Effective nuclear charge and electron repulsion

Question

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?

Answer 1

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.

Answer 2

$Z_\mathrm{eff}$ is an empirically derived value, commonly estimated by Slater's Rules. In the Slater system, a shielding parameter $S$ is calculated for a given electron by adding up the shielding constants for all other electrons on the atom with $n$ equal or less than the electron of interest.

If the electron of interest is in an $s$ or $p$ orbital, the shielding constants for the other electrons are given as:

1.00 for any electron in a shell with $n$ less than or equal to 2 less than the electron of interest. (Example: For a 3p electron, the shielding constant of a 1s electron is 1, because $3-1\geq 2$)

0.85 for any electron in a shell with $n$ that is 1 less than the electron of interest.

0.35 for any electron in the same shell as the electron of interest (except 1s, which fits better to a value of 0.30)

If the electron of interest is in a d or f orbital, the shielding constants for the other electrons are given as:

1.00 for any electron in a lower shell

0.35 for any electron in the same shell as the electron of interest

To calculated the parameter $S$ for a given electron from these constants, one simply adds the constants for all other electrons on the atom. Then the effective charge is calculated as $$Z_\mathrm{eff} = Z_\mathrm{actual} - S.$$