Pair Density Matrix

For the systems under study, the interactions consist of pairwise additive potentials, $V({\bf R}) = \sum_{i<j} V_{ij}({\bf r}_i-{\bf r}_j)$. The full density matrix given by the Feynman-Kac relation,

$$\left< e^{- \int_0^\tau dt \; V({\bf R}(t)) } \right>_{\rm ... ...^\tau dt \; V_{ij}({\bf r}_{ij}(t)) } \right>_{\rm BRW} \quad,$$ (39)

which, in the limit of sufficiently small $\tau$, can by approximated by a product of pair density matrices

$$\left< e^{- \int_0^\tau dt \; V({\bf R}(t)) } \right>_{{\rm... ...}({\bf r}_{ij}(t)) } \right>_{{\rm BRW}_{{\bf r}_{ij}}} \quad.$$ (40)

This is known as the pair approximation. It means that the correlation of two particles becomes independent of other particles within a sufficiently small time interval. The derivation of the pair density matrix will be discussed in the remaining part of this section.

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 $k$-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.