Crude adiabatic basis

In Eq. (11b), we observed that since the crude adiabatic basis is used S = 0, for kzQ. Therefore the degeneracy is lifted at first order in the Q-space only, which is therefore used to identified the branching space. The first-order result is... 

For a conical intersection of states k, I at X , define a crude adiabatic basis ... 

Note that each point of conical intersection requires a distinct crude adiabatic basis. The eigenstates are expanded in the crude adiabatic basis... 

Apart from special models, Eq. (14) is satisfied only when the whole space of interacting electronic states is considered in the ADT matrix S (in diatomics, with only a single nuclear degree of freedom, there is no curl-condition). This, however, is contradicting the spirit of choosing a small subset of electronic states in the ADT matrix and would lead one back to the crude adiabatic basis discussed in the previous section. Therefore strictly diabatic electronic states, satisfying rigorously Eqs. (12) and (13) do not exist in the multidimensional case. ... 

Formally, the off-diagonal appearance of W" in the diabatic basis is similar to the crude adiabatic basis [as is the (nearly) diagonal form of Tjv]. However, it should be kept in mind that the form of Eq. (18) refers to a small electronic subspace only, whereas all other electronic states are treated in the adiabatic representation. By contrast, the matrix W of Eq. (8) in the... 

A connection between the crude adiabatic basis and the diabatic basis can be established, e.g. with the aid of quasi-degenerate perturbation the-ory.51-53 indices i and j refer to the subset (a) of strongly coupled... 

Suppose that the matrix elements Wy, Win, etc. are expanded in a Taylor series around the reference configuration Qo of the crude adiabatic basis (as is frequently done in standard JT theory and within the linear vibronic coupling scheme described in Chapter 7 of this book). Then the off-diagonal elements Wij (i j) and Wj are (at most) of first order in these displacements 6Q. Hence, the linear term in Wij cannot be affected by the coupling Wj between blocks (a) and (/ ) and we have ... 

Such a condition is obeyed, for instance, by a set of adiabatic states determined for a given molecular geometry Q and kept unchanged as the nuclei move. Such a set of states, the so-called crude adiabatic states, can be taken as a basis for the expansion of the adiabatic states at any other geometry Q. The expansion is expected to be very poorly convergent, so that the crude adiabatic basis is not useful in real calculations, but it is a good starting point for qualitative discussions of symmetry properties in spectroscopy. In the present context, the crude adiabatic basis provides an example of a trivially diabatic set of states. 

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