Hilbert spaces

According to the Helmholtz theorem the Hilbert space of 2-D vector fields p x, y) with the inner product... 

By assuming the Hilbert space of dimension N, one can easily establish the relation between coupling matrices and by considering tbe (i/)tb matrix element of V ... 

The Untruncated Hilbert Space An Alternative Derivation General Implications... 

For the Berry phase, we shall quote a definition given in [164] ""The phase that can be acquired by a state moving adiabatically (slowly) around a closed path in the parameter space of the system. There is a further, somewhat more general phase, that appears in any cyclic motion, not necessarily slow in the Hilbert space, which is the Aharonov-Anandan phase [10]. Other developments and applications are abundant. An interim summai was published in 1990 [78]. A further, more up-to-date summary, especially on progress in experimental developments, is much needed. (In Section IV we list some publications that report on the experimental determinations of the Berry phase.) Regarding theoretical advances, we note (in a somewhat subjective and selective mode) some clarifications regarding parallel transport, e.g., [165], This paper discusses the projective Hilbert space and its metric (the Fubini-Study metric). The projective Hilbert space arises from the Hilbert space of the electronic manifold by the removal of the overall phase and is therefore a central geometrical concept in any treatment of the component phases, such as this chapter. 

The symbol M represents the masses of the nuclei in the molecule, which for simplicity are taken to be equal. The symbol is the Kionecker delta. The tensor notation is used in this section and the summation convention is assumed for all repeated indexes not placed in parentheses. In Eq. (91) the NACT appears (this being a matrix in the electronic Hilbert space, whose components are denoted by labels k, m, and a vector with respect to the b component of the nuclear coordinate R). It is given by an integral over the electron coordinates... 

The mixed, v t — % notation here has historic causes.) The Schrodinger equation is obtained from the nuclear Lagrangean by functionally deriving the latter with respect to t /. To get the exact form of the Schrodinger equation, we must let N in Eq. (95) to be equal to the dimension of the electronic Hilbert space (viz., 00), but we shall soon come to study approximations in which N is finite and even small (e.g., 2 or 3). The appropriate nuclear Lagrangean density is for an arbitrary electronic states... 

Then, two things (that are actually interdependent) happen (1) The field intensity F = 0, (2) There exists a unique gauge g(R) and, since F = 0, any apparent field in the Hamiltonian can be transformed away by introducing a new gauge. If, however, condition (1) does not hold, that is, the electronic Hilbert space is truncated, then F is in general not zero within the tmncated set. In this event, the fields A and F cannot be nullified by a new gauge and the resulting YM field is true and irremovable. 

Attention is directed to a previous discussion of what happens when the electronic basis is extended to the complete Hilbert space, [79] p. 60 especially Eqs. (2.17)-(2.18). It is shown there that in that event the full symmeti of the invariance group is regained (in effect, through the cancellation of the... 

Projecting the nuclear solutions Xt( ) oti the Hilbert space of the electronic states (r, R) and working in the projected Hilbert space of the nuclear coordinates R. The equation of motion (the nuclear Schrddinger equation) is shown in Eq. (91) and the Lagrangean in Eq. (96). In either expression, the terms with represent couplings between the nuclear wave functions X (K) and X (R). that is, (virtual) transitions (or admixtures) between the nuclear states. (These may represent transitions also for the electronic states, which would get expressed in finite electionic lifetimes.) The expression for the transition matrix is not elementaiy, since the coupling terms are of a derivative type. 

When the wave function is completely general and pennitted to vary in the entire Hilbert space the TDVP yields the time-dependent Schrodinger equation. However, when the possible wave function variations are in some way constrained, such as is the case for a wave function restricted to a particular functional form and represented in a finite basis, then the corresponding action generates a set of equations that approximate the time-dependent Schrodinger equation. 

This analysis is heuristic in the sense that the Hilbert spaces in question are in general of large, if not infinite, dimension while we have focused on spaces of dimension four or two. A form of degenerate perturbation theory [3] can be used to demonstrate that the preceding analysis is essentially correct and, to provide the means for locating and characterizing conical intersections. 

Appendix A GBO Approximation and Geometric E hase for a Model Two-Dimensional (2D) Hilbert Space... 

APPENDIX A GBO APPROXIMATION AND GEOMETRIC PHASE FOR A MODEL TWO-DIMENSIONAL (2D) HILBERT SPACE... 

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

A. The Born-Oppenheimer Equations for a Complete Hilbert Space... 

Next, the full-Hilbert space is broken up into two parts—a finite part, designated as the P space, with dimension M, and the complementai y part, the Q space (which is allowed to he of an infinite dimension). The breakup is done according to the following criteria [8-10] ... 

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. 

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). 

This concludes our derivation regarding the adiabatic-to-diabatic tiansforma-tion matrix for a finite N. The same applies for an infinite Hilbert space (but finite M) if the coupling to the higher -states decays fast enough. 

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 ... 

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