Periodic Boundary Conditions (PBCs) are commonly utilised in many simulations of great importance in computational science, both as methods to accurately simulate perfect crystals, as well as to approximate macroscopic systems. When considering the latter case, the use of these boundary conditions restricts the wavevectors for which the Fourier components of quantities calculated from these systems can be non-zero. In particular, the minimum non-zero wavevector that can be examined, is inversely proportional to the size of the simulation cell. This means the examination of small, non-zero wavevectors can require the use of large systems, that may become prohibitively expensive to simulate. This can potentially inhibit the study of long-range contributions to the properties of the system.

In this work, Bloch boundary conditions are introduced for systems that include charges. In these, the particle positions and momenta are periodic, as in PBCs, but the charge multipoles in image cells are changed in phase controlled, by a wavevector within the first Brillouin Zone, q, that is characteristic of the boundary conditions. In such cases, the accessible wavevectors are then all shifted by q, and so the minimum accessible wavevector is now q.

A demonstration of how the Ewald summation is modified for energy and force calculations necessary for the simulation of these systems is given, explicitly for that of charges and dipoles, and a scheme to derive this for higher multipoles is given.

This framework is then applied to the longitudinal and transverse dielectric constants of dipolar fluids subject to static electric fields, both Stockmayer and polarisable dipole fluids. This is used to verify the ratio of the transverse and longitudinal susceptibility being the dielectric constant, in the limit q goes to zero. A brief analysis of how this may be applied to systems with dynamic electric fields is also given.