Rotational coupling calculations

Figure 34. Cross sections for Ar L excitation in the collision systems Si -t- Ar and S + Ar as a function of the projectile energy. The experimental data are from Schneider el The curve labeled ROT refers to 6-2v-A

Recently, we have developed a full theoretical treatment of electron capture processes involving an ab initio molecular calculation of the potential energy curves and of the radial and rotational couplings followed, according to the collision energy range concerned, by a semi-classical [21-23] orquantal [24] collision treatment. 

Fig, 19. Infrared spectra of the symmetric CHj bend mode of ethylidyne (CCHj) on Pt(l 11) at 82 K and 300 K. The dotted lines are calculated assuming a vibrational-rotational coupling. Inset shows the suggested orientation. (Reproduced by permission from Malik et... 

We found that these more sophisticated spin-coupled calculations, which used larger basis sets with polarization functions on all of the atoms and which allowed the a orbitals to relax, produced a picture of bonding in the 7t-electron system of benzene which is practically identical to that described earlier. As before, we found six equivalent spin-coupled orbitals which are transformed into one another by successive C6 rotations. The overlaps between the orbitals, ordered cpa to cp6 around the ring, are reported in Table 1. In this case, the electron correlation effects incorporated in the spin-coupled model provide an energy improvement over the SCF description of 170 kJ mol - with a further lowering of 20 kJ mol -1 on including spin-coupled ionic structures. 

Spin-coupled calculations, analogous to those for the organic heterocycles, have been carried out for both molecules. Three of the spin-coupled n orbitals for borazine are related to each other by successive C3 rotations and they take the form of fairly localized, slightly distorted N(2p) functions. The other three orbitals are also transformed into one another by successive C3 operations. These are also based on N(2p) functions but clearly show significant delocalization onto neighboring boron centres. Nonetheless, most of the electron density remains on N. 

Fig. 10.20. Rotational state distribution of CN following the photodissociation of C1CN at 191.5 nm. Comparison between exact close-coupling calculations using the full ab initio PES of Waite and Dunlap (1986) (solid curve) and the impulsive model (IM, dashed curve). Adapted from Schinke (1990).

The reliability of the newly-developed rigid-rotor potential was tested by means of close-coupling calculations of rotational state-to-state integral cross sections. The MOLSCAT code was used [66]. The results were compared with those obtained using the semiempirical potential of Buck, and with the available experimental data. 

In the second scheme to be considered, labelled case (b/ ,v), S and / are coupled to form an intermediate G, which is then coupled with N to form F. This scheme is appropriate when the hyperfine interaction between S and / is strong compared with spin-rotation coupling, and we will meet it elsewhere, most notably in the H ion. The basis kets are written in the form tj, A S, I, G G, N, F) and the hyperfine matrix elements is this basis are calculated in later chapters. The most natural extension of Hund s case (b), known as case (bpj), is that in which /, the resultant of/Vand.S coupling, is coupled with/to form/. The corresponding basis kets are rj, A N, S, J J, I, F) and we will often meet matrix elements calculated in this basis. 

Radial and rotational coupling matrix elements have been calculated in both the triplet and singlet manifolds. The radial coupling between all pairs of states of the same symmetry have been calculated by means of the finite difference technique ... 

After calculating the ground and excited mean field states of a- and y-nitrogen, we have included the correlation between the molecular motions, as well as the translational-rotational coupling, by determining the eigenvalues of the RPA matrix M(q) ]Eq. (129)]. The expansion of the potential in the translational displacements (u/ ) of the molecules [see Eq. 

Table III with the preceding columns. The new results calculated with the ab initio potential agree very well with the frequencies from inelastic neutron scattering (Kjems and Dolling, 1975) and from infrared and Raman spectroscopy (Thi6ry and Fabre, 1976 Fondire et al., 1981) for all types of modes. Also the phonon dispersion relations, displayed in Fig. 4, are in good agreement with the neutron-scattering data. Since most of the lattice modes are actually mixed libron-phonon modes, this indicates that the translation-rotation coupling is correctly included in the RPA formalism.

Results and Predictions. Detailed close coupling calculations for "real" Av<0 vibrational predissociation of weak-coupling systems such as the hydrogen-inert gas complexes are more difficult and computationally more expensive than those for predissociation by internal rotation. The computational expense arises simply from the very large increase in the nvmber of channels which must be included in order to obtain converged results. The difficulty, on the other hand, arises from the fact that these resonances have very small widths, usually 10 cm , %jhich makes them very difficult to find. 

For the prototype level (n,Jl,v, j, J)=(0,0,1,0,0T"of H2-, D2-and HD-Ar, the results of close coupling calculations for the partial and total predissociation widths are listed in Table V. Tests showed that the angular basis sets used to obtain these results were fully converged (25) they included diatom rotation states up to j 8 for v l and up to j" 10 for v -O. The absence of odd-j" dissociation products for H2- and D2-Ar merely reflects the fact that their potentials have no odd-j anisotropy terms. 

Fig. 16. Comparison of calculated (dashed curve) and measured (full curve) smoothed cross sections at three different energies, showing the improvement obtained by including rotational coupling. Upper pictures without rotational coupling [/f12(/ c) = 005eV L12 = 0] and lower pictures with rotational coupling [H 12(/ c) = 0-065 eV L12 = 0-04/i]. Notice the different vertical scales.

One such link between semiempirical theory and experiment that appeared about that time was the development of calculational methods for optical rotatory dispersion. Moffitt s theoretical work with Kronig—Kramers transforms coupled with Djerassi s experimental data on steroids gave rise to rules for the prediction of the sign of optical rotation. Computer calculations with semiempirical methods played a role. i Wavefunctions of at least an approximate sort were needed for the dipole and dipole velocity matrix elements of the theory. 

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