# Is there a general / simple formula for Slater's rules?

I just learned how Slater's rules work on Wikipedia. These rules really are very simple. But the presentation of the rules seemed not very efficient. I would think there would be someway to set up and equation of some sort such that all you have to do is plug in the relevant quantum numbers and the element number and use some sort of formula to just compute Slater's rules more easily.

Is there such an equation / formula? If so, I'd like to see it. It would be more convenient than memorizing the rules in verbal form.

Sadly, no. It wouldn't be too hard to program a "get Slater shielding" script. But it wouldn't be terribly useful.

First off, let's back up briefly. Slater's rules allow you to estimate the effective nuclear charge $Z_{eff}$ from the real number of protons in the nucleus and the effective shielding of electrons in each orbital "shell" (e.g., to compare the effective nuclear charge and shielding for say $3d$ vs. $4s$ in transition metals).

Slater wanted to use these for quantum calculations, such that the radial wavefunction would be something like:

$$\psi_{ns}(r) = r^{n-1}e^{-\frac{(Z-\sigma)r}{n}}$$

So the rules he devised were fairly simple and produced fairly accurate predictions of things like the electron configurations of transition metals and ionization potentials.

Later, others performed better optimizations of $\sigma$ and $Z_{eff}$ using variational Hartree-Fock methods. For example, Clementi and Raimondi published "Atomic Screening Constants from SCF Functions." J Chem Phys (1963) 38, 2686–2689. (That article covers elements up to Kr.)

It's pretty obvious that no simple formula will cover these curves.

These days, basis set optimization is done by computer, and I doubt many people even glance at the exponents.

In my mind, Slater's rules serve two main purposes at this point:

• They're relatively simple to teach and learn and allow quick qualitative predictions, particularly for electron configurations in the transition metals.
• They emphasize that exponents for multi-electron atoms must be empirically derived.

I still teach them, particularly, for the first reason. To a lesser degree, they also reinforce the idea of effective nuclear charge and provide some semi-quantitive basis for that concept.

• So basically it is a slightly verbose / tedious rule of thumb that while very useful isn't physically accurate enough to be used for things requiring high precision. Correct? – Stan Shunpike Mar 13 '15 at 3:45
• I wouldn't say it's tedious. You can generally get the shielding constant in a minute or two. Considering he published in 1930, it's exceedingly good. Check the optimized exponents by Clementi, which aren't that different. Not many things in science stand up to 85 years. – Geoff Hutchison Mar 13 '15 at 3:48
• But no, for high precision, no one uses Slater rules because no one uses minimal basis sets like this anymore. With Moore's Law, I can probably do QM on a smart watch. – Geoff Hutchison Mar 13 '15 at 3:49
• Fair enough. I used it today. Maybe it just felt tedious for the 3 minutes it took me. I have never worked with minimal basis sets. Do you know a good introductory textbook that discusses them? I googled it, but I'd be interested in learning about them and related topics at a similar complexity level. I'm more familiar with physics, but I read chem too. – Stan Shunpike Mar 13 '15 at 3:53
• I'd recommend inorganic chemistry textbooks. We use Miessler and Tarr's Inorganic Chemistry which covers this. – Geoff Hutchison Mar 13 '15 at 15:45