Double adiabatic approximation

Note that since the profile of the lower adiabatic potential energy surface for the proton depends on the coordinates of the medium molecules, the zeroth-order states and the diabatic potential energy surfaces depend also on the coordinates of the medium molecules. The double adiabatic approximation is essentially used here the electrons adiabatically follow the motion of all nuclei, while the proton zeroth-order states adiabatically follow the change of the positions of the medium molecules. 

The main point of the preceding discussion is an assumption about the adiabaticity of intramolecular motion with reference to the intermolecular one, that is, division of the system into two subsystems the fast one involving electrons and intramolecular vibrations and the slow one incorporating the intermolecular vibrations. This division of the motions was called a double adiabatic approximation (DAA) and was applied earlier in the theory of proton transfer in the reorganizing medium (see, e.g., refs. 10, 14, 28, and 31-33). The wavefunctions in DAA are presented as the products of the wavefunctions of the fast and slow subsystems ... 

In the adiabatic approximation, particles starting out in the remote past in the gjpund state remain on the lowest eigenvalue at all times. These particles experience stance scattering by a double-barrier potential. It is known that in this situation jp Is one-energy point with unity tunneling probability, irrespective of the details ijp potential [380-385], This point occurs when the incident energy is near a id state of the well contained within the barriers. Similar phenomena have been... 

This has the form of a double-well oscillator coupled to a transverse harmonic mode. The adiabatic approximation was discussed in great detail from a number of quantum-mechanical calculations, and it was shown how the two-dimensional problem could be reduced to a one-dimensional model with an effective potential where the barrier top is lowered and a third well is created at the center as more energy is pumped into the transverse mode. From this change in the reactive potential follows a marked increase in the reaction rate. Classical trajectory calculations were also performed to identify certain specifically quanta effects. For the higher energies, both classical and quantum calculations give parallel results. 

We first employ the adiabatic approximation and restrict the theoretical treatment to the electronic ground state. This is equivalent to the (one-dimensional) double minimum problem discussed in Section VI.A. Thus, the control field is determined as... 

Detailed numerical studies of the Born-Oppenheimer approximation have been performed in the context of studies of baryons with double charm [7,73]. The method works quite well for ccq configurations, as expected, but also for the ssq or even qqq cases. In table 7.1, we display a comparison of the extreme and uncoupled adiabatic approximations with exact results for the mmm system with masses m = l and m = 0.2,0.5 and 1, bound by the smooth 2 S r potential. The quality of the approximation is impressive for both the energy of the first levels and the short-range correlation. 

Casida first pointed out that double excitations appear to be miss-ing from TDDFT linear response within any adiabatic approximation. Experience shows that, as in naphthalene, adiabatic TDDFT will... 

If we consider ET in the ground state, we shall deal with an adiabatic case where the occurrence of charge transfer depends on the activation energy between the double-minimum potential (Meyer, 1978). When defining the activation energy AG+, one typically uses the approximations of harmonic... 

The choice of an adiabatic picture leads to difficulties when one of the potentials has a double minimum (see Fig. 3.5). The vibrational level separations of such a curve do not vary smoothly with vibrational quantum number, as do the levels of a single minimum potential. In the separate potential wells (below the barrier), the levels approximately follow two different smooth curves. However, above the potential barrier the separation between consecutive energy levels oscillates. The same pattern of behavior is found for the rotational constants below and above the potential barrier. In addition, the rotational levels above the barrier do not vary as BVJ(J + 1). An adiabatic deperturbation of the (E,F+G,K) states of H2 has been possible (Dressier et al., 1979) only because the adiabatic curves were known from very precise ab initio calculations. 

If the adiabatic Bom-Oppenheimer approximation were exact, photochemistry and photophysical processes would be rather straightforward to describe. Molecules would be excited by the incident radiation to some upper electronic state. Once in this electronic state, the molecules could radiate to a lower electronic state, or they could decompose or isomerize on the upper electronic potential energy surface. No transitions to other electronic states would be possible. The spectroscopy of the systems would also be greatly simplified, as there would no longer be any phenomena such as lambda doubling, etc., which lifts degeneracy of some energy levels of the clamped-nucleus electronic Hamiltonian, //,. 

But, generally, such a cycle with adiabatic and isothermal irreversible processes may be realized with real gas (or even liquid). Those with real gas approximate the reversible Carnot cycle with ideal gas by a double limiting process as follows (i.e., we form the ideal cyclic process from set A (and also B and C), see motivation of postulate U2 in Sect. 1.2) running this cycle slower and slower... 

The rate of the electronically non-adiabatic process which appears to remove HCI(v = 1) should approximately double when HQ is raised to =2. In addition, the endothermic reaction ( -65) becomes energetically possible and results that confirm that this reaction takes place are referred to in Section 1. However, there does appear to be a discrepancy between the various kinetic results that have been obtained for... 

The situation is quite different in bimolecular reactions with an activation energy (E >0). In particular, the "diatomic" model is certainly a bad approximation for radical-radical rebinding along a double bond in which the maximum of the effective potential (35 IV) lies near the saddle-point of the potential energy surface /141/, In this case no central forces govern the nuclear motion hence, the total angular momentum is not a constant, which means that the reaction cannot be rotationally adiabatic. Therefore, in this situation the statistical theory cannot correctly reproduce the results of the simple collision theory. 

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