Diabatic potential matrix

In this section, we prove that the non-adiabatic matiices have to be quantized ( similar to Bohr-Sommerfeld quantization of the angulai momentum) in order to yield a continous, uniquely defined, diabatic potential matrix W(i). In another way, the extended BO approximation will be applied only to those cases that fulfill these quantization rules. The ADT matrix A(s,so) transforms a given adiabatic potential matiix u(i) to a diabatic matiix W(s, so)... 

However, in order to obtain a uniquely defined diabatic potential matrix, it is not necessary for the A matiix to be uniquely defined throughout CS. Still, we ignore this difficulty and go ahead to derive A by a direct integration of Eq. (62),... 

For a two-state system, the eigenfunctions of the diabatic potential matrix of Eq. (63) in terms of its elements are... 

In Section III.D, we shall investigate when this happens. For the moment, imagine that we are at a point of degeneracy. To find out the topology of the adiabatic PES around this point, the diabatic potential matrix elements can be expressed by a hrst order Taylor expansion. 

For diabatic calculations, the equivalent expression uses the diabatic potential matrix elements [218]. When the value of this coupling becomes greater than a pre-defined cutoff, the tiajectory has entered a non-adiabatic region. The propagation is continued from this time, ti, until the trajectoiy moves out of the region at time f2-... 

H3 (and its isotopomers) and the alkali metal triiners (denoted generally for the homonuclears by X3, where X is an atom) are typical Jahn-Teller systems where the two lowest adiabatic potential energy surfaces conically intersect. Since such manifolds of electronic states have recently been discussed [60] in some detail, we review in this section only the diabatic representation of such surfaces and their major topographical details. The relevant 2x2 diabatic potential matrix W assumes the fomi... 

XI. The Necessary Conditions for a Rigorous Minimal Diabatic Potential Matrix... 

With these definitions we can now look for the necessary condition(s). Thus, we assume that at each point sq iu configuration space the diabatic potential matrix W(/l) [= W(s,so)] fulfills the condition ... 

XI. THE NECESSARY CONDITIONS FOR A RIGOROUS MINIMAL DIABATIC POTENTIAL MATRIX... 

For example one forms, within a two-dimensional (2D) sub-Hilbert space, a 2x2 diabatic potential matrix, which is not single valued. This implies that the 2D transformation matrix yields an invalid diabatization and therefore the required dimension of the transformation matrix has to be at least three. The same applies to the size of the sub-Hilbert space, which also has to be at least three. In this section, we intend to discuss this type of problems. It also leads us to term the conditions for reaching the minimal relevant sub-Hilbert space as the necessary conditions for 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. 

We consider a 2D diabatic framework that is characterized by an angle, P(i), associated with the orthogonal transformation that diagonalizes the diabatic potential matrix. Thus, if V is the diabatic potential matrix and if u is the adiabatic one, the two are related by the orthogonal transformation matrix A [34] ... 

Minimal diabatic potential matrix, non-adiabatic coupling, 81-89... 

H + D2 reaction, quasiclassical trajectory calculation, 167-168 minimal diabatic potential matrix, noninteracting conical intersections, 81-89... 

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