Born-Oppenheimer approximation symmetry properties

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. 

Qualitative information about molecular wave functions and properties can often be obtained from the symmetry of the molecule. By the symmetry of a molecule, we mean the symmetry of the framework formed by the nuclei held fixed in their equilibrium positions. (Our starting point for molecular quantum mechanics will be the Born-Oppenheimer approximation, which regards the nuclei as fixed when solving for the electronic wave function see Section 13.1.) The symmetry of a molecule can differ in different electronic states. For example, HCN is linear in its ground electronic state, but nonlinear in certain excited states. Unless otherwise specified, we shall be considering the symmetry of the ground electronic state. 

---