For the systems under study, the interactions consist of pairwise
full density matrix given by the Feynman-Kac relation,
Electrons and protons interact via the Coulomb potential. For the two particle problem, the eigenfunctions of the Hamiltonian can be expressed in terms of special functions and the pair density matrix can be calculated by performing the sum over all states (Pollock, 1988). However, the states are only known analytically in an infinite volume. For the purpose of a simulation in a periodic cell, the Coulomb potential is broken up into a short range part in real space and a long range part in -space using an optimized Ewald break-up (Ewald, 1917) developed by (Natoli and Ceperley, 1995). For both parts separately, a pair action will be derived as discussed in the following sections. It should be noted that the break up of the potential is an approximation, which is made in order to calculate the pair density matrix corresponding to a long-range pair potential in a periodic system. Ideally, one would calculate the full action and then perform an Ewald break-up.