Adiabatic-to-diabatic transformation matrix

B. The Necessary Condition for Having a Solution for the Adiabatic-to-Diabatic Transformation Matrix... 

IV. The Adiabatic-to-Diabatic Transformation Matrix and the Line Integral Approach... 

Appendix A The Jahn-Teller Model and the Longuet-Higgins Phase Appendix B The Sufficient Conditions for Having an Analytic Adiabatic-to-Diabatic Transformation Matrix I. Orthogonality II. Analyticity... 

IV. THE ADIABATIC-TO-DIABATIC TRANSFORMATION MATRIX AND THE LINE INTEGRAL APPROACH... 

In Section IV.A, the adiabatic-to-diabatic transformation matrix as well as the diabatic potentials were derived for the relevant sub-space without running into theoretical conflicts. In other words, the conditions in Eqs. (10) led to a.finite sub-Hilbert space which, for all practical purposes, behaves like a full (infinite) Hilbert space. However it is inconceivable that such strict conditions as presented in Eq. (10) are fulfilled for real molecular systems. Thus the question is to what extent the results of the present approach, namely, the adiabatic-to-diabatic transformation matrix, the curl equation, and first and foremost, the diabatic potentials, are affected if the conditions in Eq. (10) are replaced by more realistic ones This subject will be treated next. 

Once there is an estimate for the error in calculating the adiabatic-to-diabatic tiansfomiation matrix it is possible to estimate the error in calculating the diabatic potentials. For this purpose, we apply Eq. (22). It is seen that the error is of the second order in , namely, of 0( ), just like for the adiabatic-to-diabatic transformation matrix. 

Obviously, the fact that the solution of the adiabatic-to-diabatic transformation matrix is only perturbed to second order makes the present approach rather attractive. It not only results in a very efficient approximation but also yields an estimate for the error made in applying the approximation. 

One of the main outcomes of the analysis so far is that the topological matrix D, presented in Eq. (38), is identical to an adiabatic-to-diabatic transformation matrix calculated at the end point of a closed contour. From Eq. (38), it is noticed that D does not depend on any particular point along the contour but on the contour itself. Since the integration is carried out over the non-adiabatic coupling matrix, x, and since D has to be a diagonal matrix with numbers of norm 1 for any contour in configuration space, these two facts impose severe restrictions on the non-adiabatic coupling terms. 

The derivation of the D matrix for a given contour is based on first deriving the adiabatic-to-diabatic transformation matrix, A, as a function of s and then obtaining its value at the end of the arbitrary closed contours (when s becomes io). Since A is a real unitary matrix it can be expressed in terms of cosine and sine functions of given angles. First, we shall consider briefly the two special cases with M = 2 and 3. 

We intend to show that an adiabatic-to-diabatic transformation matrix based on the non-adiabatic coupling matrix can be used not only for reaching the diabatic fi amework but also as a mean to determine the minimum size of a sub-Hilbert space, namely, the minimal M value that still guarantees a valid diabatization. 

In this section, diabatization is formed employing the adiabatic-to-diabatic transformation matrix A, which is a solution of Eq. (19). Once A is calculated, the diabatic potential matiix W is obtained from Eq. (22). Thus Eqs. (19) and (22) form the basis for the procedure to obtain the diabatic potential matrix elements. 

The matiix as well as the matrix ate called the Wigner matrices and are the subject of this section. Note that if we are interested in finding a relation between the adiabatic-to-diabatic transformation matrix and Wigner s matrices, we should mainly concentrate on the matiix. Wigner derived a fomiula for... 

It is expected that for a certain choice of paiameters (that define the x matrix) the adiabatic-to-diabatic transformation matrix becomes identical to the corresponding Wigner rotation matrix. To see the connection, we substitute Eq. (51) in Eq. (28) and assume A( o) to be the unity matrix. 

The main difference between the adiabatic-to-diabatic transformation and the Wigner matrices is that whereas the Wigner matiix is defined for an ordinary spatial coordinate the adiabatic-to-diabatic transformation matrix is defined for a rotation coordinate in a different space. 

APPENDIX B THE SUFFICIENT CONDITIONS FOR HAVING AN ANALYTIC ADIABATIC-TO-DIABATIC TRANSFORMATION MATRIX... 

In this appendix, we discuss the case where two components of Xm, namely, x p and XMg (p and q are Cartesian coordinates) are singular in the sense that at least one element in each of them is singular at the point B p = a,q = b) located on the plane formed by p and q. We shall show that in such a case the adiabatic-to-diabatic transformation matrix may become multivalued. 

In Section V.B, we discussed to some extent the 3x3 adiabatic-to-diabatic transformation matrix A(= for a tri-state system. This matrix was expressed in terms of three (Euler-type) angles Y,y,r = 1,2,3 [see Eq. (81)], which fulfill a set of three coupled, first-order, differential equations [see Eq. (82)]. 

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