Sudden approximations

Often a degree of freedom moves very slowly for example, a heavy-atom coordinate. In that case, a plausible approach is to use a sudden approximation, i.e. fix that coordinate and do reduced dimensionality quantum-dynamics simulations on the remaining coordinates. A connnon application of this teclmique, in a three-dimensional case, is to fix the angle of approach to the target [120. 121] (see figure B3.4.14). 

Figure B3.4.14. The infmite-order-sudden approximation for A+ BC AB + C. In this approximation, the BC molecule does not rotate until reaction occurs.

By using this approach, it is possible to calculate vibrational state-selected cross-sections from minimal END trajectories obtained with a classical description of the nuclei. We have studied vibrationally excited H2(v) molecules produced in collisions with 30-eV protons [42,43]. The relevant experiments were performed by Toennies et al. [46] with comparisons to theoretical studies using the trajectory surface hopping model [11,47] fTSHM). This system has also stimulated a quantum mechanical study [48] using diatomics-in-molecule (DIM) surfaces [49] and invoicing the infinite-onler sudden approximation (lOSA). 

This idea, christened later as the sudden approximation, has recently been developed in detail by Levine et al. [1989], in connection with dissipative tunneling in the framework of quantum... 

In fact, in the sudden approximation one looks for the minimum of the barrier action taken on a certain class of paths, each consisting of two straight segments. If the actual extremal path is close to one of the paths from this class - and this is indeed the case for low enough O - then the sudden approximation provides accurate results. In particular, the sudden approximation has been shown [Hontscha et al. 1990] to provide accuracy within less than 10% for the rate constant at... 

The situation presented in fig. 29 corresponds to the sudden limit, as we have already explained in the previous subsection. Having reached a bend point at the expense of the low-frequency vibration, the particle then cuts straight across the angle between the reactant and product valley, tunneling along the Q-direction. The sudden approximation holds when the vibration frequency (2 is less than the characteristic instanton frequency, which is of the order of In particular, the reactions of proton transfer (see fig. 2), characterised by high intramolecular vibration frequency, are being usually studied in this approximation [Ovchinnikova 1979 Babamov and Marcus 1981]. 

Fig. 33. Three-dimensional instanton trajeetories of a partiele in a symmetrie double well, interaeting with symmetrieally and antisymmetrieally eoupled vibrations with eoordinates and frequeneies q, to, and ru, respeetively. The curves are 1, ru, ru, P ojq (MEP) 2. to, (u, < (Oq (sudden approximation) 3. ru, < cOq, oj, P ojo 4. to, > (Oq, < (Oq.

A calculation of tunneling splitting in formic acid dimer has been undertaken by Makri and Miller [1989] for a model two-dimensional polynomial potential with antisymmetric coupling. The semiclassical approximation exploiting a version of the sudden approximation has given A = 0.9cm" while the numerically exact result is 1.8cm" Since this comparison was the main goal pursued by this model calculation, the asymmetry caused by the crystalline environment has not been taken into account. 

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... 

Snider R. F., Coombe D. A. Kinetic cross sections in the infinite order sudden approximation, J. Phys. Chem. 86, 1164-74 (1982). 

Goldflam R., Green S., Kouri D. J. Infinite order sudden approximation for rotational energy transfer in gaseous mixtures, J. Chem. Phys. 67, 4149-61, (1977). 

Infinite-order sudden approximation (IOSA), electron nuclear dynamics (END), molecular systems, 345-349 Initial relaxation direction (IRD), direct molecular dynamics, theoretical background, 359-361 Inorganic compounds, loop construction, photochemical reactions, 481-482 In-phase states ... 

The evaluation of elements such as the M n,fin s is a very difficult task, which is performed with different levels of accuracy. It is sufficient here to mention again the so called sudden approximation (to some extent similar to the Koopmans theorem assumption we have discussed for binding energies). The basic idea of this approximation is that the photoemission of one-electron is so sudden with respect to relaxation times of the passive electron probability distribution as to be considered instantaneous. It is worth noting that this approximation stresses the one-electron character of the photoemission event (as in Koopmans theorem assumption). 

The switch-over point does not have to be specified since this is a sudden approximation. All that matters is that a value rc exists for which (i) the field due to the core has vanished except for the Coulomb component and (ii) the Bom-Oppenheimer approximation is still valid. 

At high velocities, v/ve 1, these ion-atom collisions can be described by the Born approximation, and this description leads to a reasonable description of the observed Na nd —> ni results.4 For example, the results of Percival and Richards,4 shown in Fig. 13.3, are obtained by calculating the Born cross section for the nd — nf transition and using a sudden approximation to estimate the redistribution of population among the nearly degenerate higher angular momentum states. 

The integral in (4.33) may be evaluated by the steepest descent method, which leads to an optimum value of Q = Q. This amounts to minimization of the total action 5, + S2 over the positions of the bend point Q. In fact, in the sudden approximation one looks for the minimum of the barrier action taken on a certain class of paths, each consisting of two straightforward segments. If the actual extremal path is close to one of the paths from this class—and this is indeed the case for low enough ft—then the sudden approximation provides accurate results. In particular, the sudden approximation permits calculation of the rate constant to an accuracy of 10% at V /co0 = 3, ft = 0.1, C 0.05 [Hon-tscha et al., 1990], For the cubic parabola (n = 1 in (4.29)) at small coupling parameter the rate constant in the sudden approximation may be evaluated analytically by using the one-dimensional instanton result (3.68) for k D ... 

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. 

Figure 5.2. Three-dimensional instanton trajectories of a particle in a symmetric double well, interacting with symmetrically and antisymmetrically coupled vibrations with coordinates and frequencies q%, a>s and qa, a>0 (MEP) 2, a>a, 0 (sudden approximation) 3, 0,...

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