An electron pair emitted from a crystalline surface system is represented by a relativistic two-electron scattering state, which contains the pair correlation mediated by a screened Coulomb interaction. This state is obtained from the solutions of two one-electron Dirac equations with potentials, which incorporate the electron¯electron interaction as a dynamical screening of the usual effective one-electron potentials. Numerical applications to the electron pair emission due to low-energy electron impact on a He atom and on a clean W(001) surface demonstrate that pair correlation effects can be quite strong and significantly improve the agreement of calculated results with experimental data.