Non-adiabatic coupling terms

Hence, in order to contract extended BO approximated equations for an N-state coupled BO system that takes into account the non-adiabatic coupling terms, we have to solve N uncoupled differential equations, all related to the electronic ground state but with different eigenvalues of the non-adiabatic coupling matrix. These uncoupled equations can yield meaningful physical... 

It is important to note that the non-adiabatic coupling terms have a direct effect on the momentum of the nuclei, which is the reason it is called a dynamic coupling. By substituting Eq. (B.13) in Eq. (B.9), we get... 

When the non-adiabatic coupling terms x and x are considered negligibly small and dropped from Eq. (B.15), we get the uncoupled approximate Schrbdinger equation... 

In a different field, location, and characteristics of ci s on diabatic potential surfaces have been recognized as essential for the evaluation of dynamic parameters, like non-adiabatic coupling terms, needed for the dynamic and... 

THE ELECTRONIC NON-ADIABATIC COUPLING TERM IN MOLECULAR SYSTEMS ... 

XIV. A Theoretic-Numeric Approach to Calculate the Electronic Non-Adiabatic Coupling Terms... 

The next question asked is whether there are any indications, from ab initio calculations, to the fact that the non-adiabatic transfonnation angles have this feature. Indeed such a study, related to the H3 system, was reported a few years ago [64]. However, it was done for circular contours with exceptionally small radii (at most a few tenths of an atomic unit). Similar studies, for circular and noncircular contours of much larger radii (sometimes up to five atomic units and more) were done for several systems showing that this feature holds for much more general situations [11,12,74]. As a result of the numerous numerical studies on this subject [11,12,64-75] the quantization of a quasi-isolated two-state non-adiabatic coupling term can be considered as established for realistic systems. 

In other words, the non-adiabatic coupling terms between P and Q states are all assumed to be zero. These requirements will later be reconsidered for a relaxed situation where these coupling terms are assumed to be not necessarily identically zero but small, that is, of the order s in regions of interest. 

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. 

In our introductory remarks, we said that this section would be devoted to model systems. Nevertheless it is important to emphasize that although this case is treated within a group of model systems this model stands for the general case of a two-state sub-Hilbert space. Moreover, this is the only case for which we can show, analytically, for a nonmodel system, that the restrictions on the D matrix indeed lead to a quantization of the relevant non-adiabatic coupling term. 

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