  Most treatments of electron-electron correlations in dense plasmas either
ignore them entirely (random phase approximation) or neglect the role of ions
(jellium approximation). In this work, we go beyond both these approximations
to derive a new formula for the electron-electron static structure factor which
properly accounts for the contributions of both ionic structure and
quantum-mechanical dynamic response in the electrons. The result can be viewed
as a natural extension of the quantum Ornstein-Zernike theory of ionic and
electronic correlations, and it is suitable for dense plasmas in which the ions
are classical and the conduction electrons are quantum-mechanical. The
corresponding electron-electron pair distribution functions are compared with
the results of path integral Monte Carlo simulations, showing good agreement
whenever no strong electron resonance states are present. We construct
approximate potentials of mean force which describe the effective screened
interaction between electrons. Significant deviations from Debye-H\"uckel
screening are present at temperatures and densities relevant to high energy
density experiments involving warm and hot dense plasmas. The presence of
correlations between conduction electrons is likely to influence the
electron-electron contribution to the electron and thermal conductivity. It is
expected that excitation processes involving the conduction electrons (e.g.,
free-free absorption) will also be affected.
