(untitled) (CSIAM Transactions on Applied Mathematics, 2022)
We discuss the expansion of interaction kernels between anisotropic rigid molecules. The expansion decouples the correlated orientational variables, which is the 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 molecules can be fully recognized. Thus, writing down the form of expansion consistent with the symmetries is significant. Symmetries of two types are considered. First, we examine the symmetry of an interacting cluster, including the translation and rotation of the whole cluster, and label permutation within the cluster. The expansion is expressed by symmetric traceless tensors, with the linearly independent terms identified. Then, we study the molecular symmetry characterized by a point group in O(3). The proper rotations determine what symmetric traceless tensors can appear. The improper rotations decompose these tensors into two subspaces and determine how the tensors in the two subspaces are coupled. For each point group, we identify the two subspaces, so that the expansion consistent with the point group is established.