Relative motion, energy eigenvalue

Whereas the group jr and its representations are relevant and sufficient for problems which are completely defined by relative nuclear configurations (RNCs) of a SRM, primitive period isometric transformations have to be considered as nontrivial symmetry operations in all those applications where the orientation of the NC w.r.t. the frame and laboratory coordinate system is relevant, e.g. the rotation-internal motion energy eigenvalue problem of a SRM. Inclusion of such primitive period operations leads to the internal isometric group ( ) represented faithfully by... 

Electrons are much lighter than nuclei and move much faster. Thus the electrons can adjust to the movement of the nuclei which means that the electronic states are at any moment essentially the same as if the nuclei were fixed. This is the basis of the Born-Oppenheimer approximation which assumes fixed nuclei. The wave function can be expressed as a product of an electronic wavefunction with the nuclei assumed fixed and a nuclear wavefunction describing the relative nuclear motion. Energy eigenvalues for the electronic Schrodinger equation, solved for different nuclear separations, form a potential, that is inserted into the nuclear Schrodinger equation together with the nuclear repulsion term. 

The energy eigenvalues for the relative motion are the same as in the Bohr theory. 

For most systems, where the velocity of motion of the nuclei is slow relative to the electron velocity, this decoupling of electronic and nuclear motion is valid and is referred to as the adiabatic approximation. Equation (II.3) thus defines an electronic eigenstate (rn,Rv), parametric in the nuclear coordinates, and a corresponding eigenvalue Ek(RN) that is taken to represent the potential-energy curve or surface corresponding to state k. 

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