Exchange and correlation contributions to superconducting electron pairing

The screening of the electron-phonon and Coulomb interactions in an electron gas is considered with use of various forms of the exchange correction term G(q). A solution of the Eliashberg equations including G(q) is achieved with conventional approximations, and the resulting superconducting transition temperatures $T_c$ are found to be very sensitive to the form of exchange. Smooth functional forms of G(g), such as the Hubbard model, yield only small deviations from the random-phase approximation, which greatly overestimates $T_c$ for simple metals such as Na and Li. However, an exchange function with a peak near q≊2$k_F$ (e.g., the forms proposed by Devreese and independently by Overhauser) greatly reduces $T_c$ and may explain the mysterious absence of superconductivity in simple metals such as Na, Li, and Rb. These many-body effects may also be significant in superconductors with more complex energy-band structures.