Adiabatic potential, vibrationally

The obtained PES forms the basis for the subsequent dynamical calculation, which starts with determining the MEP. The next step is to use the vibrationally adiabatic approximation for those PES degrees of freedom whose typical frequencies a>j are greater than a>o and a>. Namely, for the high-frequency modes the vibrationally adiabatic potential [Miller 1983] is introduced,... 

The adiabatic approximation in the form (5.17) or (5.19) allows one to eliminate the high-frequency modes and to concentrate only on the low-frequency motion. The most frequent particular case of adiabatic approximation is the vibrationally adiabatic potential... 

Figure 5-3. Active site and calculated PES properties for the reactions studied, with the transferring hydrogen labelled as Hp (a) hydride transfer in LADH, (b) proton transfer in MADH and (c) hydrogen atom transfer in SLO-1. (i) potential energy, (ii) vibrationally adiabatic potential energy, (iii) RTE at 300K and (iv) total reaction path curvature. Reproduced with permission from reference [81]. Copyright Elsevier 2002...

It is readily seen that when /3 is sufficiently large so that the hyperbolic sines in (4.19) can be replaced by exponentials, the effect of the prefactor B, is to replace the potential V%s) by the vibrationally adiabatic potential ... 

Figure 8.2. Vibrationally adiabatic potential for formaldehyde conversion in excited electronic (a) A A2 and (b) a3A2 states. The levels of the vt vibration are indicated. (From Jensen and Bunker [1982].)...

Figure 3.1 A schematic diagram showing the relationship of reactive resonances to the vibrationally adiabatic potential curve. The upper panel illustrates a Feshbach resonance trapped in a well the lower panel shows a barrier resonance or QBS.

Fig. 6.4. Schematic illustration of the multi-dimensional reflection principle in the adiabatic limit. The left-hand side shows the vibrationally adiabatic potential curves en(R). The independent part of the bound-state wavefunction in the ground electronic state is denoted by

Figure 6 (a) Vibrationally adiabatic potential energy curves with v-, = 0 for the... 

Quantum number of the effective vibrationally adiabatic potential. 

A more accurate treatment of the reaction uses variational TST, in which the dividing surface is allowed to move off the saddle point, or equivalently, uses the adiabatic theory as described in Section 27.2. The vibrationally adiabatic potential... 

In the absence of curvature coupling, this reduces to a simple one-dimensional Hamiltonian for motion along a vibrationally adiabatic potential,... 

With all of the v, and v2 assignments made, some interesting trends in the fitted parameters kt and W7 appear. From Table 2, we see that the significant deviations of kt from unity are generally found for the highly bend excited transition state levels. Therefore, most of the breakdown of transition state theory appears to be associated with only those few levels. In addition, we see in Table 2 that W7 becomes larger as v2 is increased from 0 to 2 for a given value of v,. This is consistent with the discussion below Eq. (13) since the vibrationally adiabatic potentials become narrower as v2 increases (8,16). This same trend in W7 is even more apparent in the results discussed below for H + H2, 7=1, where both even and odd values of v2 are allowed. 

The energies of the nine features in Table 4 all correspond closely to the energies of maxima in the vibrationally adiabatic potential curves (8). Table 5 illustrates the agreement between the energies predicted by the spectroscopic constants and maxima, max, in the quantal density of reactive states. 

Figure 4 Vibrationally adiabatic potential curves for O + H2 with v, = 1, J = 0, and v2 = 0, 2, 4, and 6. From the bottom up the curves correspond to the [10°], [12°], [14°], and [16°] states. (Reprinted with permission from Ref. 14.)...

The vibrational adiabatic potential Vadiab was created for a given set of the vibrational quantum numbers Vk, fixed during the reaction process. Therefore, it is impossible to exchange energy between the vibrational modes (we assume, therefore, that the Coriolis coupling constants Bkk = 0), as well as between the vibrational modes and the reaction path (we assume that the curvature coupling constants Bks = 0). This would mean a change of Vk s. 

Quantally, the vibrationally adiabatic potential energy surface, En(u) is defined by the energy eigenvalue of the n th vibrational state at u. Semiclassically, the n th eigenvalue is determined by the condition that the action in the v direction, at fixed u be equal to (n+l/2)h. This determines an energy Ej +i/2( ) Thus the action m(u) is defined as... 

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