Temperature-dependent Coulomb interactions in hydrogenic systems

In a hydrogenic plasma, the effective potentials $V_{\mathrm{ij}}\mathrm{}\left(r\right)\mathrm{}$ of electron-proton, electron-positron, and electron-electron systems are considered. The inclusion of quantum-mechanical effects leads to a finite value of $V_{\mathrm{ij}}\mathrm{}\left(0\right)\mathrm{}$ at the origin. For each system simple and analytical expressions are obtained at high temperature and small interparticle distance. Barker's numerical computations are extended to the high-temperature domain, small-$r$ region, and electron-positron system. The symmetry effects are included for the electron-electron system. Various simple approximate formulas are obtained which describe $V_{\mathrm{ij}}\mathrm{}\left(r\right)\mathrm{}$ at all distances. Some of them are in excellent agreement with the numerical results.