Effective nuclear charge and electron repulsion 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?
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?


No, that's pretty much it. The Wikipedia entry has a partial periodic table with some $Z_\mathrm{eff}$ values. If you go further down to larger atoms, it becomes obvious that $Z_\mathrm{eff}$ decreases quite a lot towards the outer shells, so you could argue that the concept has its merits in capturing a quite significant effect.