The long range part of the potential leads to long range action. As a
first step, it can be calculated from the primitive approximations as
used in (Shumway, 1999). Alternatively, one can use the random phase
approximation (RPA) (Pines and Nozieres, 1989) to obtain an improved long range
action (for details see (Magro, 1996)), which is constructed in such
a way that the *local energy* is minimized,

(48) |

where is the Fourier transform of the potential between particle of type and denoted by T and T. Setting , leads to three body terms, which are approximated by the RPA. The resulting first order differential equations, which are integrated numerically in imaginary time from to with the initial condition . The calculated coefficients then enter PIMC simulations as long range action in form of Eq. 2.42.