Space-fixed coordinates, permutational

Now, the permutation of the nuclei P 2 has no effect on the positions of the electrons measured in the space-fixed coordinate system. Thus for an electron at (X, Yt, Z, ) we have... 

When separating the centre of mass motion from the Schrodinger problem for a system, some apparently reasonable choices of space fixed coordinates have transformation properties under nuclear permutations that mix variables that are formally nuclear with those that are formally electronic variables. This renders the idea of a potential surface expressed in such coordinates, problematic. The problems are discussed and some solutions suggested. 2001 by Academic Press. 

For a system composed of Na atoms of type A, Nb atoms of type B, etc., the Hamiltonian is invariant to all permutations of equivalent nuclei and to inversion of all coordinates through a space-fixed origin. The number of permutation-inversion isomers of any given configuration could therefore be as large as... 

The symmetry group of the molecular Hamiltonian consisting of all permutations of identical nuclei, inversion of all coordinates through a space-fixed origin, and all combinations thereof. 

For a secure account to be given in terms of the separation (O Eq. 2.43), which is what is really required if one is to use the clamped nuclei electronic Hamiltonian, it would be necessary to consider more than one coordinate space. On the manifold at least two coordinate spaces are required to span the whole manifold. The internal coordinates within any coordinate space are such that it is possible to construct two distinct molecular geometries at the same internal coordinate specification, so that a potential expressed in the internal coordinates cannot be analytic everywhere (CoUins and Parsons 1993). It would therefore seem to be a very tricky job. But even if it were to be accomplished it seems very unlikely that a multiple minima argument could be constructed to account for point group symmetry in this context It is possible to show (see Section IV of Sutcliffe (2000)) that in the usual Eckart form of the Hamiltonian for nudear motion, permutations can be such as to cause the body-fixed frame definition to fail completely. 

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