Representation generalized permutation

For a set of equivalent nuclei in general site the matrices 11(G) are identical with the right regular representation matrices2 lf the nuclear position vectors of all K nuclei of a SRM are included in the basis Xkd), 11(G) denotes a K by K permutation matrix. In addition to the matrix groups (2.49) and (2.49 ) the set... 

The permutational behaviour of states having Si[Pg.112]

In this particular case, the point group can be described as a set of permutations of the labels 0,1,2,3 of the comers of the tetrahedron or as symmetry operations of this polyhedron. However, we should carefully note that the representation of symmetry operations as permutations of comers, or atoms, is not generally unique. For example, in the case of a planar molecule both the reflection in the plane and the identity mapping are represented by the identity permutation, although they are different symmetry operations. consists of all 24 permutations of the corners of the tetrahedron,... 

The most well known invariants describing atomic neighbourhoods are the set of bond-order parameters proposed by Steinhardt et al. [4]. These have been successfully used as order parameters in studies of nucleation [5], phase transitions [6] and glasses [7]. In the following sections we show that the bond-order parameters actually form a subset of a more general set of invariants called the bispectrum. We prove that the bispectrum components indeed form a rotational and permutational invariant representation of atomic environments. The formally... 

The general problem is now clear the quantities i /,. p are tensor components, with respect to the group U(m), and we want to find linear combinations of these components that will display particular symmetries under electron permutations and hence under index permutations. Each set of symmetrized products, with a particular index symmetry, will provide a basis for constructing spin-free CFs (as in Section 7.6) for states of given spin multiplicity and in this way the full-CI secular equations will be reduced into the desired block form, each block corresponding to an irreducible representation of U(m). It is therefore necessary to study both groups U(m), which describes possible orbital transformations, and which provides a route (via the Young tableaux of Chapter 4) to the construction of rank-N tensors of particular symmetry type with reject to index permutations. 

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