Isotropic interaction approximation

If molecules are involved, isotropic potential functions are in general not adequate and angular dependences reflecting the molecular symmetries may have to be accounted for. In general, up to five angular variables may be needed, but in many cases the anisotropies may be described rigorously by fewer angles. We must refer the reader to the literature for specific answers (Maitland et al., 1981) and mention here merely that much of what will interest us below can be modeled in the framework of the isotropic interaction approximation. 

Molecules generally interact with anisotropic forces. The accounting for the anisotropy of intermolecular interactions introduces substantial complexity, especially for the quantum mechanical treatment. We will, therefore, use as much as possible the isotropic interactions isotropic interaction approximation (IIA), where the Hamiltonian is given by a sum of two independent terms representing rotovibrational and translational motion. The total energy of the complex is then given by the sum of rotovibrational and translational energies. The state of the supermolecule is described by the product of rotovibrational and translational wavefunc-tions, with an associated set of quantum numbers r and t, respectively. 

For the rototranslational spectra, within the framework of the isotropic interaction approximation, the expressions for the zeroth and first moments, Eqs. 6.13 and 6.16, are exact provided the quantal pair distribution function (Eq. 5.36) is used [314]. A similar expression for the binary second translational moment has been reported [291],... 

The quadrupole-induced components A1A2AL = 0223,2023 are the most important components of the spectrum. Figure 6.3 compares the positive-frequency wings of the various spectral functions, Eq. 6.55, at the temperature of 77 K. These consist of free — free and bound — free components, the first and third terms to the right of Eq. 6.55. The dimer structures were suppressed in Fig. 6.3 but are shown (as obtained in the isotropic interaction approximation) in Fig. 6.4. The low-frequency end of the profiles, Fig. 6.3, may be considered a low-resolution rendition (as may be obtained with a monochromator of low resolution, 10 cm-1 pressure broadening would similarly flatten the spectral dimer structures). 

Fig. 6.4. The dimer structures in the isotropic interaction approximation, in the iow-frequency portion of the bound —< free 0223,2023 components, compared with the free — free component, at 120 K [282].

Comparison of theory and measurement. For a comparison of theory with measurements, rototranslational absorption spectra were computed in the isotropic interaction approximation and compared with low-resolution ( 10 cm-1) spectra, dimer structures are not discernible in the measurement. The frequencies range from 0 to 2250 cm-1. Temperatures were chosen... 

Figure 6.14 compares the results of line shape computations based on the isotropic interaction approximation with the measurement by Hunt [187], This spectrum does not have many striking features because of the relatively high temperature of 300 K. We notice only a broad, unresolved Q branch and a diffuse Si(l) line of H2 is seen other lines such as Si(J) with J = 0, 2, 3,. .. are barely discernible. Various dips of the absorption at 4126, 4154 and 4712 cm-1 are caused by intercollisional interference, a many-body effect which is not accounted for in a binary theory. Roughly 90% of the Q branch (in the broad vicinity of 4150 cm-1) arises from the isotropic overlap induced dipole component (XL = 01). The anisotropic overlap component (XL = 21) is a little less than one-half as intense as the quadrupole induced term (XL = 23). These two components with X = 2 are responsible for the Si line structures superimposed on the broad isotropic induction component which is of roughly comparable intensity near the Si line center. 

Practically all computations shown above were undertaken in the framework of the isotropic interaction approximation. For the examples considered, agreement of calculated and observed spectra was found. The most critical comparisons between theory and measurement were made for the H2-X systems whose anisotropy is relatively mild. Nevertheless, some understanding is desirable of what the spectroscopic effects of the anisotropy are. Furthermore, other important systems like N2-N2 and CO2-CO2 are more anisotropic than H2-X. The question thus remains as to what the spectroscopic significance of anisotropic interaction might be. In this Section, an attempt is made to focus on the known spectroscopic manifestations of the anisotropy of the intermolecular interaction. 

Figure 6.15 shows the rototranslational spectrum of H2-H2 at 77 K as an example. The diffuse lines are nearly indistinguishible from those obtained in the isotropic interaction approximation, Fig. 6.6. Strikingly different are the various dimer structures near the H2 So(0) and So(l) rotational transition frequencies.

Dimers. It is well known that H2 pairs form bound states which are called van der Waals molecules. The discussions above based on the isotropic interaction approximation have shown that for the (H2)2 dimer a single vibrational state, the ground state (n = 0), exists which has two rotational levels f = 0 and 1). If the van der Waals molecule rotates faster ( > 1), centrifugal forces tear the molecule apart so that bound states no longer exist. However, two prominent predissociating states exist which may be considered rotational dimer states in the continuum (/ = 2 and 3). The effect of the anisotropy of the interaction is to split these levels into a number of sublevels. 

The rototranslational and fundamental absorption spectra of the H2-H complex have been obtained from first principles, for temperatures from 200 to 2500 K [21, 103]. Close-coupled and isotropic interaction approximation calculations give nearly identical values at frequencies from 0 to 6000 cm-1. No laboratory measurements exist for comparison with the calculations. The H2-H system is of considerable interest in stellar environments at such temperatures. 

The hfs tensors of ligand nuclei in the first coordination sphere of a metal complex are usually dominated by the isotropic interaction, i.e. the transition probabilities may be approximated by the formulae given in Table 2.1 for aiso > 2v . 

We note that a computational study of the dimer features is involved. It must account for the anisotropy of the interaction as this was done for the pure rotational bands of hydrogen pairs [355, 357], Whereas a treatment based on the isotropic potential approximation may be expected to predict nearly correct total intensities of the free-bound, bound-free, and bound-bound transitions involving the (H2)2 van der Waals molecule (and, of course, the free-free transitions which make up more than 90% of the observed intensities), the anisotropy of the interaction causes elaborate fine structure that is of considerable interest for the measurement of the anisotropy [248]. 

In this review we will first describe two approaches which we have used to represent atomic and molecular systems without resorting to the B-0 approximations. Next, we will describe two numerical applications of the theory, which led to determining interesting non-adiabatic contributions. In the last section we will consider future theoretical work on a general non-adiabatic approach to an N-particle system with any isotropic interaction potential, including coulombic interaction, which is presently being developed in our group. 

If we now transfer our two interacting particles from the vacuum (whose dielectric constant is unity by definition) to a hypothetical continuous isotropic medium of dielectric constant e > 1, the electrostatic attractive forces will be attenuated because of the medium s capability of separating charge. Quantitative theories of this effect tend to be approximate, in part because the medium is not a structureless continuum and also because the bulk dielectric constant may be an inappropriate measure on the molecular scale. Eurther discussion of the influence of dielectric constant is given in Section 8.3. 

The Slater-Kirkwood equation (Eq. 39) was selected with N = 4 for carbon and N = 1 for hydrogen. The success of the equivalent calculation for the intermolecular interaction of CH4 molecules was mentioned in the previous section. Atoms, rather than bonds, were chosen as the basis for the calculation because the location of the atom centers is unambiguous and the approximation of isotropic polarizability is better for an atom than for a bond. Possible deviations from isotropic polarizability are discussed in Section V. Ketelaar19 gives for the atomic polarizabilities of hydrogen and carbon a = 0.42 and 0.93x 10-24 cm3, respectively. The resulting equation for the London energy is... 

The Lines approximation is expected to be quite accurate for the description of the exchange interaction between a strongly axial doublet and an arbitrary isotropic spin. For all other cases, the Lines model [84] is a reasonable approximation. Efficient implementation of the Lines model was done in the program POLY ANISO. 

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