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Appendix B: Polarization Formalism For Ellipsoids

Stratton [52] defines a set of numbers A_i which often arise in the discussion of the properties of ellipsoids. These numbers are defined by the integrals,

where

and the constants a₁, a₂, and a₃ are the ellipsoid semi-axes lengths. The A_i parameters obey the simple sum rule,

Eqn. (73) is useful since it reduces the integrals that need to be computed for a general ellipsoid from three to two. Various combinations and ratios of these numbers are required for the intrinsic viscosity calculation and are given in Appendix C. Note that Stratton uses the notation A_i for these quantities, while Haber and Brenner [14] and Scheraga [120] use α_i. We have followed Stratton's notation to avoid confusion with our notation for the polarizability.

The A_i integrals can be expressed as combinations of the standard elliptic integrals. However, for numerical purposes, it is just as simple to evaluate the integrals directly using Gauss-Legendre quadrature. It is first useful to transform the integrals by letting x = tan²(θ), so that 0 < θ < π/2. An integration mesh can be easily set up and enough points chosen to achieve convergence. For the smaller values of the ratios a₃/a₁ and a₃/a₂ (on the order of 50 or less) we needed less than 100 points in the quadrature mesh to achieve 5 to 6 significant figure accuracy, while for the highest values of these ratios studied (on the order 1,000) we needed 10,000 points in the integration mesh to achieve the same accuracy.

For ellipsoids of revolution we have a₁ = a₂ so that a₃/a₁ = a₃/a₂ = x, the aspect ratio and the number of integrals needed reduces to one. In this case we have A₁ = A₂ = (2- A₃)/2 and A₃ is given by,