Atomic one- and two-electron subshell moments in position and momentum spaces

Abstract.

For 357 subshells of the 53 neutral atoms He through Xe in their ground states, the two-electron intracule (relative motion) <u ^k> _nl and extracule (center-of-mass motion) <R ^k> _nl subshell moments in position space are examined as well as their counterparts <v ^k> _nl and <P ^k> _nl in momentum space, where n and l are the principal and azimuthal quantum numbers of the atomic subshell, respectively. It is clarified that between the intracule and extracule moments the “2^k-rule” is strictly valid, which means <u ^k> _nl = 2^k<R ^k> _nl and <v ^k> _nl = 2^k<P ^k> _nl for any nl subshell. Theoretical analysis also proves that for a particular case of k = +2, two relations <u ²> _nl = (N _nl −1)<r ²> _nl and <v ²> _nl = (N _nl −1)<p ²> _nl hold exactly, where N _nl (≥2) is the number of electrons in the subshell nl, and <r ^k> _nl and <p ^k> _nl are the familiar one-electron subshell moments in position and momentum spaces, respectively. The latter equality establishes a new and rigorous relation between the second electron-pair moments in momentum space and the total energy of an atom through the virial theorem. For k=+1, −1, and −2, the numerical Hartree-Fock results for the 357 subshells show that there are approximate but accurate linear relations between <u ^k> _nl and <r ^k> _nl and between <v ^k> _nl and <p ^k> _nl , in which the proportionality constant in each space depends on n,l, and k.