Quantum geometric phase

The presence of the half-odd quantum number j in Eq. (69) is potentially a physically measurable consequence of geomehic phase, which was first claimed to have been detected in the spectrum of Naa [16]. The situation is, however, quite complicated and the first unambiguous evidence for geometric phase in Nas was reported only in 1999 [17],... 

In Figure 1, we see that there are relative shifts of the peak of the rotational distribution toward the left from f = 12 to / = 8 in the presence of the geometiic phase. Thus, for the D + Ha (v = 1, DH (v, f) - - H reaction with the same total energy 1.8 eV, we find qualitatively the same effect as found quantum mechanically. Kuppermann and Wu [46] showed that the peak of the rotational state distribution moves toward the left in the presence of a geometric phase for the process D + H2 (v = 1, J = 1) DH (v = 1,/)- -H. It is important to note the effect of the position of the conical intersection (0o) on the rotational distribution for the D + H2 reaction. Although the absolute position of the peak (from / = 10 to / = 8) obtained from the quantum mechanical calculation is different from our results, it is worthwhile to see that the peak... 

The relative shift of the peak position of the rotational distiibution in the presence of a vector potential thus confirms the effect of the geometric phase for the D + H2 system displaying conical intersections. The most important aspect of our calculation is that we can also see this effect by using classical mechanics and, with respect to the quantum mechanical calculation, the computer time is almost negligible in our calculation. This observation is important for heavier systems, where the quantum calculations ai e even more troublesome and where the use of classical mechanics is also more justified. 

Figure 2. Quantum classical cross-sections for the reaction D-I-Ha (r — l,j — 1) DH (v — l,/)-l-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 geometric phase effect included using either a complex phase factor (dashed) or a vector potential (dotted).

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. 

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],... 

Potential fluid dynamics, molecular systems, modulus-phase formalism, quantum mechanics and, 265—266 Pragmatic models, Renner-Teller effect, triatomic molecules, 618-621 Probability densities, permutational symmetry, dynamic Jahn-Teller and geometric phase effects, 705-711 Projection operators, geometric phase theory, eigenvector evolution, 16-17 Projective Hilbert space, Berry s phase, 209-210... 

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