Adiabatic corrections

The simplest way to add a non-adiabatic correction to the classical BO dynamics method outlined above in Section n.B is to use what is known as surface hopping. First introduced on an intuitive basis by Bjerre and Nikitin [200] and Tully and Preston [201], a number of variations have been developed [202-205], and are reviewed in [28,206]. Reference [204] also includes technical details of practical algorithms. These methods all use standard classical trajectories that use the hopping procedure to sample the different states, and so add non-adiabatic effects. A different scheme was introduced by Miller and George [207] which, although based on the same ideas, uses complex coordinates and momenta. 

For the discharge of compressible fluids from the end of a short aiping length into a larger cross section, such as a larger pipe, vessel, or atmosphere, the flow is considered adiabatic. Corrections are applied to the Darcy equation to compensate for fluid property changes due to the expansion of the fluid, and these are known as Y net expansion factors [3]. The corrected Darcy equation is ... 

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... 

The principal advantage of SCS in comparison with IOS is that the adiabaticity of collisions may be taken into account. The difference between actual cross-sections and their purely non-adiabatic estimation is not large but increases with rotational frequency. As shown in Fig. 5.5 the adiabatic correction improves even qualitatively the high-frequency alteration of cross-sections by minimizing the discrepancy between SCS... 

Fig. 5.5. Comparison of CC results ( ) with SCS without adiabatic correction (o) and with it (+) [191].

Fig. 5.6. Collisional broadening of N2 rotational components, (a) In Q-branch, calculated by purely non-adiabatic theory at 300 K (1) and with adiabatic corrections at 300 K (2) and at 100 K (3) [215]. (b) In S-branch, calculated in [191] with adiabatic corrections using the recipe of Eq. (5.56). The experimental data (+) are from [214].

When calculating the rate constants, two potentials were used the anisotropic 6-12 Lennard-Jones from [209] and the anisotropic Morse [216] for comparison. The results appeared to be very similar, thus indicating low sensitivity of the line widths to the potential surface details. The agreement with experimental data shown in Fig. 5.6(h) is fairly good. Moreover, the SCS approximation gives a qualitatively better approach to the problem than the purely non-adiabatic IOS approximation. As is seen from Fig. 5.6 the significant decrease of the experimental line widths with j is reproduced as soon as adiabatic corrections are made [215]. 

This difference is primarily an effect of partial adiabaticity of collision. If it is completely ignored as in the J-dififusion limit then decay is practically mono-exponential so that oE = 10.04 A and aE = 10.07 A are almost the same. However, these cross-sections are nearly twice those represented in Eq. (5.64), which proves that adiabatic correction of the. /-diffusion model (IOS approximation) is significant, at least at T = 300 K. 

The pressure being higher, all features of rotational structure disappear and the difference between spectra of various spin modifications becomes so smooth that any of them practically reproduces the whole contour shape. In Fig. 5.14 the theoretical contours are shown calculated with and without adiabatic correction of the impact operator for an ideal nitrogen solution in Ar. They are compared with the experimental one related to the same value of... 

Fig. 5.15. Theoretical dependences of HWHM on the rate of rotational energy relaxation perturbation theory asymptotics (1), classical weak-collision. /-diffusion model (2), quantum theory without (3) and with (4) adiabatic correction.

In Fig. 4 we compare the adiabatic (dotted line) and the stabilized standard spectral densities (continuous line) for three values of the anharmonic coupling parameter and for the same damping parameter. Comparison shows that for a0 1, the adiabatic lineshapes are almost the same as those obtained by the exact approach. For aG = 1.5, this lineshape escapes from the exact one. That shows that for ac > 1, the adiabatic corrections becomes sensitive. However, it may be observed by inspection of the bottom spectra of Fig. 4, that if one takes for the adiabatic approach co0o = 165cm 1 and aG = 1.4, the adiabatic lineshape simulates sensitively the standard one obtained with go,, = 150 cm-1 and ac = 1.5. 

The only difficulty with using this method is the lack of heat capacity data. With the wide spread use of the Picker et al. (129) heat capacity calorimeter one can usually find published heat capacities for most systems of interest (3) at 25°C. Since the Cp does not attribute much to the adiabatic correction, this is not a serious limitation. 

A detailed discussion of the theoretical evaluation of the adiabatic correction for a molecular system is beyond the scope of this book. The full development involves, among other matters, the investigation of the action of the kinetic energy operators for the nuclei (which involve inverse nuclear masses) on the electronic wave function. Such terms are completely ignored in the Born-Oppenheimer approximation. In order to go beyond the Born-Oppenheimer approximation as a first step one can expand the molecular wave function in terms of a set of Born-Oppenheimer states (designated as lec (S, r ))... 

Fig. 2.1 The adiabatic correction to the Born-Oppenheimer approximation for H2 and HD schematic, not to scale AC = C(H2)-C(HD). In each case the uncorrected potential lies to the left, the corrected to the right...

In Chapter 4 we will learn to calculate the equilibrium constant for an exchange reaction like Equation2.15 using the Born-Oppenheimer approximation. If, in addition, the adiabatic correction is included, the equilibrium constant calculated in the Born-Oppenheimer approximation must be multiplied by a correction factor containing the energy difference AAC. 

To give the reader some further appreciation of the adiabatic correction, we next discuss the so-called Rydberg correction of the hydrogenic atom. The kinetic energy of the electron in the hydrogen atom is expressed by Equation 2.5, where mj is set equal to the electron mass, me. It is well known that the Schrodinger equation for the hydrogen atom separates into two equations, one of which deals with the motion of the center of mass of the system and the other with the motion of the electron... 

The meaning of adiabatic correction is that the addition of C to the ground state energy calculated with Roo should yield a value equal to (close to) the exact ground state energy of the hydrogen atom... 

Table 2.1 Adiabatic corrections (cm ) for H-D and Mu-H isotope effects (Bardo,R. D., Kleinman, L. I., Raczkowski, A. W. and Wolfsberg, M., J. Chem. Phys. 69, 1106, 1978) ...

The BO approximation, which assumes the potential surface on which molecular systems rotate and vibrate is independent of isotopic substitution, was discussed in Chapter 2. In the adiabatic regime, this approximation is the cornerstone of most of isotope chemistry. While there are BO corrections to the values of isotopic exchange equilibria to be expected from the adiabatic correction (Section 2.4), these effects are expected to be quite small except for hydrogen isotope effects. 

There have been no previous direct non-BO studies of the response of H2 and its isotopomers to electric fields. The ground-state dipole moment of HD has been determined experimentally by Nelson and Tabisz [81] to be 0.000345 a.u. There have been several theoretical studies of the dipole moment of HD, all within the BO approximation but including adiabatic corrections. The calculated values by Wolniewicz, 0.000329 [83], Ford and Browne, 0.000326 [82], and Thorson et al., 0.000334 [84], aU agree well with the experimental value, although they are all about 5% too small. This is an extremely difficult experiment to carry out, and because all theoretical studies agree on the value, it... 

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