Molecular potential permutation symmetry

HCCS radical, Renner-Teller effect, tetraatomic molecules, II electronic states, 633-640 H2D molecule, non-adiabatic coupling, two-state molecular system, 107-109 HD2 molecule, permutational symmetry isotopomers, 713-717 potential energy surfaces, 692-694 Heaviside function ... 

Potential fluid dynamics, molecular systems, modulus-phase formalism, quantum mechanics and, 265—266 Pragmatic models, Renner-Teller effect, triatomic molecules, 618-621 Probability densities, permutational symmetry, dynamic Jahn-Teller and geometric phase effects, 705-711 Projection operators, geometric phase theory, eigenvector evolution, 16-17 Projective Hilbert space, Berry s phase, 209-210... 

Just as permutational symmetry was not considered in the work of Bom and his collaborators, neither is it considered in the later work. With the choice of transla-tionally invariant coordinates made above, it is a simple matter to incorporate electronic permutational symmetry and, without any diminution of mathematical generality, to require that the electronic part of the wave function includes spin and be properly antisymmetric. On this understanding, it is perfectly reasonable to assume that the potential at the minimum should not be degenerate. If it seems sufficient to treat the nuclei as distinguishable particles, then it can confidently be asserted that the Bom-Oppenheimer approach offers a perfectly satisfactory account of molecular wave functions whose energy is close to a minimum in the potential. 

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. 

Since the related Hamiltonian needs to remain invariant under all the symmetry operations of the molecular symmetry (point) group, the potential energy expansion, see equation (5), may contain only those terms which are totally symmetric under all symmetry operations. Consequently, a simple group theoretical approach, based principally on properties of the permutation groups can be devised, " which yields the number and symmetry classification of anharmonic force constants. The burgeoning number of force constants at higher orders can be appreciated from the entries given in Table 4. 

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