  Correlation effects of an electron gas in an external potential are derived
using an Effective Action functional method. Corrections beyond the random
phase approximation (RPA) are naturally incorporated by this method. The
Effective Action functional is made to depend explicitly on two-point
correlation functions. The calculation is carried out at imaginary time. For a
homogeneous electron gas, we calculate the effect of exchange on the ring
diagrams at zero temperature and show how to include some of the ladder
diagrams. Our results agree well with known numerical calculations. We conclude
by showing that this method is in fact a variant of the time dependent density
functional method and suggest that it is suitable to be applied to the study of
correlation effects in the non-homogeneous case.
