Jacobi Coordinates on Hyper-tori and Geometric Factors in the Volume Dependencies

We derive the volume dependence of bound states from a cluster-cluster picture with nucleon degrees of freedom. To achieve this, we demonstrate how to construct Jacobi coordinates on the lattice under the periodic boundary. A constant factor called ``Geometric factor'' appears in the generalization from point-like particles to clusters. We validate our derivation using many-body calculations, specifically, we find this factor to be essential in extracting asymptotic normalization constants from lattice calculations of \isotope[16]{O} ground state.