Question

Why are 7 electrons so much worse in shielding the nuclear charge than 8?

What IS electron shielding, anyway (I suppose I ask for a qualitative quantum mechanical explanation)?

Answer 1

Well, first, they're worse simply because there's only 7 of them, instead of 8. Simple electrostatics. And 8 electrons aren't necessarily a lot better than the one extra electron would suggest -- it's only if they form a closed shell that things might improve a little, and the reason for that is again simple electrostatics: a closed shell implies that the angular distribution of electron density is spherically symmetric. If the shell is NOT closed, there will be some favored directions for the electron density. Electrostatics tells you that the maximum possible shielding by a charge distribution is achieved when it is spherically symmetric about the charge you're trying to shield.

You don't need QM to understand shielding. It's a purely classical effect, and it's just that if I interpose an electron between the nucleus and another (valence) electrons, the shielding electron will repel the valence electrons, cancelling out some of the attraction from the nucleus. We describe it as an *effective* reduction of the charge on the nucleus, but that's just a convenient shorthand. What's actually going on is force cancellation: the nucleus attracts, the intervening inner electrons repel, the outer electrons, and the repulsion from the inner electrons (or actually electrons in the same shell, too) cancels part of the attraction from the nucleus. The net effect is, at the most crude level of approximation, similar to what would happen if the charge on the nucleus were weakened and you ignored the repulsion from the other electrons.

A lot of descriptions of the atom are like this: what physicists call "mean field" descriptions, or theories. The idea is that it's hard to wrap your head around a dozen or two particles all orbiting around and attracting or repelling each other. Hard to get intuition about such a "many body" problem. So what we do is (over)simplify a bit: we consider only one particle at a time, or perhaps two (e.g. a valence electron and the nucleus), and we average out the influence of all the other particles, and represent their influence with a "mean" (meaning "average") field that no longer depends on the exact position at a given time of the other particles.

In the case of the atom and its electrons, when we consider just one electron, the crudest possible mean field approximation of the influence of all the other electrons is just to say that they "renormalize" the charge on the nucleus -- and then we treat the valence electron as responding to the Coulomb force from this renormalized (lower) charge. That's a very crude approximation, but it gets us in the right direction, and helps explain a lot of the chemistry of the elements.

We can certainly do better: we could renormalize the entire electric potential generated by the nucleus, insted of just its charge. It would no longer be a potential that fell off with 1/r, because the density of the other electrons is not constant. In fact, it would be a field that fell off like Z/r at short distances (Z being the nuclear charge), Zeff/r at large distances, and exponentially (roughly) in between. A complicated shape. Then we could compute the orbit of the valence electron in this renormalized field, and we'd get a much better description of its behaviour.

In fact, we can go one step further: we could start with some *arbitrary* spherically-symmetric field, q(r). Compute the orbitals of each electron using q(r). Now use this information to construct the density of electrons and *recompute* the mean field they exert on each other -- which gives us a better q(r). Now go back and recompute the orbitals, then construct the density, and improve q(r) again. Around and around, until things stop changing. This is called self-consistent field (SCF) theory, and it is the first and most important way in which the properties of electrons in atoms (or any many-body situation) are computed when you are trying for high precision.

Answer 2

I asked for QM because I had an image of the electrons both repelling AND reducing the photons from the Nucleus pulling on the outer electron. Thanks for clearing that up, and for the aside about SCF.

Anyway, thanks again.

Answer 3

Anything explainable by forces alone is explainable by classical mechanics, because QM changes nothing about the nature of forces and energy. There are only three things you ever need QM for: (1) to explain why various observables of a bound state are quantized, e.g. why the spectrum of H is a line spectrum, or why the H atom doesn't collapse, or why the SHO has zero-point energy, or the structure of the Periodic Table; (2) why some observables cannot be simultaneously measured to arbitrary precision, e.g. why it isn't possible to know all components of the angular momentum of an electron in an atom; and (3) why certain transitions can take place that appear to violate conservation laws, e.g. tunneling.