Inhomogeneous relativistic electron gas: Correlation potential

The correlation energy of a relativistic electron gas based on a sum of ring diagrams including the transverse-photon contributions is numerically calculated. From this, a local, relativistic correlation potential is deduced which appears in the density-functional theory of inhomogeneous systems where relativistic effects make important contributions. Comparison with its nonrelativistic counterpart shows that the use of nonrelativistic results is very poor. The correlation contribution to the potential is found to be about 5% of the exchange for $\beta =2.5\mathrm{}$ to about 2.5% for $\beta =1\mathrm{} (\beta =\frac{k_F}{m})$.