Legendre polynomials, intermolecular

The anisotropy of the liquid crystal phases also means that the orientational distribution function for the intermolecular vector is of value in characterising the structure of the phase [22]. The distribution is clearly a function of both the angle, made by the intermolecular vector with the director and the separation, r, between the two molecules [23]. However, a simpler way in which to investigate the distribution of the intermolecular vector is via the distance dependent order parameters Pl+(J") defined as the averages of the even Legendre polynomials, PL(cosj r)- As with the molecular orientational order parameters those of low rank namely Pj(r) and P (r), prove to be the most useful for investigating the phase structure [22]. 

The pair interaction potential integrated over the intermolecular vector ri2 and the short molecular axes bi and b2 only depends on the coupling between the long axes (ai a2), and thus can be expanded using Legendre polynomials Pn(ai a2). Taking into account the first non-polar term we obtain ... 

Maier and SaupeS first pointed out the importance of attractive orientational interactions such as the induced dipolar forces. Later work included both the hard core repulsion and Van der Waals attraction in a mean field approximation. The total effective pseu-dopotiential is found by expansion of the intermolecular interaction in spherical harmonics for cylindrical rods. The anisotropic part therefore includes all interactions with the symmetry of the quadrupolar Legendre polynomial P2(Cos 0). 

The vibrationaUy averaged intermolecular potential V(i, 6) for an atom-diatom system is conventionally expanded in Legendre polynomials. 

Realistic intermolecular interaction potentials for mesogenic molecules can be very complex and are generally unknown. At the same time molecular theories are often based on simple model potentials. This is justified when the theory is used to describe some general properties of liquid crystal phases that are not sensitive to the details on the interaction. Model potentials are constructed in order to represent only the qualitative mathematical form of the actual interaction energy in the simplest possible way. It is interesting to note that most of the popular model potentials correspond to the first terms in various expansion series. For example, the well known Maier-Saupe potential JP2 (Sfli )) is just the first nonpolar term in the Legendre polynomial expansion of an arbitrary interaction potential between two uniaxial molecules, averaged over the intermolecular vector r,-, ... 

It was first noticed by Gelbart and Gelbart [16] that the predominant anisotropic interaction in nematics results from a coupling between the isotropic attraction and the anisotropic hard-core repulsion. This coupling is represented by the effective potential Veff(l, 2) = V(l, 2) Q(ji 2 z)- This potential can be averaged over all orientations of the intermolecular vector and then can be expanded in Legendre polynomials. The first term of the expansion has the same structure as the Maier-Saupe potential /(r,2)72((fli 02)) but with the coupling constant J determined... 

The Gegenbauer polynomials arise quite naturally when one wants to expand any inverse power of the distance into a series in terms of angular variables. Such problems are very frequent when dealii with intermolecular forces, which are almost always expressed as such inverse powers of the intermolecular distances. One can say that Gegenbauer polynomials have the same importance in this field as Legendre polynomials in electrostatics. 

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