (TDVP), general nuclear dynamics, 334-337 Electronic wave function, permutational symmetry, 680-682 Electron nuclear dynamics (END) degenerate states chemistry, xii-xiii direct molecular dynamics, structure and properties, 327 molecular systems, 337-351 final-state analysis, 342-349 intramolecular electron transfer,... 

A fundamental property of the wave function is that it can be used as basis for irreducible representations of the point group of a molecule [13], This property establishes the connection between the symmetry of a molecule and its wave function. The preceding statement follows from Wigner s theorem, which says that all eigenfunctions of a molecular system belong to one of the symmetry species of the group [14],... 

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