Born-Oppenheimer approximation symmetry operators

It is often convenient to use the symmetry coordinates that form the irreducible basis of the molecular symmetry group. This is because the potential-energy surface, being a consequence of the Born-Oppenheimer approximation and as such independent of the atomic masses, must be invariant with respect to the interchange of equivalent atoms inside the molecule. For example, the application of the projection operators for the irreducible representations of the symmetry point group D3h (whose subgroup... 

From the general considerations presented in the previous section, one can expect that the many-body non-adiabatic wave function should fulfill the following conditions (1) All particles involved in the system should be treated equivalently (2) Correlation of the motions of all the particles in the system resulting from Coulombic interactions, as well as from the required conservation of the total linear and angular momenta, should be explicitly incorporated in the wave function (3) Particles can only be distinguishable via the permutational symmetry (4) The total wave function should possess the internal and translational symmetry properties of the system (5) For fixed positions of nuclei, the wave functions should become equivalent to what one obtains within the Born-Oppenheimer approximation and (6) the wave function should be an eigenfunction of the appropriate total spin and angular momentum operators. 

Effects due to the geometric phase (GP) have been reported by Joubert-Doriol, Ryabinkin and Izmaylov.In particular they report on symmetry breaking and spatial localisation, and on GP effects studied with the multi-dimensional LVQ model. In the first study by Ryabinkin and Izmaylov the ground state dynamics is considered of a two-state system approximated by (a) a Hamiltonian of a two-state Cl model, (b) the Born-Oppenheimer (BO) model and (c) a BO model augmented with an explicit GP dependence in the kinetic energy operator. It is demonstrated that... 

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