Dynamical correlations in a two-dimensional electron gas: First-order perturbation theory

First-order Feynman diagrams for the proper polarizability of a two-dimensional electron liquid in the jellium model, valid for arbitrary wave vector and frequency, are evaluated, their singularities and analytical properties are examined, and a reliable numerical procedure for calculating them is established. Quantities such as the dielectric function ε(→k,ω), the dynamic-complex-local-field factor G(→k,ω), dynamic structure factor S(→k,ω), static structure factor S(→k), and the pair correlation function $g(r)$ have been calculated and compared with the corresponding quantities in the random-phase approximation (RPA). Comparison has also been made with the three-dimensional case. A closed expression for the plasmon dispersion within the RPA has been derived. The first-order theory interestingly enough predicts a large enhancement of the static polarizability for k=2kF, which could play an important role in predicting charge-density-wave instabilities in metallic layered compounds.