Isotropic potentials

As V0(r) decreases very rapidly with distance the major contribution to the scattering phases comes from a small part of the trajectory near the turning point. Expanding the isotropic potential in z2 = (r/rt)2 — 1 < 1 one finds [202, 203] ... 

Here e is the depth of the well and d is the distance to zero crossing point of the isotropic potential whereas at and bt are anisotropy parameters. Substituting this potential into Eq. (5.50) we get for each anisotropic term [202]... 

For an isotropic potential, the three-dimensional probability distribution is... 

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. 

Quantum theory of scattering. The differential scattering cross section for isotropic potentials is given by the scattering phase shifts r e as... 

In the isotropic potential approximation, the complete binary spectrum is obtained by superimposing basic line profiles, Ga,a2al( T), shifted by sums of molecular rotational frequencies which may be positive,... 

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]. 

The profiles of the rototranslational absorption of CH4-CH4 in the far infrared have been reported [56] see Fig. 3.22 for an example. The treatment of the spectra is based on the multipolar induction model and an advanced isotropic potential empirical overlap-induced dipole components have also been included for fitting the experimental data at several temperatures (126 through 300 K). At the lower temperatures, satisfactory fits of the measurements are possible. The analysis seems to suggest that at temperatures near room temperature a significant rotation-induced distortion of the tetrahedral frames occurs which affects the properties of the individual molecules (multipole strengths, molecular symmetry, polarizabilities, and perhaps the interaction). 

In systems involving molecules like H2, the vibrational coordinates may be considered as changing in time much faster than the translational ones, so that the motion of the latter can be determined by an isotropic potential that is the vibrational average, e.g., Fq(/ ) = (vj V(R r) vj). 

The locked permanent-dipole capture system may serve as our first example for an isotropic potential where SACM and PST coincide. In this case, the adiabatic channel potential curves have the form... 

The anisotropy of the potential shifts the position of r toward smaller values than for the isotropic potential [see Eq. (48)]. Now Q(r ) follows as... 

Thus Q = K — K implies that Q = k — k. In other words, Q is the momentum transfer to the electron. If we assume that there is no particular orientation of the perturber during the collision we can replace U (fi, fi, p) by the isotropic potential U(fi, fi, p). With this approximation in Eq. (11.5) we recognize the Bom approximation to the electron scattering amplitude... 

Fig. 40 Schematic phase diagram of the disordered phases for particles interacting with isotropic potentials (upper panel) and limited valence potentials (lower panel). In the first, standard case, the glass line hits the gas-liquid spinodal at large densities. In the limited valence case, the shrunk gas-liquid coexistence region leaves a new region in which a stable network with saturated bonding can develop. Reproduced with permission from [167]...

Now, consider another life-time , Ate, which is due to the potential energy. By taking a certain isotropic potential d> (r) - say Lennard-Jones type [15] and assuming the potential to be symmetric [16], there will be a characteristic frequency, uc, associated with the harmonic potential. Thus, Ate, can be written as ... 

To model the physical systems, i.e., write the Hamiltonian, particles are considered to be non-relativistic, charged, point masses interacting under an isotropic potential. The... 

In spite of the single form of these distance fuiK ions and the usual assumption that the angular expansion can be truncated after very few terms (for instance, only the isotropic and die first anisotropic L, Lg 0 terms), the number of parameters is mostly too large and these parameters are too strongly interdependent in affecting the measured properties, for a fully experimental determination of these parameters to be possible. Only for very simple systems such as atom- liatom systems or atom-tetrahedral molecule systems the experimental data could be used to yield a parametrized anisotropic potential of the form (1) and even there it appeared advantageous to extract part of the parameters from ab initio calculations. For other molecular systems only isotropic potentials are knovm mostly in simplified forms sudi as (6) or (7). 

For atomic systems this is of course all one needs. Especially in the case of rare gases the (isotropic) potentials are known quite accurately ... 

Quantum sum formulas based on exact pair distribution functions (obtained in the isotropic potential approximation) are also known for n = 0,1,2, and 3 [318,319] we mention also unpublished work by J. D. Poll. Levine has given detailed estimates based on classical moments, assessing the bound dimer contributions [302]. 

In the case of isotropic potential energy surfaces, such as appropriate (approximately) for ion A - induced dipole B systems, the situation is even simpler. In this case, the external rotational levels of A and B transfer unchanged into W(E,T) and g becomes a centrifugal partition function... 

This factor 77 may be called the decoherence function since it describes the effective loss of coherence in the fullerene state. For elastic scattering with an isotropic potential and the gas initially in a thermal state it reads [Homberger 2003 (b)]... 

Notes Calculations indicate that the Cj, structure is preferable. Isotropic potential... 

Malescio G., Franzese G., Skibinsky A., Buldyrev S., Stanley H. (2005) Liquid-liquid phase transition for an attractive isotropic potential with wide repulsive range, Phys. Rev. E, 71(6), 061504-061512. 

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