Sub-Hilbert space

B. The Born-Oppenheimer-Huang Equadon for a (Finite) Sub-Hilbert Space TIT. The Adiabatic-to-Diabatic Transformation... 

From now on, the index M will be omitted and it will be understood that any subject to be treated will refer to a finite sub-Hilbert space of dimension M. 

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. 

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. 

In Section n.B, it was shown that the condition in Eq. (10) or its relaxed form in Eq. (40) enables the construction of sub-Hilbert space. Based on this possibility we consider a prescription first for constmcting the sub-Hilbert space that extends to the full configuration space and then, as a second step, constructing of the sub sub-Hilbert space that extends only to (a finite) portion of configuration space. 

By following Section II.B, we shall be more specific about what is meant by strong and weak interactions. It turns out that such a criterion can be assumed, based on whether two consecutive states do, or do not, form a conical intersection or a parabolical intersection (it is important to mention that only consecutive states can form these intersections). The two types of intersections are characterized by the fact that the nonadiabatic coupling terms, at the points of the intersection, become infinite (these points can be considered as the black holes in molecular systems and it is mainly through these black holes that electronic states interact with each other.). Based on what was said so far we suggest breaking up complete Hilbert space of size A into L sub-Hilbert spaces of varying sizes Np,P = 1,..., L where... 

Before we continue with the construction of the sub-Hilbert spaces, we make the following comment Usually, when two given states fomr conical intersections, one thinks of isolated points in configuration space. In fact, conical intersections are not points but form (finite or infinite) seams that cut through the molecular configuration space. However, since our studies are carried out for planes, these planes usually contain isolated conical intersection points only. 

The Pth sub-Hilbert space is defined through the following two requirements ... 

At this point, we make two comments (a) Conditions (1) and (2) lead to a well-defined sub-Hilbert space that for any further treatments (in spectroscopy or scattering processes) has to be treated as a whole (and not on a state by state level), (b) Since all states in a given sub-Hilbert space are adiabatic states, stiong interactions of the Landau-Zener type can occur between two consecutive states only. However, Demkov-type interactions may exist between any two states. 

As we have seen, the sub-Hilbert spaces are defined for the whole configuration space and this requirement could lead, in certain cases, to situations where it will be necessary to include the complete Hilbert space. However, it frequently happens that the dynamics we intend to study takes place in a given, isolated, region that contains only part of the conical intersection points and the question is whether the effects of the other conical intersections can be ignored ... 

Figure 2. A schematic picture describing the three consecutive sub-Hilbert spaces, namely, the (P — l)th, the Pth, and the (P + l)th. The dotted lines are separation lines.

In case of three conical intersections, we have as many as eight different sets of eigenfunctions, and so on. Thus we have to refer to an additional chai acterization of a given sub-sub-Hilbert space. This characterization is related to the number Nj of conical intersections and the associated possible number of sign flips due to different contours in the relevant region of configuration space, traced by the electronic manifold. 

The general formula and the individual cases as presented in Eq. (97) indicate that indeed the number of conical intersections in a given snb-space and the number of possible sign flips within this sub-sub-Hilbert space are interrelated, similar to a spin J with respect to its magnetic components Mj. In other words, each decoupled sub-space is now characterized by a spin quantum number J that connects between the number of conical intersections in this system and the topological effects which characterize it. 

This Section considers one of the more important dilemmas in moleculai physics Given a Bom-Oppenheimer-Huang system, what is the minimal sub-Hilbert space for which diabatization is still valid. 

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. 

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. ... 

According to Section VI, the size M of the sub-Hilbert space is determined whether the respective M states form an isolated set of states fulfilling Eqs. (91). In this case, diabatization is always valid for this subsystem. However, it can happen that under certain geometrical situations the size of the sub-Hilbert space for which diabatization is valid is even smaller than this particular M... 

The group of states, for which Eq. (137) is expected to be valid, fomis a sub-Hilbert space that is isolated with respect to other portions of the Hilbert space following the definition in Eqs. (40). 

In what follows, we assume that indeed the group of states foiin an isolated sub-Hilbert space, and therefore have a Yang-Mills field that is zero or not will depend on whether or not the vaiious elements of the t matiix are singular. 

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. 

Another subject with important potential application is discussed in Section XIV. There we suggested employing the curl equations (which any Bohr-Oppenheimer-Huang system has to obey for the for the relevant sub-Hilbert space), instead of ab initio calculations, to derive the non-adiabatic coupling terms [113,114]. Whereas these equations yield an analytic solution for any two-state system (the abelian case) they become much more elaborate due to the nonlinear terms that are unavoidable for any realistic system that contains more than two states (the non-abelian case). The solution of these equations is subject to boundary conditions that can be supplied either by ab initio calculations or perturbation theory. 

The question to be asked is Under what conditions (if at all) do the components of X fulfill Eq. (B.8) In [34] it is proved that this relation holds for any full Hilbert space. Here, we shall show that this relation holds also for the P sub-Hilbert space of dimension M, as defined by Eq. (10). To show that we employ, again, the Feshbach projection operator foraialism [79] [see Eqs. (11)]. 

Born-Oppenheimer-Huang equation Hilbert space, 44-45 sub-Hilbert space, 46-47 equations, 186-191... 

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