What an electron actually feels

Hydrogen has one electron and one proton, and every hydrogenic orbital is a solution for exactly that situation. Every other atom has more, and the extra electrons get in the way.

That interference has a name — screening, or shielding — and a shape. It is not a small correction to be waved at: for a 2p electron in neon it removes more than four tenths of the nuclear charge, and what remains is what decides the size of the atom, the energy needed to remove an electron from it, and the order in which its shells fill.

The charge that is left #

An electron far from the nucleus of a lithium atom sees a nucleus of charge three with two electrons already wrapped around it. If those two screened perfectly, the outer electron would feel a charge of one and lithium’s outer orbital would be hydrogen’s.

They do not screen perfectly. The outer electron spends part of its time inside them, where they screen nothing at all, and the amount of time depends on the shape of its radial distribution — the quantity where the electron is is about — rather than on any count of electrons.

So the honest statement is a subtraction with an unknown in it:

$$Z_{\text{eff}} = Z - \sigma,$$

where $\sigma$ is the screening constant and nothing so far says what it is.

Slater’s answer, and what kind of answer it is #

In 1930 John Slater published a set of rules for $\sigma$ that are still in every textbook, and they are worth stating precisely because their form says what they are.

Sort the electrons into groups — 1s, then 2s and 2p together, then 3s and 3p together, then 3d, then 4s and 4p, and so on. For an electron in one of those groups:

• every other electron in the same group screens by 0.35, except within 1s where it is 0.30;
• for an s or p electron, each electron in the shell one below screens by 0.85, and each electron further in screens by 1.00;
• for a d or f electron, every electron in an earlier group screens by 1.00;
• electrons in later groups screen by nothing at all.

Those numbers are fitted. They were chosen to reproduce measured ionisation energies and atomic sizes, and different fits give different numbers — Clementi and Raimondi’s, published in 1963 from Hartree–Fock calculations, disagree with Slater’s by a few tenths almost everywhere.

That does not make them useless. It makes them a model with a parameter, which is a different sort of object from arccos(−1/3), and this site’s rule is that the difference has to be stated rather than blurred. Everything in this essay computed from the rules is exact; the rules themselves are a fit.

What the rules give, and what follows exactly #

Applying them is arithmetic once the configuration is known, and the configuration is itself computed here rather than typed in: shells are filled in order of $n + l$, lowest first, with $n$ breaking ties.

For carbon’s 2p electron the sum runs: three other electrons in the same group at 0.35, two 1s electrons at 0.85. That is 2.75, so $Z_{\text{eff}} = 3.25$ against a bare charge of six. Counting the inner electrons at one apiece — the version most often reached for — would have given 2.00, and it is wrong by a third.

From that number everything about the orbital’s extent follows in closed form, with no further fitting:

$$\langle r\rangle = \frac{3n^2 - l(l+1)}{2Z_{\text{eff}}}, \qquad \langle r^2\rangle = \frac{n^2\left[5n^2 + 1 - 3l(l+1)\right]}{2Z_{\text{eff}}^2}.$$

Carbon’s 2p mean radius comes out at 1.54 bohr; neon’s at 0.85. The second-period contraction is a factor of nearly three between lithium and neon, and that is what the figure above draws.

Raise the charge to what a carbon 2p feels and the same function contracts by a factor of three and a quarter, with every angular feature exactly where it was.

At an oxygen’s effective charge it has contracted again by nearly a factor of two, and the sequence is linear in the reciprocal of the charge rather than in anything about the element.

The comparison that decides the periodic table #

The first three rules above are ordinary. The fourth is the one that does the work: a d electron is screened by everything inside it, completely, while an s electron in the shell above is screened by the shell below it only at 0.85.

Follow the consequence at scandium, the first transition metal. Its 3d electron feels 3.00. Its 4s electron also feels 3.00. The two numbers are equal, and the orbitals they belong to are not remotely the same size: the closed form gives 3.50 bohr for the 3d and 8.00 for the 4s.

By zinc the numbers have separated. The 3d electron feels 8.85 and sits at 1.19 bohr; the 4s electron feels 4.35 and sits at 5.52. The 3d orbital is now less than a quarter of the size of the 4s — and it is still the 4s that was filled first and the 4s that is ionised first.

That is not a contradiction, and treating it as one is the source of a great deal of confusion. Size and energy are different questions. The 4s orbital is large, but its radial distribution has a small inner peak that reaches inside the argon core, where the screening fails and the full nuclear charge is briefly felt. That peak is small in area and enormous in effect, and it is drawn in the radial distribution across the periodic table. It exists because the 4s function changes sign three times on the way out, and the sign changes are what nodes counts.

The same treatment applied to a shell one row down gives the number that decides which orbital an atom fills first, and it is not the number the row would suggest.

The test the model can fail #

A fitted model earns its place by predicting something it was not fitted to, so it is worth asking what these rules get right beyond the sizes they were tuned on.

The clean test is the ionisation energy, because the hydrogenic expression for it,

$$I \approx \frac{Z_{\text{eff}}^2}{2n^2} \text{ hartree},$$

involves nothing but the screened charge and the principal quantum number. For lithium’s 2s electron the rules give $Z_{\text{eff}} = 1.30$, and the expression returns 0.211 hartree, or 5.75 electron volts, against a measured 5.39. For sodium’s 3s, $Z_{\text{eff}} = 2.20$ gives 6.58 eV against a measured 5.14. For potassium’s 4s, 2.20 again gives 4.11 eV against 4.34.

Three predictions, all within about a quarter of the measured value, from a rule that takes ten seconds to apply and knows nothing about any of the three elements except how many electrons they have. That is a good model.

It is also visibly a model rather than a theory. Sodium’s error is twenty-eight per cent in one direction and potassium’s is five per cent in the other, and no amount of care in applying the rules will fix that, because the failure is in the assumption that a screened hydrogenic function is the right shape. The same pattern runs through this site: VSEPR computed predicts the tetrahedral angle exactly and the water angle badly, for the same kind of reason.

The contraction nobody expects #

One consequence of the fourth rule is worth following because it is responsible for a large part of inorganic chemistry.

Across the first transition series, ten electrons are added to the 3d shell, and each of them screens the 4s electrons at only 0.85 — but screens the other 3d electrons at 0.35. So the 3d shell contracts sharply as it fills, from 3.50 bohr at scandium to 1.19 at zinc, while the 4s contracts far more gently, from 8.00 to 5.52.

By the time the shell is full, the 3d orbitals have withdrawn so far inside the 4s that they barely reach the ligands of a complex at all. That is why the d-orbital participation in bonding is small at the end of the series, why zinc behaves like a main-group element with a filled core, and why the argument in hypervalency without d orbitals is available to make: the d orbitals of a main-group element like sulfur are further out of reach still.

The same arithmetic run through the f block gives the lanthanide contraction, which is why hafnium and zirconium are almost the same size and why separating them is a well-paid speciality.

What was computed, and how #

Three things here are computed and one is quoted, and the boundary matters.

Quoted: Slater’s screening constants — 0.35, 0.30, 0.85, 1.00 — and the grouping they apply to. They are a fit to 1930 spectroscopy.

Computed: the electron configuration of every element, by filling shells in $n+l$ order rather than reading a table; the screening sum for any shell of any neutral atom, by applying the rules to that configuration; and the resulting mean and mean-square radii, from the closed forms above.

The last of those is checked rather than trusted. Integrating the radial distribution numerically on a forty-thousand-point grid has to agree with the closed form to two parts in a thousand for every orbital drawn. A quadrature that agreed with a closed form for 1s and drifted for 4s would be exactly the sort of plausible failure that never announces itself.

Two further claims are checked. The first is that the screened charge never falls as protons are added across a series — if it did, the rules would have been applied to the wrong configuration, which is the mistake most often made when they are applied by hand. The second is the closed-form check just described.

The refusals matter as much. Slater’s rules asked for the 3d electrons of an atom that has none must give no answer rather than a number, and a configuration asked for an atom past the end of the table must fail rather than silently lose electrons.

Where the model stops #

Everything drawn here is a one-electron function with a modified charge, which is not the same object as a many-electron atom and does not become one by adjusting a parameter. There is no exact orbital for carbon at all, as orbitals are not where the electron is sets out, and the screened hydrogenic function is a convenient stand-in for something that has no exact form.

Three specific limits are worth naming.

The rules have no angular dependence within a shell. Slater gives the 2s and 2p electrons of carbon the same $Z_{\text{eff}}$ of 3.25, and they are not the same — a distinction that matters again in Bent’s rule, against the substituent series, where the s and p characters of a hybrid are traded against each other: the 2s penetrates and the 2p does not, and their ionisation energies differ by about two electron volts. Better fits, including Clementi’s, separate them.

The radial function is still hydrogenic. Dividing lengths by $Z_{\text{eff}}$ shrinks the orbital uniformly, and a real screened orbital is not a uniformly shrunken hydrogen orbital — the inner peak contracts less than the outer one, because the inner peak is inside the screening.

Nothing here is an energy. The rules give a charge, and turning a charge into an energy needs $-Z_{\text{eff}}^2/2n^2$, which is the hydrogenic expression applied to a system it was not derived for. It gets ionisation trends broadly right and individual values wrong by electron volts.

The generalisation #

The useful habit this leaves behind is the separation of two questions that look like one.

How big is it is a question about the wavefunction’s extent, and there are several answers depending on which measure of extent is meant — enough that how big is an orbital is an essay of its own.

How tightly is it bound is a question about the energy, and the energy depends on where the density is relative to the other electrons, not on how far it reaches. An orbital with a small inner peak and a large outer one can be simultaneously the biggest orbital in the atom and among the most tightly bound.

Almost every surprising fact about the transition metals follows from those two answers coming apart: why the 4s fills first and empties first, why the 3d contracts so sharply across the series, why the second and third rows of transition metals are so similar in size, and why ionic radii are so much smaller than atomic ones.

Which quantity the effective charge was fitted to #

The correction is a fit rather than a derivation is the essay’s own caution, and it understates the difficulty by one step, because a fit has a target and the target is almost never stated.

An effective nuclear charge is defined by requiring that a hydrogenic expression, evaluated at that charge, reproduce something about the real atom. Which something is a choice, and the choices do not agree.

Fit the energy. Set the hydrogenic energy $-Z_{\text{eff}}^2/2n^2$ equal to the orbital’s actual energy and solve. That is what Slater’s rules approximate, and it is the right choice if the number is going to be used to estimate an ionisation energy.

Fit the size. Set the hydrogenic mean radius $n^2/Z_{\text{eff}}$ equal to the orbital’s actual mean radius. That gives a different number, because a real orbital’s energy and radius are not related to each other the way a hydrogenic orbital’s are — the real function has a different shape, and one hydrogenic parameter cannot reproduce two of its properties at once.

Fit the density at the nucleus. Different again, and it is the one that matters for hyperfine couplings and for electron-capture rates.

Three targets, three effective charges, one orbital. The differences are not small: an orbital whose energy is matched by one charge is typically the wrong size by a noticeable fraction at that charge, and the discrepancy grows for the orbitals with the most structure.

So a tabulated $Z_{\text{eff}}$ is a number with an unstated purpose attached, and the standard tabulations differ partly for that reason and not only because their fitting procedures differ.

Which sets the scope of everything computed from one. A comparison is safe when both sides of it use the effective charge for the quantity it was fitted to. Using an energy-fitted charge to compute a size, or a size-fitted one to compare energies down a group, is asking a one-parameter substitution to carry two properties — and the essay’s own arithmetic, which uses the energy expression throughout, is on the right side of that line only because it never converts to a radius.

There is a reason the practice survives despite all of that, and it is a familiar one. The three effective charges disagree about magnitudes and agree about order: whichever target is fitted, the charge rises across a period, falls down a group, and is larger for a penetrating orbital than for a diffuse one at the same shell. So an argument that uses $Z_{\text{eff}}$ to say which of two orbitals is more tightly held is safe under any of the three, and an argument that uses it to say by how much is not — which is exactly the division the essay’s own conclusions fall on.

Who found it, and when #

Slater’s rules are from a 1930 paper in the Physical Review, written the year after the Hartree method appeared and intended as a way of getting useful screened wavefunctions without doing a self-consistent calculation at all — a shortcut whose usefulness outlived the shortage of computing power that motivated it.

Enrico Fermi and Llewellyn Thomas had a statistical account of screening slightly earlier, treating the electrons as a gas rather than individually; it gives the right general shape and not the shell structure. Clementi and Raimondi’s 1963 tables replaced Slater’s fit with values extracted from Hartree–Fock calculations, which is where the numbers in most modern textbooks come from — although the rules, being teachable in five minutes, are what everyone remembers.

The 4s-before-3d question has a longer and less settled history than its confident textbook treatment suggests, and the modern statement of it — that the order in the neutral atom and the order in the ion are different, and that the neutral order is not evidence for a fixed sequence of orbital energies — was not standard until the 1990s. It is taken up in the aufbau order is not a property of the atom.

Still open: what size means #

An effective charge takes the hydrogenic orbital and gives it a parameter that pulls it towards a real atom. The next question is what “size” even means once there is more than one candidate for it, and the answer is that there are at least four, they disagree by more than a factor of two, and the one that governs chemistry is none of them.