The correction that moves three of them backwards

Closing an escape is only worth doing if the next one is named while it is still open. The metal’s contraction was closed by sweeping chromium’s 3d effective charge across every oxidation state the element has and finding the computed π/σ ratio short by factors of two to six at every point in it. The sentence that closed it also said what was left: every ligand radial function in that sweep was fixed at its neutral atom’s Slater value while three of the five donors carry a formal charge, and a fluoride’s 2p in a chromium field is not a free fluoride’s.

That is the mirror correction, it is the last one of its kind, and it turns out to be the more interesting of the two — not because it succeeds but because it fails differently.

Why the other end is a different lever #

Both sweeps change a radial function and both change the same computed quantity — the one the splitting was found proportional to — so it is reasonable to expect the same answer twice. Three things make them different, and all three are visible before any integral is run.

A donor’s window is wider. Chromium’s 3d effective charge runs from 4.60 in the metal to 6.00 in chromate, which is the whole span Slater’s rules give the element: 1.40 units of charge, a factor of 1.30. A p-block atom’s valence shell holds between one and eight electrons, and each companion screens by 0.35, so every donor here has a window of exactly 2.45 units — nearly twice as much room.

A donor’s window has a hard end. The metal’s sweep could be pushed past Slater’s range and asked what charge each ligand would need, because a larger effective charge on a 3d electron is a statement about the argon core screening imperfectly, and cores do screen imperfectly. A donor’s cannot be pushed the same way for long. Its window ends at the bare atom, where the valence shell holds one electron and the whole of the rest of the effective charge is already unscreened. Past that there is nothing left to take.

And a donor’s window need not move the ligands together. This is the one that matters, and it is worth stating why in advance rather than reporting it afterwards. Contracting the metal changes one function that appears in all ten integrals; contracting the donors changes five functions, one per complex, and each of them sits at a different distance from the metal with a different number of radial nodes. There is no reason for five such changes to share a factor.

They do not. The metal’s window moved the five ratios by 2.07, 2.09, 2.16, 2.17 and 2.18 — one factor to within five per cent, which is why the ordering across the series was frozen at every charge and why no contraction could improve one ligand at another’s expense. The donor’s window moves them by 1.57, 2.05, 3.15, 5.54 and 9.51. That is a six-fold difference between ligands, and it means this correction is of a kind that can change the pattern of discrepancies rather than merely their size.

Which makes the result worth having, because it does change the pattern and still reaches nothing.

The obvious thing first #

Before any sweep there is a single substitution to try, and it is the one the chemistry actually asks for. A chloride is an anion. Its donor chlorine has eight valence electrons, not the seven a neutral atom has, so its 2p companions screen it more and its 3p function is more diffuse than the one used earlier. The same is true of fluoride. Cyanide’s formal negative charge sits on the donor carbon, which gives that carbon five valence electrons rather than four. Water’s oxygen and ammonia’s nitrogen are formally neutral and do not move.

So three of the five radial functions were wrong in a known direction, and correcting them costs three integrals.

Chloride goes from 0.9456 to 1.4827. Its fitted parameter is 0.16. Fluoride goes from 0.3052 to 0.3373 against a fitted 0.14. Cyanide goes from 0.5013 to 0.6562 against a fitted −0.10.

Every one of them moves outwards. The discrepancy at chloride, which was 5.9-fold, becomes 9.3-fold; at fluoride 2.2-fold becomes 2.4-fold. Nothing moves the other way, because there is nothing left to move: water’s and ammonia’s donors are already at the charge they carry, and the two that were wrong were wrong in the direction that made the model look better than it is.

That is worth separating from the sweep that follows, because it is the stronger of the two statements. A sweep asks what a range of charges could buy and answers in factors. This asks what the correct charge does, and the answer is that the earlier essays were flattering the model by an amount between five and sixty per cent, at three of its five points, without anybody choosing to.

A more diffuse donor raises the ratio, which is the general fact underneath all three. A π overlap is side-on and a σ overlap is end-on, and the side-on integral lives further out in both functions — so spreading the donor helps the π channel more than the σ one. Anionic ligands are diffuse. Every correction that makes a donor look more like the ion it is therefore pushes the computed π channel up, and every fitted parameter in the series is below what the model computes.

The whole window, and what it can reach #

With the direction established, the sweep asks for the extreme. Not what the charge is but what the best charge would be: the most contracted donor the element’s own chemistry permits, which is its bare atom.

Contraction lowers every ratio, monotonically, which is the direction needed. The curves do not cross and none of them turns round. So each ligand’s best case is unambiguous: it is the left-hand end, the bare atom, and the number there is the smallest π/σ ratio the model can produce for that complex with any donor function Slater’s rules allow.

Chloride’s best is 0.2674 against a fitted 0.16. Fluoride’s is 0.2146 against 0.14. Water’s is 0.2297 against 0.10.

Stopping on one count at a time is what separates the two claims the sweep makes at once. The window says what the model could produce at any charge; a single count says what it produces at a charge somebody could defend. Every count from the stripped donor to the closed-shell anion leaves all three ligands with a positive fitted parameter above that parameter, and the count the donor actually carries — the one marked — is at or near the worst end of the window rather than the best: for chloride and fluoride it is the eighth of eight, and for water the sixth. The refusal is not a matter of the range being too narrow; it is that the range is on the wrong side.

Short by 1.67, by 1.53 and by 2.30 — at the extreme of a range that includes oxidation states none of these ligands is ever in. A chlorine atom with one 3p electron is chlorine(VI), and the ligand in the complex is chloride. The sweep is asking what the most violently oxidised version of each donor would do and the answer is that it is not enough for any of them.

The two remaining ligands are refused by the same algebra as before, and it is worth restating because nothing here touches it. Ammonia’s fitted parameter is exactly 0.00 and cyanide’s is −0.10. The model’s quantity is $S_\pi^2$ over $S_\sigma^2$, a quotient of squares, and no radial function on either centre makes that zero or negative. The metal’s sweep found that and this one finds it again, because it is a statement about the shape of the expression rather than about anything in it.

So all five are refused, and the three that had been merely out of range are now out of reach.

What each donor would need, and what a core is for #

The sharper version is the same one the metal sweep asked: solve for the charge each ligand needs on its own, and look at the numbers rather than at whether they are in the window.

Chloride needs 9.2843. Its window ends at 7.00.

The 2.28 between those two numbers is not a small extrapolation, and what it would cost is specific. Chlorine has seventeen protons. A 3p electron in it is screened, on the convention these rows use, by ten inner electrons screening completely and by its valence companions at 0.35 each — so 7.00 is what remains when every valence electron has been stripped and the core is doing its full job. Getting to 9.2843 means 2.28 of those ten core electrons screening nothing at all.

That is the same shape of statement the metal sweep ended on — water needing nine of chromium’s eighteen core electrons to stand aside — and it is a different order of violence. Two and a quarter core electrons is not obviously absurd; real cores do leak. But it is being asked of an ion that is more diffuse than the neutral atom, in the direction of less unscreening rather than more, so the required correction and the actual one point opposite ways.

Fluoride and water have no answer the arithmetic can find, and the reason is not the model this time. The overlap rule here is a quadrature checked against a second rule with a different point count and a different map, and it fails once a function becomes compact enough that neither rule resolves it. Measured per donor rather than assumed, that ceiling sits at 7.60 for fluoride’s 2p and 8.10 for water’s — barely past the windows themselves, and below the charges the two would need.

Which is a separate kind of refusal and is reported separately. Ammonia and cyanide are refused by algebra: no function produces their values. Chloride is answered with a number outside its chemistry. Fluoride and water are refused by the numerics, and that is a statement about this calculation rather than about the model — a better integrator might return a charge for them, and it would be a charge past their bare nuclei too, since the curves are monotone and already above target at the ceiling.

Three kinds of refusal across five ligands, which is one more kind than the metal’s sweep produced.

The ordering the correction can move #

The six-fold spread in sensitivity means something the metal sweep could not test, and it is the one place this essay could have gone the other way.

A correction that multiplies every ligand’s ratio by the same factor changes no ratio between them. Whatever it buys at one ligand it spends at the next, and the ordering across the series — which is the whole content of a spectrochemical parameter — survives untouched. That was the metal’s contraction and it is why that essay could say the reversal was permanent.

This one is not like that. Chloride moves by 5.54 across its window and fluoride by 1.57, so the gap between them can be opened or closed at will by choosing two charges.

With every donor at its own charge the computed ranking runs chloride, cyanide, ammonia, water, fluoride. With every donor stripped to a bare atom it runs cyanide, chloride, water, ammonia, fluoride. The two differ — at the top, where cyanide and chloride swap — so the correction genuinely does reorder the series, which is more than the metal’s could do at any charge.

The fitted ranking is chloride, fluoride, water, ammonia, cyanide.

Neither computed ordering resembles it. Both put cyanide at or near the top where the fit puts it last and negative; both put fluoride last where the fit puts it second. What the donor sweep buys is the ability to change the ordering into a different wrong one, and the reason is not subtle: the quantity being reordered is how diffuse each donor is, and the ligands are already sorted by that in the wrong direction. Cyanide’s carbon is the most diffuse donor in the set and the model rewards diffuseness with a large π channel, while the fit assigns cyanide the only negative parameter in the series.

So the freedom is real and it is pointed the wrong way, which is a more specific failure than the numbers are too small.

What was computed, and how #

Every overlap is a quadrature over Slater radial functions: chromium’s 3d at the effective charge Slater’s rules give chromium(III), the donor’s 2p or 3p at the charge being swept, at each complex’s measured bond length. The σ integral is $3d_{z^2}$ against $p_z$ and the π one is $3d_{xz}$ against $p_x$, which is the only difference between the two channels.

The donor’s effective charge is not written down twice. Each row in the series carries a Slater expression for its neutral atom, and the sweep recovers the unscreened part of that expression by adding back 0.35 for each valence companion — so at the neutral count the sweep reproduces exactly the number used earlier, and a check confirms that it does rather than trusting it.

Nine things are checked: that each donor’s window spans 2.45 units of charge; that the sweep reproduces the earlier number at the neutral count; that the five spans differ by more than five-fold, which is what separates this sweep from the metal’s; that all three ligands with a positive fitted parameter have windows lying entirely above it; that the other two are refused by algebra and not by size; that every ratio which moves under the formal-charge substitution moves away from the fit; that two ligands need no charge because none exists; that every charge the search does find lies past the donor’s own bare nucleus; and that feeding each solved charge back reproduces the fitted ratio to a part in a million, so the bisection is a round trip rather than a report of where it stopped.

The quadrature ceiling is measured rather than assumed, and that is a change from the essay before it. There the cap was a single number for every ligand, because the function being contracted was the metal’s. Here the compact function is the donor’s own, a 2p gives out sooner than a 3p, and using one ceiling would have reported fluoride’s refusal at the wrong charge.

Where this stops, and what it does not touch #

The whole argument is about one claim and it is not the ligand-field model’s usefulness. A practitioner fits $e_\sigma$ and $e_\pi$ to spectra and never computes an overlap; that separation was made several essays back and nothing here disturbs it. What is being tested is the derivation that gives the model its name — that $e_\pi/e_\sigma$ is a ratio of squared overlaps — and the test is whether any radial function on either centre produces the fitted numbers.

Slater’s rules are a caricature and a different screening rule would give different windows. The windows would have to be wrong by a great deal to matter, since the best case is short by between 1.5 and 2.3 and the ceiling is a bare nucleus rather than a convention. But the fitted parameters are also not above criticism: they are the spectrochemical series’ own, fitted across many complexes rather than to these five structures, and the reference-state question raised about the denominator applies here too. A per-complex fit would give five different numbers.

And the ligand’s radial function is not the only thing held fixed. The bond lengths are the measured ones and are shared between the σ and π integrals, which the common-separation comparison showed is doing more work than it looks; the energy denominator the perturbation divides by is fixed and fails in both directions; the metal’s function is now fixed again while the donor moves. Each of these is a separate escape. Two have now been closed by sweeping them, and closing the second is what makes the list short enough to see the end of.

What remains is not another radial function. The two ligands refused by algebra are refused because the model has one interaction where the chemistry has two, and that is a structural repair rather than a parametric one.

The generalisation #

The useful habit here is to run the correct substitution before running the sweep that generalises it, and to report the two separately.

A sweep is generous by construction. It asks what a range of inputs could buy, reports a factor, and a factor that is too small is a clean refusal. But it also hides something: if the value the model should have been using all along sits on the wrong side of the value it was using, the sweep will report that as a smaller range on one side and nothing more. Running the substitution alone turns that into a statement — the earlier essays were flattering the model at three of its five points — which is a different and more specific finding than the whole window is not enough.

The corollary is about corrections with a known direction. It is tempting to treat those as safe, because the direction is physically motivated and the magnitude is small. Both halves were true here and the correction still made things worse, because physically motivated is a claim about the correction and not about which way the discrepancy lies. The cheap check is to apply it and look, and it costs three integrals.

Who found it, and when #

The angular overlap model is Schäffer and Jørgensen, 1965, and the proportionality to a squared overlap is part of its original statement. Slater’s screening rules are 1930. The effective charges, bond lengths and fitted parameters here are quoted; the overlaps, the windows, the spans and the solved charges are computed, and the arithmetic is the same arithmetic the metal’s sweep used, with the other centre moving.

The number worth carrying is not 9.28. It is that the substitution the chemistry asks for — an anion is more diffuse than a neutral atom — moves three of the five discrepancies further out, and that the best case of the entire sweep is short by a factor of one and a half at the ligand it does best on.

Still open: whether the two ends can be moved together #

The obvious open question is the joint sweep. The metal’s contraction and the donor’s have each been closed alone, and neither closing forbids their combination: contracting both shrinks both overlaps, and the σ and π integrals are not equally sensitive to which centre moves. Whether the two-dimensional search has a region the one-dimensional ones missed is a grid rather than a line, it costs twenty-five quadratures a ligand, and the direction is already known to be unhelpful — but known to be unhelpful is what the metal sweep said about the donor’s contraction, and that turned out to be the more informative of the two.

The nearer question is the one the two algebraic refusals have now made unavoidable. Ammonia and cyanide are refused because the model has one channel and pushes the metal’s d orbitals up, while an empty π* on the ligand would push them down and produce a parameter that can be negative. The overlaps that second channel needs are the same integral with a different ligand orbital in it — a metal 3d against an antibonding combination rather than against a filled p — and both of the ligands in question have a measured π* level to put in the denominator. That is a structural change to the derivation rather than another sweep of it, and it is the only repair left that could reach the two ligands no charge has touched.