Born-Oppenheimer group- adiabatic approximation

Note that the potential matrix V( (R) is a diagonal matrix by definition, in contrast to W(s)(R). Again, in analogy to the common treatment in Sec. 3.1, we call (23) the group-Born Oppenheimer adiabatic approximation or briefly the group-adiabatic approximation. This approximation assumes that the states within the manifold g are much stronger coupled to each other — e.g. via the presence of a conical intersection of the potential surfaces — than to the rest of the electronic space. 

It is well known that the Born-Oppenheimer adiabatic approximation establishes geometrical shape of a molecule. The atoms (atomic groups) constituting a molecule are imagined to be placed in the vertices of certain three-dimensional (3D) shape as illustrated in Figure 9.1. 

The gauge invariance of the group-Born-Oppenheimer approximation provides a good starting point to discuss diabatic states. In contrast to Eq. (21a), where this approximation is formulated in the adiabatic electronic basis, Eq. (26) is expressed in an arbitrary basis. Elimination of the derivative couplings appearing in the latter equation amounts to setting to zero the left hand side of Eq. (27b) ... 

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