The double hump and what removes it

The enthalpy released when a gaseous divalent ion of the first transition series is hydrated rises across the row, because the ions get smaller and bind water more tightly as the nuclear charge climbs. That is the expected trend and it is a straight line — and it is the kind of smooth background the bond that weakens as neighbours multiply computes from a coordination count rather than fits.

The measurements are not a straight line. They rise from calcium to vanadium, dip at manganese, rise again to nickel, and dip again at zinc. Two humps with a trough between them, and the trough is not small: manganese sits about 90 kJ/mol below where a line through calcium and zinc would put it.

The standard account is that the underlying trend really is the line, and that the humps are ligand field stabilisation sitting on top of it. This essay tests that account by computing the stabilisation and subtracting it.

What is computed and what is fitted #

The stabilisation is computed from the angular overlap model used for d–d spectra — the same one two models, one ratio checks against a point-charge calculation and finds agreeing on every ratio to eight decimal places. For each $d^n$ configuration it fills the split levels in the high-spin order and sums their energies measured from the barycentre, so a full shell gives exactly zero by construction.

That produces one number per ion in units of the octahedral splitting: $0$, $0.4$, $0.8$, $1.2$, $0.6$, $0$, $0.4$, $0.8$, $1.2$, $0.6$, $0$ for $d^0$ to $d^{10}$. Those are the multiples of two fifths that a course gives as a table and here are outputs of a filling rather than entries in one.

Then three parameters are fitted at once to ten measurements: an intercept, a slope in the number of d electrons, and the splitting itself in kilojoules per mole. All three enter linearly, so it is a single normal-equation solve.

The shape of the correction is computed. Its scale is fitted. That distinction is the whole of whether this exercise is circular, and the next section is the reason it is not.

The result #

The bare line gives $R^2 = 0.732$. The line with the computed stabilisation added gives $R^2 = 0.965$.

The slope comes out at $40.9$ kJ/mol per d electron and the intercept at $1623$ kJ/mol, which is calcium’s value to within its own experimental scatter — a check that nothing put in.

The number that makes it a two-route measurement #

The fitted splitting is $163$ kJ/mol, which is $13{,}600$ cm⁻¹.

That quantity is measured independently, by looking at where these ions absorb light. Where a d–d band falls turns a splitting into a wavelength; run it backwards on the aqua ions and the splittings are $12{,}600$ cm⁻¹ for V²⁺, $13{,}900$ for Cr²⁺, $7{,}800$ for Mn²⁺, $10{,}400$ for Fe²⁺, $9{,}300$ for Co²⁺, $8{,}500$ for Ni²⁺ and $12{,}600$ for Cu²⁺.

So a number fitted to a table of thermodynamic measurements lands inside the range a spectrometer gives for the same ions. Two experiments that share nothing — a calorimeter and a monochromator — and one model between them.

It lands at the top of that range, and that is worth saying rather than smoothing over. A thermodynamic fit absorbs every effect that varies with the d count in the same way the stabilisation does, and there are others: the ionic radius does not shrink smoothly across the series either, and the same partly filled d shell that produces the stabilisation also produces the irregularity in the radii. So the fitted parameter is an upper bound on the stabilisation rather than a measurement of it, and the agreement is the right kind of agreement — good enough to say the model is describing something, not good enough to be a determination.

The refusal, which is a different filling #

The check that keeps this from being a curve fit with a chemical story attached is to compute the stabilisation for a filling these ions do not have.

Fill the levels low spin — pairing electrons in the lower set before using the upper one — and the pattern changes completely: $0$, $0.4$, $0.8$, $1.2$, $1.6$, $2.0$, $2.4$, $1.8$, $1.2$, $0.6$, $0$. That is a single hump rather than two, peaking at $d^6$.

Fitting with that shape gives $R^2 = 0.751$ — barely better than the bare line, and far worse than the high-spin fit’s $0.965$.

So the fit is sensitive to which pattern is used, which means it is testing the pattern rather than merely absorbing the data into three parameters. Aqua complexes of the first row are high spin — water sits low in the series the spectrochemical series is not electrostatics measures — which the pairing energy decides the moment establishes from a competition between the splitting and the pairing energy, and the thermodynamics agrees with the magnetism.

Why the dips are where they are #

The two dips are at manganese and zinc, and both are configurations with exactly zero stabilisation.

Manganese(II) is high-spin $d^5$: one electron in each of the five orbitals, three below the barycentre and two above, and $3 \times (-0.4) + 2 \times (+0.6) = 0$ exactly. Zinc is $d^{10}$: every orbital doubly occupied, and a full shell has no shape at all — the same statement a filled shell has no shape makes about the density.

Calcium is $d^0$ and is the third zero, which is why the line can be anchored at both ends of the series by ions that need no correction at all. Three points on the line, seven off it, and the seven displaced by amounts the model computes without seeing them.

That is the structure that makes the argument work. If every ion had a correction the exercise would be a three-parameter fit to ten points and would prove very little; with three ions requiring exactly zero correction, the line is nearly determined before the correction is applied.

What the correction is not #

Three things it is worth being clear about, because ligand field stabilisation is routinely asked to carry more than it can.

It is not the bond energy. The stabilisation is a few tens of kilojoules; the hydration enthalpy is two thousand. It is a correction of a few per cent to a quantity dominated by electrostatics, and the humps are visible only because the underlying trend is smooth.

It is not a prediction of stability, in the sense eighteen is a count is careful about: a count or a stabilisation is a permission and a preference, not a rate or a yield. The Irving–Williams order of complex stabilities has the same shape and is often explained the same way, and the same caution applies: the ligand field term is a small part of a large number, and it happens to be the part that varies with the d count.

And it says nothing about why water binds at all. The barycentre convention means the model computes only differences between fillings; the absolute stabilisation of any filling is zero by construction. What the model supplies is the shape of the variation, and the fit supplies the scale.

What the residuals still say #

The fit is good and it is not perfect, and the leftover is worth a look because $R^2 = 0.965$ on ten points still leaves structure.

The largest residuals after the correction are at cobalt and nickel, and they are of opposite sign: cobalt sits below the corrected line and nickel above it. That is the shape a splitting that varies across the series produces — the correction has been applied with one common value, and the ions whose real splitting is furthest from that value are the ones left over.

The optical numbers say the same thing. Nickel’s splitting is $8{,}500$ cm⁻¹ against the fitted $13{,}600$, so its stabilisation has been over-corrected; chromium’s is $13{,}900$, almost exactly the fitted value, and chromium sits on the corrected line.

There is a temptation to fit each splitting separately and drive the residuals to nothing. That would be ten parameters for ten points, and the exercise would stop testing anything — which is the same trap the force field is not in the spectrum records for a vibrational fit, where four constants and three frequencies means the constants are chosen rather than determined.

The splitting whose scale is the one fitted parameter is computed from a reduction rather than taken from a table, and everything about the shape of the correction — which levels, how many, how far above and below the barycentre — comes out of that reduction with nothing fitted. What is fitted is a single number: how many kilojoules a mole the splitting is worth.

Why the shape of the correction is worth more than its size #

The exercise has a structure that recurs whenever a model is tested against a trend, and it is worth naming.

A correction with a fitted scale and a computed shape is a testable claim; a correction with both fitted is a curve fit. The stabilisation pattern used here — $0$, $0.4$, $0.8$, $1.2$, $0.6$, $0$, repeating — has zeros at three of the ten points and a specific pattern of maxima between them, and none of that came from the enthalpies. Had the measurements not dipped at manganese, or had they dipped at iron instead, no choice of scale would have rescued the fit.

That is what the low-spin comparison demonstrates directly. The same data, the same three parameters, a different computed shape, and $R^2$ falls from $0.965$ to $0.751$. The fit is discriminating between shapes rather than absorbing the data.

And the scale is checkable elsewhere, which is rarer. Most corrections of this kind end with a fitted parameter and no way to test it: the fit is better, therefore the model is right, and the parameter is whatever it had to be. Here the parameter is a physical quantity that a different experiment measures, and comparing the two is a check the fit could have failed. It came out at the top of the optical range rather than in the middle of it, which is itself informative — it says the fitted parameter is absorbing something else as well.

Both halves of that are what separate this from the general run of “explained by ligand field stabilisation”, which is a phrase attached to almost every irregularity in the first transition series and is usually attached without either check.

Where the model stops #

The enthalpies are quoted. All ten come from thermochemical cycles rather than from direct measurement, and they carry uncertainties of a few kilojoules — which is small against a 90 kJ/mol trough and not negligible against the residuals after the correction. That is the ordinary position for this field and this site’s rule for it has not changed: a measurement is quoted and marked as quoted, and everything computed from it is computed here.

The splitting is treated as constant across the series. It is not: the optical values above vary by nearly a factor of two. Letting it vary would add parameters and improve the fit for no additional insight, which is why it is not done here.

High spin, octahedral, and six-coordinate. All three are true for these aqua ions and none is derived here.

No pairing energy. The stabilisation computed here counts only the level energies, not the cost of pairing electrons, which changes between free ion and complex and is what the pairing energy decides the moment puts a number on. That term also varies across the series and is absorbed into the fit.

The same argument, one place it is weaker #

The hydration enthalpies are the textbook case and there is a second one usually given beside them: the lattice energies of the transition-metal dihalides, which show the same double hump.

The argument transfers and one of its supports does not. In a solid the metal ion is octahedrally coordinated by halides, the stabilisation is computed the same way, and the humps come out at the same configurations. What is missing is the clean anchor: a lattice energy also depends on the ionic radius through a Madelung sum, and the radii of these ions are themselves irregular across the series for the same reason the enthalpies are.

So the correction and the background are correlated in a way they are not for hydration, where the background is a smooth function of nuclear charge. The fit still improves, and it improves for two reasons that cannot be separated with these data.

That is worth recording because the two cases are usually presented as equally good evidence, and they are not: one has three ions requiring exactly zero correction and a background with an independent argument behind it, the other has neither. The lattice sum whose answer depends on the order of adding is where the Madelung half of that problem is set out.

One last observation about the three zeros, because they do more work than their number suggests. Calcium, manganese and zinc require no correction at all, and they are spaced across the series at $d^0$, $d^5$ and $d^{10}$ — the two ends and the middle.

Three points spread that way nearly determine a straight line on their own. So the line in the figures is not fitted to the corrected data in any circular sense: it passes close to three measurements that the model says need nothing done to them, and the other seven then have to be displaced from it by amounts the model computes without looking. That is as close to a controlled test as a table of ten thermodynamic measurements is going to allow.

The same shape in a length #

The correlated-background problem in the lattice energies has a cleaner version elsewhere, because the quantity that correlates with them is itself measured and shows the same hump.

The ionic radii of the first-row divalent ions do not fall smoothly across the series. In six-coordinate high-spin compounds they run, in picometres, calcium 100, titanium 86, vanadium 79, chromium 80, manganese 83, iron 78, cobalt 75, nickel 69, copper 73, zinc 74 — down, up, down, up. A double hump in a length rather than in an energy, with maxima at manganese and zinc and the sequence starting from calcium.

The three points at the top of the humps are the same three ions that need no ligand field correction at all: $d^0$, $d^5$ and $d^{10}$. Everything between them is smaller than the smooth trend would give.

The mechanism is the same one, seen structurally. Electrons in the lower set point between the ligands, so they shield the nuclear charge from them poorly and let them approach; electrons in the upper set point at the ligands and push them away. A configuration with more of the first and fewer of the second is a smaller ion.

So the effect extracted here from a table of hydration enthalpies is visible in a table of radii determined by diffraction, with the same three null points and the same shape. That matters for the lattice-energy case specifically: the correlation between the correction and the Madelung background is not a coincidence to be apologised for, it is the same correction appearing twice in one calculation — once through the radii and once through the stabilisation — which is why the two cannot be separated with those data and why the hydration enthalpies remain the clean test.

What a thermodynamic table adds #

Ligand field theory establishes that the splitting is a symmetry statement before it is an energy, computes its size two ways from models sharing only the ligand directions, measures the spectrochemical series against charge and against a π parameter, and turns a splitting into the wavelength a complex absorbs.

This one takes the same computed splitting to a table of thermodynamic measurements that has nothing optical in it. The double hump is removed by a correction whose shape is computed and whose scale is one fitted number; the fit improves from $R^2 = 0.732$ to $0.965$; the fitted scale lands inside the range a spectrum measures for the same ions; and a filling these ions do not have fits barely better than nothing. Two experiments, one model, and a number that appears in both.

The open question is the term this one absorbed — the pairing energy, which also varies across the series and which is the other half of every high-spin against low-spin argument.