Bounds to average cosines 〈cosθ12〉 in the Hartree–Fock theory of atoms

We show that the expectation value 〈cosθ12〉 and its subshell-pair components 〈cosθ12〉nl,n′l′ are bounded from below and above in the Hartree–Fock theory of atoms, where θ12 is the interelectronic angle subtended by the position vectors r1 and r2 of a pair of electrons, and n and l are the principal and azimuthal quantum numbers. The lower bounds to 〈cosθ12〉nl,n′l′ are −l/[2(2l−1)(2l+1)] for ll′ pairs with l=l′+1, while they are 0 for the other ll′ pairs, independently of n and n′. A weighted sum of these subshell-pair bounds gives a lower bound to 〈cosθ12〉 . The upper bounds are 0 in all the cases.