Phase effects

If we make use of the fact that the interferogram as represented in Eq. 2.12 is symmetrical about 8 = 0 but the first data point is actually sampled before the zero retardation point, at 8 = —e the interferogram takes the form 

For this representation, 5(8) and B( are said to be linked through the complex Fourier transform. To recover Blffj from 5(8), the inverse complex transform 

If there were simply a sampling error in the collection of the interferogram, the phase spectrum would be linear, but in general the phase spectrum contains higher-order terms, for example. 

A more detailed discussion of phase correction is given in Chapter 4. At this point it is sufficient to say that if the phase angle 0 in an interferogram measured with a continuous broadband source varies only slowly with wavenumber (as is 

Most modem FT-IR spectrometers that operate at a resolution of 1 cm or poorer collect full double-sided interferograms (i.e., they are measured from —A2 to +A2). In this case, the phase spectmm is calculated from the entire interferogram, hence no approximations are made in the calculation of 0y. Chase [11] demonstrated the improvement in photometric accuracy when symmetrically double-sided interferograms are acquired. The advantage of collecting a double-sided 

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In order to segregate the theoretical issues of condensed phase effects in chemical reaction dynamics, it is usefiil to rewrite the exact classical rate constant in (A3.8.2) as [5, 6, 7, 8, 9,10 and U]... 

A3.8.2 THE ACTIVATION FREE ENERGY AND CONDENSED PHASE EFFECTS... 

Many of the condensed phase effects mentioned above have been studied computationally using the PI-QTST approach outlined in die first part of the last section. One such study 48 has focused on the model synnnetric... 

The molecular phase effects are especially important when the system has some type of synnnetry. Nevertheless, the typical treatment of non-adiabatic effects ignores the adiabatic phase, although, as cautioned, this is a problematic step. 

Kuppermann A and Wu Y S M 1993 The geometric phase effect shows up in chemical reactions Chem. 

Gonnella G, Orlandini E and Yeomans J M 1997 Spinodal decomposition to a lamellar phase effect of hydrodynamic flow Phys. Rev. Lett. 78 1695... 

Mead and Truhlar [10] broke new ground by showing how geometric phase effects can be systematically accommodated in scattering as well as bound state problems. The assumptions are that the adiabatic Hamiltonian is real and that there is a single isolated degeneracy hence the eigenstates n(q-, Q) of Eq. (83) may be taken in the form... 

Figure 1. Quasiclassical cross-sections for the reaction D -I- H2 (w — 1,2 — 1) DH (v — 1, /) -f H at 1.8-eV total energy as a function of/. The solid line indicates results obtained without including the geometric phase effect. Boxes show the results with the geometric phase included using either 9o = 0 (dashed) or 9o = 11.5 " (dotted).

As demonstrated in [53] it is convenient to incorporate the geometrical phase effect by adding the vector potential in hyperspherical coordinates. Thus we found that the vector potential gave three terms, the first of which was zero, the second is just a potential term... 

In order to incorporate the geometric phase effect in a formulation based on an expansion in G-H basis functions we need to consider the operation of the momentum operator on a basis function, that is, to evaluate terms as... 

Since the geometric phase effect is related to the angle 4> we express as... 

Note that in this TDGH-DVR formulation of quantum dynamics, the inclusion of the geometric phase effects through the addition of a vector potential is veiy simple and the calculations can be carried out with about the same effort as what is involved in the ordinary scattering case. 

Figure 3 shows the results with and without including the geometric phase effect for the D + H2 reaction. The basis set is taken as 1,1,1,15,15,15, that is. 

We end this section of phase effects in complex states by reflecting on how, in the first place, we have arrived at a complex description of phenomena that... 

The phase-change nale, also known as the Ben phase [101], the geometric phase effect [102,103] or the molecular Aharonov-Bohm effect [104-106], was used by several authors to verify that two near-by surfaces actually cross, and are not repelled apart. This point is of particular relevance for states of the same symmetry. The total electronic wave function and the total nuclear wave function of both the upper and the lower states change their phases upon being bansported in a closed loop around a point of conical intersection. Any one of them may be used in the search for degeneracies. 

C. Dynamical Jahn-Teller and Geometric Phase Effects... 

The full quantum mechanical study of nuclear dynamics in molecules has received considerable attention in recent years. An important example of such developments is the work carried out on the prototypical systems H3 [1-5] and its isotopic variant HD2 [5-8], Li3 [9-12], Na3 [13,14], and HO2 [15-18], In particular, for the alkali metal trimers, the possibility of a conical intersection between the two lowest doublet potential energy surfaces introduces a complication that makes their theoretical study fairly challenging. Thus, alkali metal trimers have recently emerged as ideal systems to study molecular vibronic dynamics, especially the so-called geometric phase (GP) effect [13,19,20] (often referred to as the molecular Aharonov-Bohm effect [19] or Berry s phase effect [21]) for further discussion on this topic see [22-25], and references cited therein. The same features also turn out to be present in the case of HO2, and their exact treatment assumes even further complexity [18],... 

Vibronic Species of the Vibrational States of Li3 with Consideration of Geometric Phase Effect ... 

Figure 9. Vibrational levels of the first-excited state Lis calculated without consideration (NGP) and with consideration (GP) of geometric phase effect [12].

This relation will be used to study geometrical phase effects within the diabatic framework for the FI3 system and its two isotopic analogues. What is meant by... 

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