An invariant expansion of the two-body statistical correlation function of a fluid is proposed. This expansion does not depend on any particular reference frame used to define the orientation of the molecules, and therefore can be reduced to the expansions of the literature in a simple way. The new expansion permits a rather convenient way of including the effects of molecular symmetry into it. The expressions for a few thermodynamic properties in terms of this expansion are obtained. The equations for x-ray, neutron, and light scattering are somewhat simpler using this expansion. The Ornstein—Zernike equation has a very convenient form, and is given in Fourier transformed form in terms of 6j angular recoupling coefficients.
The irreducible representations of the three-dimensional rotation group are obtained directly from the irreducible representations of its infinitesimal generators (the spin matrices). parametrized in terms of the rotation angle and the direction of the rotation axis. Expressions are given for the rotation operator exp(i IjIn . S) in terms of two different bases of 2j + I elements for spin j. The results are related to the spectral decomposition of the rotation operator and expressions obtained for spin projection operators along any spatial direction for arbitrary spin.is useful for this purpose only if the series can be summed in some way. There are two well-known cases in which this can be readily done, namely, the twodimensional (spin i) and the three-dimensional (spin 1)representations.
The expansion of the exponential of a tensorial expression such as the interaction or pair correlation function between two nonspherical molecules 1, 2 is of the form ∑mnl λmnlΦmnl(12), where Φmnl(12) are invariant tensorial expressions that depend only on the orientation of 1 and 2. The generating function e−∑mnl λmnlΦmnl =∑pqt ipqt(λ) Φpqt defines a generalized Bessel function (GBF). We discuss integral representations and recurrence relations for the GBF. The first GBFs for dipolar and linear quadrupolar exponents, which are of interest in the theory of ionic solutions are computed explicitly.
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