Adiabatic invariance method

Quantum effects become important only at veiy low temperatures, i.e. for BIkT 1. The best candidates for observing these effects thus are hydrides. Quantum effects become more pronounced when A and/or B are species with open electronic shells see e.g. the differences between the associations N2 + N2 —> N4 and Q2 + Q2 —> 4 at temperatures below 10 K [10], Before leaving this section, it should be mentioned that alternative approaches such as the ACCS A treatment Irom [18], the perturbed rotational state treatment finm [19] and the semiclassical adiabatic invariance method Irom [20] all represent simplified variants of the here described SACM/CT approach and, like the... 

A large number of approximate theories have been proposed for ion-dipole reactions. Some of these include the average dipole orientation (ADO) approximation and its extension to include conservation of angular momentum (the AADO method ), various transition-state theories involving variational and statistical modifications, the semiclassi-cal perturbed rotational state (PRS) approximation, classical trajectory studies, the adiabatic invariance method, and the statistical adiabatic channel model (SACM). 

It can be seen that the agreement between all three methods is excellent. Furthermore, the adiabatic invariance method of Bates and Morgan [15] also gives good agreement with these theories for this temperature range as does the perturbed rotational state approximation [13]. Our quantum-mechanical results thus verify the trajectory formula of equation (6) and we recommend that it be used in interstellar simulations for reactions that go at the capture rate. However, to be sure that a reaction does go at the capture rate it is important to have experimental data, at least at room temperature. 

Table 2. Maximum variation in the vibrational energy Ef and the adiabatic invariant Jf using the energy conserving method (6).

We have derived time-reversible, symplectic, and second-order multiple-time-stepping methods for the finite-dimensional QCMD model. Theoretical results for general symplectic methods imply that the methods conserve energy over exponentially long periods of time up to small fluctuations. Furthermore, in the limit m —> 0, the adiabatic invariants corresponding to the underlying Born-Oppenheimer approximation will be preserved as well. Finally, the phase shift observed for symmetric methods with a single update of the classical momenta p per macro-time-step At should be avoided by... 

The adiabatic invariants for other systems can be determined, theoretically, in a similar way. Still, this direct method is in general extremely troublesome, and one may well ask whether there is no simpler method of discovering invariants. We shall now show how... 

If we set r = 1 this method is equivalent to the method of averaging over the angles, namely the usual procedure leading to adiabatic invariance. Hence, the theory of complete stability due to Birkhoff is actually a strengthening of the adiabatic invariance. 

Unfortunately these methods are seldom directly applicable to van der Waals spectra, but the present paper shows how the underlying ideas may be adapted. One major difficulty is that Franck-Cordon restrictions almost invariably preclude the observation of a progression of van der Waals stretching bands. Hence the sequence of vibrational term v ues required by the conventional methods is unavailable. Secondly, although the HX motions in A— HX complexes have useful analogies with electronic motions, their adiabatic separation from the van der Waals modes is often far from complete. Consequently inversion for the full potential function may require a preliminary deperturbation of the spectrum. Nevertheless remarkable results may be obtained in favourable cases, as exemplified below for the Ar...HF system. 

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