Perturbation Calculation of the Uniform Electron Gas with a Transcorrelated Hamiltonian

With a transcorrelated Hamiltonian, we perform a many body perturbation (MBPT) calculation on the uniform electron gas in the high density regime. By using a correlation factor optimised for a single determinant Jastrow ansatz, the second order correlation energy is calculated as $\frac{1-\ln2}{\pi^{2}}\ln(r_{s})-0.05075$. This already reproduces the exact logarithmic term of the random phase approximation (RPA) result, while the constant term is roughly $7\%$ larger than the RPA one. The close agreement with the RPA method demonstrates that the transcorrelated method offers a viable and potentially efficient method for treating metallic systems.