Three-state

At a given temperature and pressure, a pure compound can exist in one, two or three states. The compound exists at three different states at the triple point and at two different states along the curves of vaporization, freezing and sublimation. Refer to Figure 4.6. 

B. The Quantization of the Three-State Non-Adiabatic Coupling Matrix... 

Thus from the D matrix, it is easy to say that for three states n will be an integer and for two states n will be one-half of an odd integer. 

Now, we are in a position to present the relevant extended approximate BO equation. For this purpose, we consider the set of uncoupled equations as presented in Eq. (53) for the = 3 case. The function icq, that appears in these equations are the eigenvalues of the g matrix and these are coi = 2 (02 = —2, and CO3 = 0. In this three-state problem, the first two PESs are u and 2 as given in Eq. (6) and the third surface M3 is chosen to be similar to M2 but with D3 = 10 eV. These PESs describe a two arrangement channel system, the reagent-arrangement defined for R 00 and a product—anangement defined for R —00. 

VIII. An Analytical Derivation for the Possible Sign Rips in a Three-State System... 

B. The Study of a Real Three-State Molecular System Strongly Coupled (2,3) and (3,4) Conical Intersections... 

Appendix C On the Single/Multivaluedness of the Adiahatic-to-Diahatic Transformation Matrix Appendix D The Diabatic Representation Appendix E A Numerical Study of a Three-State Model Appendix F The Treatment of a Conical Intersection Removed from the Origin of Coordinates Acknowledgments References... 

We prove our statement in two steps First, we consider the special case of a Hilbert space of three states, the two lowest of which are coupled strongly to each other but the third state is only weakly coupled to them. Then, we extend it to the case of a Hilbert space of N states where M states are strongly coupled to each other, and L = N — M) states, are only loosely coupled to these M original states (but can be stiongly coupled among themselves). 

It is important to emphasize that this analysis, although it is supposed to hold for a general three-state case, contradicts the analysis we perfonned of the three-state model in Section V.A.2. The reason is that the general (physieal) case applies to an (arbitrary) aggregation of conical intersections whereas the previous case applies to a special (probably unphysical) situation. The discussion on this subject is extended in Section X. In what follows, the cases for an aggregation of conical intersections will be tenned the breakable situations (the reason for choosing this name will be given later) in contrast to the type of models that were discussed in Sections V.A.2 and V.A.3 and that are termed as the unbreakable situation. 

This conclusion contradicts the findings discussed in Sections V.A.2 and V.A.3. In Section V.A.2, we treated a three-state model and found that functions can n ver flip signs. In Section V.A.3, we treated a four-state case and found that either all four functions flip their sign or none of them flip their sign. The situation where two functions flip signs is not allowed under any conditions. 

In Section XIV.B, this derivation is extended to a three-state system. 

We ended Section XV.A by claiming that the value a(r q = 0.4 A) is only 0.63ic instead of it (thus damaging the two-state quantization requirement) because, as additional studies revealed, of the close locations of two (3,4) conical intersections. In this section, we show that due to these two conical intersections our sub-space has to be extended so that it contains three states, namely, the second, the third, and the fourth states. Once this extension is done, the quantization requirement is restored but for the three states (and not for two states) as will be described next. 

In Section IV, we introduced the topological matrix D [see Eq. (38)] and showed that for a sub-Hilbert space this matrix is diagonal with (-1-1) and (—1) terms a feature that was defined as quantization of the non-adiabatic coupling matrix. If the present three-state system forms a sub-Hilbert space the resulting D matrix has to be a diagonal matrix as just mentioned. From Eq. (38) it is noticed that the D matrix is calculated along contours, F, that surround conical intersections. Our task in this section is to calculate the D matrix and we do this, again, for circular contours. 

APPENDIX E A NUMERICAL STUDY OF A THREE-STATE MODEL... 

As much as the results in the last section (Appendix D) are interesting the rather more interesting case is the one for Ag = 0, namely, the case where the three states degenerate at one point. Here we find that even in this case D is not... 

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