What the screening model cannot see

Every account of the periodic table leans on one sentence: an electron in a many-electron atom does not feel the full nuclear charge, because the other electrons get in the way. The sentence is true, the correction it names is large, and the standard way of estimating it is a set of rules John Slater published in 1930 that can be applied in a minute with no calculation at all.

Those rules are used in what an electron actually feels to settle the transition-metal question they were partly invented for. This essay runs them at the one place they are most often invoked and finds that they say nothing whatever.

One group, one charge #

Slater’s groups are (1s), (2s, 2p), (3s, 3p), (3d), (4s, 4p), and so on. The bracketing is the part of the rules most often lost in the retelling, and it is doing real work in two directions.

It is what makes the transition-metal answer come out: the 3d is a group of its own, screened at $1.00$ by everything inside it, while the 4s that follows is screened by that 3d at only $0.85$. That asymmetry is the whole reason a computed 3d ends up smaller and lower than a 4s at the same element.

And it is what makes the second-period answer come out empty. The 2s and the 2p are one group, so every electron screens a 2s electron and a 2p electron by identical amounts. Running the rules on the computed ground-state configuration of each element gives, for both orbitals:

Not approximately equal. Equal, by construction, at every element, and the same is true of 3s against 3p from silicon to argon — $4.15$, $4.80$, $5.45$, $6.10$, $6.75$, identical in each row.

What the ratio of radii is, and why it is fixed #

The mean radius of a hydrogenic orbital has a closed form, $\langle r\rangle = [3n^2 - l(l+1)]/2Z_{\text{eff}}$, so once the screened charge is fixed the size follows with no further computation. For $n = 2$ that is $6/Z$ when $l = 0$ and $5/Z$ when $l = 1$.

The screened charge enters as an overall factor. So if the two orbitals share it, their sizes are in the ratio $6:5$ at every element in the period, regardless of the element, and the numbers confirm it: $2.308$ against $1.923$ bohr at boron, $1.846$ against $1.538$ at carbon, $1.026$ against $0.855$ at neon. The ratio prints as $1.2000$ six times.

That is a striking prediction and it is striking in the wrong way. A model whose answer for a whole period is one number repeated is a model that has been asked a question it does not contain.

Notice also which way the inequality runs. The 2s is the larger orbital and it is the one that lies lower. Whatever separates the two in energy, it is not that one is drawn in tighter.

What actually separates them #

The distinguishing feature is visible in the radial functions and it is not their extent.

It is the shape near the nucleus. A 2s has one radial node, which cuts its density into an inner lobe and an outer shell with the inner lobe sitting very close in; a 2p has no radial node, one shell, and a density that goes to zero at the nucleus faster than the s does — as r² rather than as a constant, because the angular part of a p function vanishes at the origin. The radial distribution across the table draws both functions and both distributions.

Integrating each radial distribution out to a stated radius makes the difference a number. Within half a bohr of the nucleus, a hydrogenic 2s carries $0.969$ per cent of its density and a 2p carries $0.017$ per cent. The 2s has fifty-six times as much of itself in that region.

Within one bohr the figures are $3.43$ and $0.36$ per cent, a ratio of $9.4$. Within one and a half, $5.00$ and $1.86$, a ratio of $2.7$. And by two bohr the two have crossed: $5.265$ against $5.274$ per cent, equal to three figures.

So the whole of the difference is concentrated inside about two bohr, it is a factor of tens where it exists, and beyond that radius the 2p is very slightly the more compact of the two. That inner region is where the nuclear potential is deepest, which is why a small amount of density there is worth a large amount of energy.

The same construction on the 2p is the comparison, and the difference is in the first inch of the curve rather than anywhere later.

The two curves agree closely from about two bohr outward and disagree completely inside it, which is the whole of the penetration difference stated as a pair of integrals rather than as a pair of pictures. And the same failure of the screening rules repeats one shell further out, which is what makes it structural rather than a mis-set constant.

And the first period is the control, because there the two shells the rules are asked about differ in n rather than in l.

The third shell, where the ordering has three members #

One period further on the same integrals give a three-way ordering, and it is the ordering the whole of the transition series is built on.

Within half a bohr of a nucleus, a hydrogenic 3s carries $0.285$ per cent of its density, a 3p carries $0.0061$ per cent, and a 3d carries less than $0.0001$ per cent — which is to say nothing. Within one bohr the figures are $0.985$, $0.125$ and $0.0006$ per cent. The 3s has fifty times the 3p’s density in the inner region and the 3p has two hundred times the 3d’s.

Two things follow, and both are chemistry rather than bookkeeping.

The first is that the s-below-p-below-d ordering within a shell is not a small effect that happens to run one way. It is a sequence of factors of tens, arising from the $r^l$ behaviour at the origin: each unit of angular momentum multiplies the wavefunction by another power of $r$ near the nucleus, so each unit removes the orbital another order of magnitude further from the deepest part of the potential.

The second is that a 3d orbital, having essentially nothing of itself in the penetrating region, behaves in an atom almost as a hydrogenic orbital at whatever charge is left over after complete screening by everything inside it. That is exactly what Slater’s rules encode when they screen a d electron at $1.00$ by every inner electron — the rules capture the d case because the d case has no penetration to capture, and fail in the s-against-p case for the same reason in reverse.

By three bohr the order has reversed again: $2.26$ per cent for the 3s, $5.27$ for the 3p, $0.45$ for the 3d. The penetrating orbital is the one that has already spent its inner density and is on its way out; the crossings are why no single radius summarises this and why the integrals have to be done.

Why the rules cannot contain it #

Slater’s rules assign each electron a number and add the numbers up. The screening constant is a property of which shell an electron is in, and the shells are grouped by principal quantum number with s and p together.

Penetration is not a property of a shell. It is a property of the shape of a radial function — specifically of whether the function is non-zero at small radius, which is governed by $l$ through the $r^l$ behaviour at the origin and by the number of radial nodes. Every s function starts at a finite value at the nucleus; every p function starts at zero; every d function starts flatter still. A model that tabulates one constant per group has nowhere to put that.

It is worth being precise about what this does and does not indict. Slater fitted his constants to reproduce measured energies, and they do a creditable job of it for a one-parameter-per-group scheme. The complaint is narrower: the rules are routinely quoted as the explanation of the s-below-p ordering, and they compute the two as identical. The ordering is real and its explanation is elsewhere.

Where the rules do separate two shells #

The contrast is sharp one period further on, and it is what makes the failure above a specific one rather than a general complaint about the method.

At scandium the two effective charges happen to come out equal, at $3.00$ apiece, and the orbitals still differ in size by more than a factor of two — because $n$ differs and $\langle r\rangle$ contains $3n^2$. By iron the charges are $6.25$ and $3.75$, by zinc $8.85$ and $4.35$, and the 3d has contracted to $1.19$ bohr against the 4s at $5.52$.

That separation is what makes the transition-metal argument in what an electron actually feels work, and it exists because Slater’s grouping puts a d shell in its own group. Within a group, there is nothing.

What cannot be computed here #

The honest position on the s–p gap is that the penetration can be measured here and the energy cannot be computed.

The energy difference between a 2s and a 2p in a real atom is a difference of total energies of two many-electron states, and getting it requires the two-electron integrals — the electron-electron repulsion, evaluated between orbitals — which a one-electron treatment does not contain. Everything drawn here is hydrogenic: one electron, one nucleus, an exact solution, with the screening applied as a change of $Z$ and nothing else.

What is available is the experimental side, which is not small. Methane’s photoelectron spectrum, computed as a two-band prediction in character tables and reduction and measured at $12.7$ and $23.0$ electronvolts, is the carbon 2s and 2p regions ten electronvolts apart with the hydrogen 1s functions mixed in. Water’s four valence bands, treated in water’s lone pairs are not a pair, span twenty electronvolts for the same reason. The gap is large, it grows across a period, and it is what makes the 2s a core-like function in the heavier second-period elements and a valence function in the lighter ones.

One negative result is worth stating because it closes off the obvious alternative account. Take four different measures of size for the 2s and the 2p at Z = 1 — the mean radius, the root-mean-square radius, the most probable radius and the sphere holding ninety per cent — and how big is an orbital computes all four. On every one of them the 2s is the larger, and the 2s is the orbital that lies lower. So the ordering is not a size effect at all, and no measure of extent could have produced it.

What this does to the rest of the subject #

Three familiar arguments are downstream of the penetration difference rather than of any screening constant, and it is worth naming them because each is usually told with a screening constant in it.

Hybridisation costs energy. Mixing 2s character into a bonding function means promoting density out of the penetrating orbital into the non-penetrating one, and the price is the s–p gap. Hybridisation does not explain makes the case that the mixing is a change of basis rather than a physical event; the promotion energy is the reason the change of basis is not free when the occupation changes with it.

Electronegativity tracks s character. An orbital with more s in it holds an electron more tightly because the s part penetrates, which is the mechanism behind Bent’s rule as Bent’s rule, against the substituent series measures it, and behind the s-character budget that the angle does not fix the hybridisation computes from bond angles.

The filling order is not the energy order of an isolated atom. The aufbau order is not a property of the atom shows every one-electron estimate putting 3d below 4s at every element while the filling goes the other way. The same distinction applies here in a milder form: what a rule of thumb ranks and what a Hamiltonian ranks are different lists.

Drawn at ninety per cent of their own densities on one scale, the two are almost the same size: the 2s surface is a shell inside a shell reaching 9.13 bohr, and the 2p reaches the same 9.13 and holds its density in one lobe pair — say what it encloses draws both. The pictures show the shapes and say nothing about the energies, which is the honest state of this argument.

A failure of structure rather than of fitting #

The model gives the 2s and the 2p the same number, and it is worth being precise about why no better fitting could repair that — because the usual response to a rule that gets something wrong is to adjust its constants.

Slater’s rules assign electrons to groups, and the s and p electrons of a shell are one group. That is a decision about the form of the model, not a value of a parameter. Whatever screening constant the group is given, both orbitals receive it, and the difference between them is identically zero at every element. There is no set of numbers that fixes it, because the quantity being predicted has been defined away.

So the failure is not that the rules are crude. It is that they have no term of the right shape, and a model with no term of the right shape produces a zero rather than a poor estimate.

What the missing term would have to represent is the difference in radial node structure. A 2s function has one radial node and therefore amplitude close to the nucleus, inside the 1s shell, where the screening has not taken effect; a 2p has none and no such inner region. The 2s is held more tightly because it penetrates, and penetration is a property of the count $n - l - 1$, which is exactly the quantity a rule organised by shell cannot see.

The size of what is being missed is worth stating, because it is not a correction. The separation between the 2s and 2p levels is several electronvolts in every element from boron to neon — comparable to a chemical bond, and larger than most of the effects computed carefully elsewhere.

It is also, of the two quantities in the neighbourhood, the more consequential one.

Across a period the effective charge rises, and the rules capture that: they get the trend that makes atoms smaller and ionisation energies larger from left to right.

Within a shell the s lies below the p, and the rules give nothing. That separation is what makes the s and p blocks of the periodic table distinct, what makes hybridisation a meaningful operation — mixing two orbitals of different energy, with the mixing costing something — and what underlies every argument about s character in this collection.

So the model gets the variation the periodic table is arranged by and misses the variation that decides what happens inside each square of it. That is a more damaging division than a uniform inaccuracy would have been, and it is the reason the rules are a teaching device rather than a working tool.

The repair that works is available and is worth naming, because it shows what the fix costs. Effective charges obtained by fitting to full atomic calculations — one per orbital rather than one per group — do separate the 2s from the 2p, and they are tabulated for every element. What they give up is the thing that made Slater’s rules worth teaching: they cannot be worked out on paper from the configuration, they carry no argument inside them, and a reader using them is using somebody else’s calculation rather than following a construction.

That is the ordinary trade in this collection and it is worth stating rather than resolving. A rule simple enough to derive has a structure, and a structure has things it cannot represent. Slater’s cannot represent the difference between an s and a p in one shell, and the honest response is to say so and use it for the trend it does capture, rather than to adjust constants that are not where the difficulty lies.

Who found it, and when #

Slater’s paper is Atomic Shielding Constants, Physical Review, 1930, and its purpose was to supply analytic one-electron functions that could be written down rather than computed — the whole apparatus of Hartree’s numerical self-consistent field being, at the time, weeks of hand calculation per atom. The constants were fitted to reproduce measured energies and sizes across the table, and the grouping of s with p was one of the fitted choices rather than a derived result.

The penetration argument is older than the rules and was in the spectroscopic literature well before quantum mechanics: the quantum defect, the amount by which an alkali metal’s terms depart from hydrogen’s, was known empirically from the 1890s and depends strongly on $l$ — large for s terms, smaller for p, nearly zero for d and f. That empirical ordering is the same fact this essay computes as an integral, seen thirty years earlier through a spectrograph.

So the two accounts were never in competition. The rules estimate the charge and the quantum defect measures the shape, and only the second has an $l$ in it.

One number per object #

This essay and its neighbours are about quantities that look like one number and are not. How big is an orbital found four measures of size disagreeing by a factor of two and a half; one level is not one comparison found the contour level failing to be a measure of size at all; and this one finds the effective charge failing to separate two orbitals that every chemical argument needs separated.

The pattern is worth stating because it recurs across chemistry: a model that returns one number per object cannot express a difference that lives in the object’s shape. The ligand-field parameters have the same character, and so does a force constant. What makes those models useful is that they are checked against something that does see the shape, which is what spectra are for.