Statistical angular correlation coefficients and second electron-pair moments for atoms

Statistical correlation coefficients τ[f] introduced by Kutzelnigg et al. [Phys. Rev. 172, 49 (1968)] provide overall measures of the difference between the electron-pair density and the product of single-electron densities, where f = f(r) is a probe function. It is shown that the angular correlation coefficient τ[r], a particular case of τ[f] for f(r) = r, is simply expressible in terms of the generalized electron-pair moments 〈q^2〉(a,b) with two real-valued parameters a and b. Especially, the relation for a = b = 1 means that τ[r] is nothing but the difference between the center-of-mass motion and relative motion contributions in the second electron-pair moments. Conversely, the electron-pair moments 〈q^2〉(a,b) are obtained from the single-electron moment 〈{∑_i}{(r_i)^2}〉 and the correlation coefficient τ[r]. The same is also true in momentum space