To the best of my knowledge, the effective nuclear charge $Z_\mathrm{eff}$ is usually defined such that the potential energy of an electron in an atom, $\langle V \rangle$, can be expressed as
$$\langle V \rangle = -Z_\mathrm{eff}\langle\psi|\frac{1}{|r - r_a|}|\psi\rangle$$
The Hartree-Fock expression for $\langle V \rangle$, meanwhile, is
$$\langle V \rangle = - Z\langle\psi | \frac{1}{|r - r_a|} | \psi\rangle + \sum_i \langle\psi |\hat{J_i} |\psi\rangle - \sum_{i \in \sigma} \langle\psi | \hat{K}_i | \psi\rangle$$
Is there any relation between these expressions? It was taught in college Chemistry that $Z_\mathrm{eff}$ accounts for electron shielding, which is simply electron repulsion effects. Is the effective nuclear charge just a semiempirical value fitted to approximate the total potential energy of an electron in an orbital, or is there another meaning behind the number?
Based on what I know, Clementi et al’s values for $Z_\mathrm{eff}$ are not evaluated based on this principle. Could I inquire why this is not the case?