Isotropic interactions

Predictions of theory for rods with axial ratio x = 100 are shown by the curves in Fig. 8. Here % is plotted as ordinate against the volume fractions Vp and Vp in the coexisting phases the ordinate may alternatively, be regarded as an (inverse) measure of temperature. The narrow biphasic gap is httle affected by the interactions for negative values of %, as was noted above. If, however, % is positive, a critical point emerges at = 0.055. For values of % immediately above this critical limit, the shallow concave curve delineates the loci of coexisting anisotropic phases, these being in addition to the isotropic and nematic phases of lower concentration within the narrow biphasic gap on the left. At x = 0.070 the compositions of two of the phases, one from each of the respective pairs, reach the same value. Three phases coexist at this triple point. 

The general features of this phase diagram have been well confirmed by experiments, especially those of Nakajima and of Miller and their co-workers. The emergence of a concentrated, well-ordered anisotropic phase when % exceeds a small positive value (that depends on the axial ratio x) is readily understood as the consequence of interactions that are effectively attractive between the rodlike particles. The dense anisotropic phase may be regarded as the prototype of a quasi-crystalline state with uniaxial order only. Comprehension of the three-dimensional order characteristic of the crystalline state is, of course, beyond the scope of the model, which does not 

---

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 . 

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],... 

We will consider the simplifications possible with the assumption of isotropic interactions. As this was done above, we assume for simplicity that we are dealing with an unmixed gas of diatomic molecules in the low-density limit. We start with Eq. 5.2,... 

In this equation, the energies , and / of the initial and final states, i) and I/), and the dipole moment all refer to a pair of diatomic molecules hcvij = Ef — Ei is Bohr s frequency condition. With isotropic interaction, rotation and translation may be assumed to be independent so that the rotational and translational wavefuntions, population factors, etc., factorize. Furthermore, we express the position coordinates of the pair in terms of center-of-mass and relative coordinates as this was done in Chapter 5. 

The above limitation to pairs not forming bound dimers is not essential and one may readily include the bound dimer contributions, thereby extending the above consideration to any dissimilar pair with isotropic interactions. For dissimilar pairs, the conclusions just obtained are not affected by the presence or absence of dimer components. 

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). 

Ht)2 dimers. The isotropic interaction potential [257] supports just one vibrational dimer level, the ground state (n = 0). The energy level of the... 

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 isotropic interaction energy—long-range forces experienced by the solute, e.g. electrostatic forces, polarization and dispersion energy. 

This hyperfine coupling is of two kinds An isotropic interaction arises from the possibility that the electronic wave-function, x , be non-zero at the nucleus, N. This is the Fermi contact term and the hyperfine coupling constant is given by ... 

TABLE VII Comparison of van der Waals Parameters Obtained from Various H(2S)-H2(X ) Potentials for Isotropic Interaction... 

As a rule, the computational difficulties in hfs calculations are connected with the isotropic interaction, since the theoretically determined anisotropic parameters in general are in good agreement with experiment, and rather insensitive to computational method as well as basis set. 

Again, as with any RPA approach, this predicts an instability in the channel with the most negative V (Q0, ) In the case of the classical crossover, the RPA susceptibility is still given by Eq. (33) and corresponds to an isotropic interacting systems. 

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. 

---