Adiabaticity parameter

ELECTROCHEMICAL ADIABATICITY PARAMETER. MEDIUM DYNAMICS VS. STATIC DISTRIBUTION... 

Under MAS the quadrupole splitting becomes time dependent, Qg = Qg (f) (see Sect. 2.3.4). This influences both the spin-locking behavior [223] and the polarization transfer [224], with the latter being further affected by the periodic modulation of the IS dipolar interaction. The effect of MAS on spin-locking of the S magnetization depends on the magnitude of the so-called adiabaticity parameter ... 

Here D(rjtj,r2t2) is the photon propagator jcv, jpv, jfw are the four-dimensional components of the operator of current for the considered particles core, proton, muon x = (vc, Vp, r, t) includes the space coordinates of the three particles plus time (equal for all particles) and y is the adiabatic parameter. For the photon propagator, it is possible to use the exact electrodynamical expression. Below we are limited by the lowest order of QED PT, i.e., the next QED corrections to Im E will not be considered. After some algebraic manipulation we arrive at the following expression for the imaginary part of the excited state energy as a sum of contributions ... 

Figure 2. Effect of the frequency < > of the perturbation by the core on an electron moving in a Bohr-Sommerfeld orbit of high eccentricity (low angular momentum). Plotted vs. the angle u, which varies by 2ir over one orbit. Note that the perturbation is localized near the core. In the inverse Bom-Oppenheimer limit (x 1) the perturbation oscillates many times during one orbit of the electron. (For further details and the formalism that describes the motion at high x as diffusive-like (see Refs. 3c and S.) For higher angular momentum / the effective adiabaticity parameter is x(l - e) xfl/2, where e is the eccentricity of the Bohr-Sommerfeld orbit. States of high / are thus effectively decoupled from the core.

A recent numerical development is the introduction of the slow or smooth variable discretization (SVD) technique [101-103]. In the diabatic-by-sector method, the basis functions to expand the total wavefunction are fixed within each sector. In the SVD method, the hyperangular basis functions are constructed using the discrete variable representation (DVR) [104], The requirement is only that the total wavefunction be smooth in the adiabatic parameter p. By expanding the hyperradial wavefunctions using DVR basis functions, a new set of hyperangular basis functions are determined and they... 

The results of HSCC calculations have proved much more rapid convergence with the number of coupled channels than the conventional close-coupling equations in terms of the independent-particle coordinates or the Jacobi coordinates based on them. This is considered to be because of the particle-particle correlations considerably taken into account already in the choice of the hyperspherical coordinate system. The results suggest an approximate adiabaticity with respect to the hyperradius p, even when the mass ratios might appear to violate the conditions for the adiabaticity, for example, for Ps- with three equal masses. Then, it makes sense to study an adiabatic approximation with p adopted as the adiabatic parameter. 

Figure 1 lUustrating diabatic (dashed lines) and adiabatic (solid lines) PE surfaces. The diabatic parameters are indicated with a subscript d), the adiabatic parameters with a subscript (a)... 

This dimensionless parameter has been called the adiabaticity parameter. When y 1, the adiabatic limit is realized, while the non-adiabatic limit is realized in the limit of y 1. As the average energy quantum hw of phonons approaches zero, y of Eq. 61 diverges, and only the adiabatic limit becomes justifiable. Bom and Oppen-... 

Figure 11. Transmission coefficient k of the reaction as a function of the adiabaticity parameter y in the normal and inverted cases.

Usually, B is of the order of several eV, while both 2 and E are at most several tenths of eV, and heo is ca. 10 meV, determined by lattice vibrations. Therefore, the adiabaticity parameter y of Eq. 80 is appreciably larger than unity, and hence the capture process is near the adiabatic limit of kx. This feature was confirmed experimentally [30, 31]. 

This form is that used by Keck and Carrier and fits the exact result of (3.27) with <3 = di, = 0 to within 20% for c5 < 10. When this form is used in (4.14) and (4.9), the energy diffusion coefficient for a homonuclear diatomic (A = B) is expressed in terms of the adiabatic parameter defined by... 

A similar expression is found for by setting Q = De in (4.19). The results for Cad and Cd are plotted in Figure 12 for several values of the adiabatic parameter y. As exjjected the adiabatic correction increases with the increasing mass of the bath molecules. Finally, it is noted that this model assumes linear coupling of the solvent to the atomic displacement. This is corrected to some extent by a kinematic factor B which is discussed in more detail in the context of vibrational deactivation in Section V. 

Vary Non-adiabatic Parameter k. If one increases k by either decreasing the molar flow rate AO or increasing the heat-exchange area, the slope increases and the ordinate intercept moves to the left as shown in Figure 8-15. for conditions of 7 < Tq ... 

Table 3.5 Comparison between Diabatic and Adiabatic Parameters...

If the diabatic coupling matrix element, He, is -independent, this d/dR matrix element between two adiabatic states must have a Lorentzian H-depen-dence with a full width at half maximum (FWHM) of 46. Evidently, the adiabatic electronic matrix element We(R) is not - independent but is strongly peaked near Rc- Its maximum value occurs at R = Rc and is equal to 1/46 = a/4He. Thus, if the diabatic matrix element He is large, the maximum value of the electronic matrix element between adiabatic curves is small. This is the situation where it is convenient to work with deperturbed adiabatic curves. On the contrary, if He is small, it becomes more convenient to start from diabatic curves. Table 3.5 compares the values of diabatic and adiabatic parameters. The deviation from the relation, We(i )max x FWHM = 1, is due to a slight dependence of He on R and a nonlinear variation of the energy difference between diabatic potentials. When We(R) is a relatively broad curve without a prominent maximum, the adiabatic approach is more convenient. When We (R) is sharply peaked, the diabatic picture is preferable. The first two cases in Table 3.5 would be more convenient to treat from an adiabatic point of view. The description of the last two cases would be simplest in terms of diabatic curves. The third case is intermediate between the two extreme cases and will be examined later (see Table 3.6). 

Equations (3.3.16 and 3.3.17) suggest one final indicator of whether the diabatic or adiabatic approach is preferable. The better approach is the one for which the sum over deperturbed functions involves fewer terms, especially if one term is dominant (for example, a > 2-1/2). An adiabaticity parameter... 

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