The electronic extracule density represents the probability density function for the center-of-mass motion of any pair of electrons. Two methods are presented for the calculation of electronic extracule moments [R-n] in position space and [P-n] in momentum space of atomic systems within the Hartree-Fock framework, where R and P are respectively the position and momentum distances between the center-of-mass of any pair of electrons and the coordinate origin. The methods;are applied to the neutral atoms from He to Xe in their ground state, and the moments [R-n] with n = -2 to +4 and [P-n] with n = -2 to +3 are reported based on the numerical Hartree-Fock calculations. These extracule moments are pointed out to have intimate relations with their partner intracule. moments. The present results, together with the previous ones for the electronic intracule moments [J. Chem. Phys., 107, 8510 (1997)], compile accurate Hartree-Fock electron-pair moments for all the 53 atoms He through Xe in both position and momentum spaces.