Adiabatic and Sudden Approximations

Since adiabatic and sudden approximations are frequently used in the study of multidimensional tunneling [4,63-69], these are briefly explained before introducing the 

Let us first consider the adiabatic approximation. The frequency cox in the x direction near the potential minimum is assumed to be much smaller than the frequency coy in the y direction. Then the x motion is frozen and the adiabatic solutions Xny y,x) that satisfy the following equation are solved first  

Let us next consider the sudden approximation. The Hamiltonian is the same as Equation (4.1), but now the frequency coy is assumed to be much smaller than co. The total wave function can be written as 

Since p is not directly related to tunneling, we approximate Equation (4.8) by the equation 

Inserting these into the Herring formula, we obtain the final expression of tunneling splitting. 

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The relevance of optical potentials to direct molecular reactions was considered by Micha (1969). Numerical results were presented for real and imaginary parts of the optical potential for H + H2, in an adiabatic approximation that included vibrational and rotational motion of H2. Distortion, adiabatic and sudden approximations to optical potentials, and their validity, have recently been described (Micha, 1974). This work also presents procedures for calculating upper and lower bounds to the second term of f P, in certain ranges of energies. The various approaches are developed in detail for atom-diatom collisions. 

The difference between the adiabatic and sudden approximations is generally not important for valence band photoelectron spectroscopy of metals because a large... 

Finally, we touch upon the adiabatic and sudden approximations in the present model. In the same way as in the case of symmetric mode coupling model, the adiabatic approximation leads to the tunneling splitting independent of Hy. This does not exhibit any characteristic behavior discussed above and can never be reliable. If we apply the sudden approximation, we also encounter a problem since the potential curve in jc direction is not symmetric except when y is zero. Thus we cannot use Equation (4.12) directly anymore. 

Alexander and DePristol concluded that an adiabatically corrected sudden approximation based on straight-line paths and the dipole-dipole interaction will provide similarly accurate results for the (ji J2) = 00 11,02,22 and 11 02 transitions at... 

M. H. Alexander and A. E. DePristo, An adiabatically corrected sudden approximation for rotationally inelastic collisions between polar molecules, J. Phys. Chem. 83 1499 (1979). 

Physically the cutting-corner trajectory implies that the particle crosses the barrier suddenly on the time scale of the slow -vibration period. In the literature this approximation is usually called sudden , frozen bath and fast flip approximation, or large curvature case. In the opposite case of small curvature (also called adiabatic and slow flip approximation), coj/coo < sin tp, which is relevant for transfer of fairly heavy masses, the MEP may be taken to a good accuracy to be the reaction path. 

In this case the parameters C and Q are of order of unity, and therefore they correspond to the intermediate situation between the sudden and adiabatic tunneling regimes. Examples are mal-onaldehyde, tropolon and its derivatives, and the hydrogen-oxalate anion discussed above. For intermolecular transfer, corresponding to a weak hydrogen bond, the parameters C, Q and b are typically much smaller than unity, and the sudden approximation is valid. In particular, carbonic acids fulfill this condition, as was illustrated by Makri and Miller [1989]. 

The original semiclassical version of the centrifugal sudden approximation (SCS) developed by Strekalov [198, 199] consistently takes into account adiabatic corrections to IOS. Since the orbital angular momentum transfer is supposed to be small, scattering occurs in the collision plane. The body-fixed correspondence principle method (BFCP) [200] was used to write the S-matrix for f — jf Massey parameter a>xc. At low quantum numbers, when 0)zc —> 0, it reduces to the usual non-adiabatic expression, which is valid for any Though more complicated, this method is the necessary extension of the previous one adapted to account for adiabatic corrections at higher excitation... 

It is not difficult to show that the inequality fi < fl, which should be met for the sudden approximation to hold, is equivalent to (2.86) if we introduce the angle 2(p between the reactant and product valleys where tan tp = Cl2/C. The borders of the regions of validity of the sudden and adiabatic approximations in the (C, 0) plane are symbolically drawn in Figure 4.7. The only physically sensible parameters are those for which... 

Figure 5.1 with both the sudden and adiabatic approximations. For the purposes of demonstration, the adiabatic barrier height has been taken to be half the one-dimensional barrier V = V0/2, so that b =, C = Cl. One sees that the sudden approximation is realized only for fairly low vibrational frequencies, while the adiabatic approximation becomes excellent for fl s 2. 

As opposed to the adiabatic limit, we assume in the sudden approximation that the internal motion is slow compared to the external (i.e., translational) motion. Most familiar is the rotational sudden approximation which is frequently exploited in energy transfer studies in full collisions (Pack 1974 Secrest 1975 Parker and Pack 1978 Kouri 1979 Gianturco 1979 ch.4). Its application to photodissociation is straightforward and will be outlined below for the model discussed in Section 3.2. 

At low temperature the classical approximation fails, but a quantum generalization of the long-range-force-law collision theories has been provided by Clary (1984,1985,1990). His capture-rate approximation (called adiabatic capture centrifugal sudden approximation or ACCSA) is closely related to the statistical adiabatic channel model of Quack and Troe (1975). Both theories calculate the capture rate from vibrationally and rotationally adiabatic potentials, but these are obtained by interpolation in the earlier work (Quack and Troe 1975) and by quantum mechanical sudden approximations in the later work (Clary 1984, 1985). 

Theoretical calculations of sticking are challenging, due to the interplay of the Coulomb and strong interactions in a non-adiabatic few-body system, yet recent predictions, including the effects of nuclear structure and the deviations from the standard sudden approximation, now converge to a few percent [36], They cannot, however, be readily compared to experiment because most measurements are primarily sensitive to final sticking which is a combination of initial... 

To calculate the parameters governing the Hamiltonian, we use an approximation that amounts to separating the transverse modes into high-frequency (HF) modes, treated adiabatically, and low-frequency (LF) modes, treated in the sudden approximation. This separation is based on the value of the zeta factor [27]... 

Figure 6. Temperature dependence of the reaction efficiency per collision for the reactions of OD + CH3Cl (open circles), 0D D20 + CH3Cl (filled circles and 0D (D20)2 + CH3Cl (half-filled circles). The reaction efficiency per collision is the experimental rate constant divided by the calculated collision rate constant, calculated by Clary using the adiabatic capture centrifugal sudden approximation (ACCSA) (28). For experimental reasons (29), the measurements were made with completely deuterated...

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