Symmetry redundancy from molecular

There are N(N-l)/2 distinct distances in a cluster of N atoms, disregarding symmetry-dictated equivalencies. This set of distances is of course redundant 3N-6 Cartesian coordinates are sufficient to determine molecular geometry, apart from the position of the center of mass and the orientation of the principle moments of inertia. 

The conclusions from this rather elementary survey of the symmetry constraint problem all point in the same general direction. The imposition of symmetry constraints (other than the Pauli principle) on a variationally-based model is either unnecessary or harmful. Far from being necessary to ensure the physical reality of the wave function, these constraints often lead to absurd results or numerical instabilities in the implementation. The spin eigenfunction constraint is only realistic when the electrons are in close proximity and in such cases comes out of the UHF calculation automatically. The imposition of molecular spatial symmetry on the AO basis is not necessary if that basis has been chosen carefully — i.e. is near optimum. Further, any breakdowns in the spatial symmetry of the AO basis are a useful indication that the basis has been chosen badly or is redundant. 

Based on path counts, it may be also possible to define a zeroth order index, °K. It follows from our path count development that k should be derived from °P fragments, a quantity that is, in fact, the count of atoms in a molecular graph. One attribute of an atom that should influence the shape of a molecule is the topological uniqueness of that atom within the molecule. One collective effect of atom topological uniqueness is the symmetry or redundancy of the molecule. 

The coordinates (XpTpZj) are redundant since they ean be determined from the eondition that the X, Y, Z) axis system has origin at the molecular centre of mass. Obviously, the translational symmetry operation diseussed above has the effect of changing the centre of mass coordinates... 

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