Symmetry-Consistent Expansion of Interaction Kernels Between Rigid Molecules (CSIAM Transactions on Applied Mathematics, 2022)
We discuss the expansion of interaction kernels between anisotropic rigidmolecules. The expansion decouples the correlated orientational variables, which isthe crucial step to derive macroscopic free energy. It is at the level of kernel expansion,or equivalently the free energy, that the symmetries of the interacting rigid moleculescan be fully recognized. Thus, writing down the form of expansion consistent with thesymmetries is significant. Symmetries of two types are considered. First, we examinethe symmetry of an interacting cluster, including the translation and rotation of thewhole cluster, and label permutation within the cluster. The expansion is expressedby symmetric traceless tensors, with the linearly independent terms identified. Then,we study the molecular symmetry characterized by a point group in $O(3).$ The properrotations determine what symmetric traceless tensors can appear. The improper rotations decompose these tensors into two subspaces and determine how the tensors inthe two subspaces are coupled. For each point group, we identify the two subspaces,so that the expansion consistent with the point group is established.
Keywords: Liquid crystals; molecular symmetry; point groups; free energy; interaction kernels; gradient expansion.