Vibrational close coupling

Because the postulated diabatic state is never computed explicitly, the NADP formalism completely avoids the conceptual difficulties associated with this state in the LCP and projection operator methods. Reduction of the vibrational computation to solution of Eq. (9.3) removes the difficult issue of vibrational completeness inherent in vibrational close-coupling theory and in the method of Schneider et al. 

The close-coupling equations are also applicable to electron-molecule collision but severe computational difficulties arise due to the large number of rotational and vibrational channels that must be retained in the expansion for the system wavefiinction. In the fixed nuclei approximation, the Bom-Oppenlieimer separation of electronic and nuclear motion pennits electronic motion and scattering amplitudes f, (R) to be detemiined at fixed intemuclear separations R. Then in the adiabatic nuclear approximation the scattering amplitude for ... 

Theory. The theory of collision-induced absorption profiles of systems with anisotropic interaction [43, 269] is based on Arthurs and Dalgamo s close coupled rigid rotor approximation [10]. Dipole and potential functions are approximated as rigid rotor functions, thus neglecting vibrational and centrifugal stretching effects. Only the H2-He and H2-H2 systems have been considered to date, because these have relatively few channels (i.e., rotational levels of H2 to be accounted for in the calculations). The... 

The close coupled scheme is described on pp. 306 through 308. Specifically, the intermolecular potential of H2-H2 is given by an expression like Eq. 6.39 [354, 358] the potential matrix elements are computed according to Eq. 6.45ff. The dipole function is given by Eq. 4.18. Vibration, i.e., the dependences on the H2 vibrational quantum numbers vu will be suppressed here so that the formalism describes the rototranslational band only. For like pairs, the angular part of the wavefunction, Eq. 6.42, must be symmetrized, according to Eq. 6.47. 

For the transition metal complexes the position is much different. Each cubic-field term, say, may be split into components by spin-orbit coupling and departure from cubic symmetry and in addition is overlaid by closely coupled vibrations. The result is that for only the simplest ions and... 

In principle, Equation (3.5) represents an infinite set of coupled equations. In practice, however, we must truncate the expansion (3.4) at a maximal channel n which turns (3.5) into a finite set that can be numerically solved by several, specially developed algorithms (Thomas et al. 1981). The required basis size depends solely on the particular system. The convergence of the close-coupling approach must be tested for each system and for each total energy by variation of n until the desired cross sections do not change when additional channels are included. Expansion (3.4) should, in principle, include all open channels (k > 0) as well as some of the closed vibrational channels (k% < 0). Note, however, that because of energy conservation the latter cannot be populated asymptotically. 

Within the close-coupling approach each partial photodissociation wavefunction (R,r Ef,n) is represented by the expansion functions Xn (R >Ef,n) and the vibrational basis functions n(r) with n and n = 0,1,2,..., n. Here, n denotes the highest state considered in expansion (3.4). It is not necessarily identical with nmox, the highest state that can be populated for a given total energy. In order to simplify the subsequent notation we consider the total of the radial functions as the elements Xn n R i Ef) of a (n + 1) x (n + 1) matrix... 

In this section we will explain the essential mechanism of vibrational predissociation by virtue of a linear atom-diatom complex such as Ar H2. Figure 12.1 illustrates the corresponding Jacobi coordinates, t In particular, we consider the excitation from the vibrational ground state of H2 to the first excited state as illustrated in Figure 12.2. The close-coupling approach in the diabatic representation, summarized in Section 3.1, provides a convenient basis for the description of this elementary process. For simplicity of presentation we assume that the coupling between the van der Waals coordinate R and the vibrational coordinate r is so weak that it suffices to include only the two lowest vibrational states, n = 0 and n = 1, in expansion (3.4) for the total wavefunction,... 

Delgado-Barrio, G., Mareca, P., Villarreal, P., Cortina, A.M., and Miret-Artes, S. (1986). A close-coupling infinite order sudden approximation (IOSA) to study vibrational predissociation of the Hel2 van der Waals molecule, J. Chem. Phys. 84, 4268-4271. 

Tennyson J, Sutcliffe BT (1982) The ab initio calculation of the vibrational-rotational spectrum of triatomic systems in the close-coupling approach, with KCN and H2Ne as examples. J Chem Phys 77 4061 1072... 

Let s now consider several examples. The simplest of all reactions is the H+H reaction. The vibrational levels are fairly widely spaced, but we must also include the rotational manifold of levels associated with each vibrational level. (See Fig. 9.) Now, it is this rotational manifold of levels (and the degeneracies of states associated with each vibration-rotation level) which ultimately breaks the bank in the size of the close coupling expansion. 

We also calculated harmonic frequencies through the second derivative of the potential around the equilibrium geometry of the dimers. From these values we obtained the energies of the vibrational levels in the harmonic approximations the values so calculated are also very different from the exact (close coupling) results, showing that although the interaction forces are very weak, the potential anhar-monicity and anisotropy are important. 

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. 

The model described is in agreement with the major features of the observations and is consistent with the data available for the predissociation of Hel van der Waals complexes. It remains to be seen if detailed close coupling calculations will provide quantitative verification of all of its features. Although there are not now data to support a generalization, it seems plausible that the zero energy orbiting resonance mechanism for efficient collision-induced vibrational relaxation will occur in all systems. It will be particularly interesting to see what selectivity of vibrational pathway exists in the case of relaxation of a polyatomic molecule. 

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