Application of Gaussian quadratures to density functional (df) theories of confined liquid crystals

We present a technique to conduct the density functional analysis of a liquid crystal system under the constraints of two parallel walls. The key feature of our approach is the way to perform multidimensional integrals in which certain regimes of the arguments are forbidden due to the effect of the wall excluded volume. This effect occurs in the surface potentials that contain orientationally dependent hard cores. The proposed calculations are based on the exact Gaussian quadratures and application of the modified limits of integration. They are quick and efficient and give reasonable orientation-density distribution functions for a relatively small number of the integration grid points. Another benefit of the proposed method is the fact that neither polynomial expansions nor interpolations of the integral elements is needed and the resulting solution for the distribution functions is only endowed by the error of the second virial approximation. Application of this algorithm can be extended to soft interactions by using the hard core cutoff of the soft potentials as the limits of integration.