Molecular Geometry & Bonding
what the shape decides
Concept

Spin state — where it appears

One of the arrangements of a partly filled shell distinguished by how many electrons are unpaired. Each has its own energy and degeneracy, and where two lie close together both are populated and the measured moment is an average.

Named by 14 essays across 4 fields — each of them below, with the objects they name alongside it.

d⁶: two states, and where they cross. The energy of the high-spin and low-spin fillings of d⁶ against the splitting, in units of the pairing energy. They cross at Δ = P exactly, with 4 unpaired electrons below it and 0 above.

The pairing energy decides the moment

Whether the sixth d electron pairs up in the lower set or goes alone into the upper one is a competition between the splitting and the cost of pairing. Run the filling rules over the whole shell and exactly four configurations have a choice — and every one of them changes state at Δ = P exactly.

applied · Magnetism A moment that is not an integer's worth of anything. The effective magnetic moment against temperature for an iron(II) complex whose two spin states lie close together, with the moments belonging to whole numbers of unpaired electrons drawn across. The curve spends its time between them and settles on neither.

A moment between two integers

A magnetic moment is celebrated as one of the few chemical measurements that returns an integer: count the unpaired electrons, feed the count into a formula, and nine first-row ions come out right. That works when one state lies far below the others. Sit a complex at its own crossover and the same measurement returns 0.30 at 80 K and 3.61 at 400 K — a quantity that counts nothing and is a temperature in disguise.

applied · Magnetism The same sample, fitted over four temperature ranges. A pair coupled at -50 cm⁻¹, its susceptibility computed exactly, fitted to a Curie–Weiss law over four ranges. The moment and the Weiss temperature the fit reports both depend on which range was used, and the quality of the fit does not warn about it.

The moment a fit invents

One coupled pair of spins, its susceptibility computed exactly, fitted to a Curie–Weiss law over four temperature ranges. The moments reported are 2.471, 2.535, 2.566 and 3.590 Bohr magnetons, and the Weiss temperatures −51, −78, −78 and −292 K — from one sample, measured perfectly, with three of the four fits agreeing with their own data to better than a part in three hundred.

applied · Magnetism One exact state, two correlation energies. The energy each of two mean fields misses, against the repulsion, for one system whose exact energy is a single smooth curve. Below U = 2 the unrestricted search returns the restricted answer and the two definitions agree to the last bit. Above it they part: at U = 32 the restricted reference reports -27.82 and the unrestricted one -0.11, a factor of 259.43, and one is growing while the other falls.

The reference decides the correlation

The correlation energy of one exact state, measured against two references that are both called Hartree–Fock, is −27.82 and −0.107 at the same repulsion — a factor of 259, on a system whose exact energy is a single smooth curve. Below the instability at U = 2 the two agree to the last bit; above it one grows without limit while the other falls, and the reference that reports almost no correlation has ⟨S²⟩ = 1.99 where a singlet is zero.

beyond · Correlation Nearly all of the error cancels, and the answer gets worse. For each repulsion: the error a spin-paired mean field makes in the total energy of one four-site system and of two two-site ones with the same number of electrons, and the error left in the difference between them. The cancellation improves from 83 to 97 per cent along the axis. The residue as a share of the quantity being computed goes the other way, from 1 to 423 per cent, because the reaction energy shrinks faster than what survives.

Two wrong numbers and a right difference

A mean field gets the total energy of a four-site system wrong by 12.11 and of two two-site systems wrong by 12.49, and 96.9 per cent of that error cancels out of the difference between them. The residue is 0.38 — and the reaction energy it is a residue of is 0.09, so the cancellation improves and the answer gets worse at the same time. Change the pair being compared to a singlet and a triplet and nothing cancels at all: the sign goes.

wrong · Approximation The coupling a fit reports, and the coupling the sample has. Exact susceptibilities of Heisenberg chains of two, four, six and eight spins, every one of them coupled at -50 cm⁻¹, each fitted with the two-spin expression over 80–600 K. The two-spin sample returns its own coupling exactly; every longer chain returns one too large, by more the longer it is, up to 20.9 per cent. Every one of those fits has an R² above 0.99, so nothing in the fit reports that anything is wrong.

The model is what is fitted

Fit a pair of coupled spins with the two-spin expression and it hands back the coupling exactly, from any temperature range. Fit a chain of eight with the same expression and it hands back −66.7 where the sample has −50, with a residual of 0.998 and a g factor of 1.973 — three numbers of which only the last says anything is wrong, and it is the one nobody looks at.

applied · Magnetism How much of a curve each extra parameter has left to work with. The singular values of the design matrix for a susceptibility curve, for two, three and four parameters fitted to the same data, on a logarithmic scale. With four they run 12.411, 2.026, 0.149, 0.025 — a span of 500 — so one per cent data fix the first two to under 3 per cent and the last to 37. Each value is what is left of the measurement after the directions above it have taken their share, so a short bar is not a hard parameter but an absent one.

How many parameters a curve is worth

A susceptibility curve routinely carries four fitted parameters and the question of whether it can support them is never asked. It has an arithmetic answer: the four directions the fit sees span a factor of five hundred, so one per cent data fix the first two to under three per cent and the last to thirty-seven — and forty points reaching two kelvin are worth more than sixteen thousand starting at twenty.

applied · Magnetism Sixteen clusters, and the couplings at which each one's sign changes. The sign of the exponent for every cluster across couplings from 2 to 1000 cm⁻¹, read at a 20–300 K window, with each change of sign marked. No cluster keeps one sign across the range. The clusters with a ground spin change sign twice below 150 cm⁻¹, at couplings that fall as the ground spin rises; the singlets first change sign above 300 cm⁻¹. The spin rule — positive for a singlet ground state, negative otherwise — is right on all sixteen at once only between 20.1 and 47.0 cm⁻¹, and the working point every earlier reading used, 50 cm⁻¹, is just outside it.

The sign rule holds between two poles

The sign of a susceptibility fit's exponent was put down to the parity of the spin count, then to the spin of the ground state, and every cluster tried had the two aligned. A star of four spins is even with a ground spin of one, and its exponent is −6.84. But swept across the coupling, every cluster's sign changes, through poles the ground spin places, and the spin rule is right on all sixteen clusters only between 20.1 and 47.0 cm⁻¹.

applied · Magnetism Where the nuisance parameters separate hardly depends on purity. For each of the seven clusters with a ground spin, the two couplings below 150 cm⁻¹ at which the monomer fraction and the temperature-independent term are uncorrelated, against the monomer fraction from half a per cent to sixteen, on logarithmic axes. Every cluster keeps both at every fraction. For ground spins of one and above they move by under three per cent; for the three doublets by about a tenth. A circle marks a pole of the third direction's exponent there and a square a zero.

Purity renames the poles

A susceptibility fit's sign poles were located at one monomer impurity, two per cent, and a real sample's impurity is rarely known. Swept from half a per cent to sixteen, the couplings where the fit's nuisance parameters separate barely move for ground spins of one and above and move a tenth for doublets. But the same separation stops being a pole and becomes a zero of the exponent at a few per cent, and above six to eight per cent every singlet acquires separations of its own.

applied · Magnetism Stars drift above the band and K₂,ₙ graphs fall below it. The upper separation coupling times (S + ½) against ground spin for every cluster computed, with the seven that defined the band shaded between 114.7 and 129.3 cm⁻¹. The stars, from four centres to eight, run 129.3, 114.7, 117.3, 127.2, 140.3 — down and then steadily up. The K₂,ₙ graphs, from K₂,₃ to K₂,₆, run 114.8, 124.0, 100.9, 90.0 — up and then steadily down. K₃,₅ sits at 113.9. Seven clusters of up to six centres happened to lie where the two families cross.

Five more clusters break the band

Seven spin clusters put their upper separation coupling times the ground spin plus a half inside a band from 115 to 129 cm⁻¹, which looked like a law of where a susceptibility fit's nuisance parameters decouple. Five clusters built to test it — two larger stars and three larger bipartite graphs — land inside it once. Stars drift above and bipartite graphs below, and neither the shape of the curve nor the first excitation places the separation instead.

applied · Magnetism The upper separation sits where the first spin-raising multiplet does. Every five-centre doublet, every six-centre triplet and the three stars, by the energy of the lowest multiplet with a larger spin than the ground state, in coupling units, and the coupling at which the fit's two nuisance parameters separate for the second time. On logarithmic axes a constant product is a line of slope minus one; the line drawn puts that multiplet 359 cm⁻¹ above the ground, and the forty-four clusters lie within a factor of one and a half of it.

The excitation that raises the spin

A susceptibility fit's upper separation — the coupling at which its monomer fraction and temperature-independent term decouple for the second time — was placed by a band that broke, and the lead it left was that at fixed ground spin the first excitation ordered it. Every connected cluster of four, five and six spin-½ centres answers that. Among twenty five-centre doublets the first excitation ranks the separations at 0.36 and fewer bonds does better; the first excitation that raises the total spin ranks them at 0.86, and its product with the separation stays within a factor of one and a half where the separation itself spans three and a half. The separation sits where the first higher-spin multiplet is about 355 cm⁻¹ above the ground.

applied · Magnetism The window's top moves every separation with it. The median of the upper separation times the first spin-raising gap, over the forty-one clusters, for windows starting at 20 K and ending between 150 and 500 K. The dashed line is the reading that it sits 1.7 kT above the ground at the window's top; the dotted line is strict proportion through the 300 K value. The medians are 189, 243, 354, 463, 569 cm⁻¹: the earlier prediction of about 240 at 200 K and 475 at 400 K comes out 243 and 463. The rise is slightly slower than proportion — as the top temperature to the power 0.92.

The window sets the constant

A susceptibility fit's upper separation, multiplied by the gap to the first spin-raising multiplet, came to about 355 cm⁻¹ across forty-one clusters, and that was read as a position: the raising multiplet sitting 1.7 kT above the ground at the top of a window ending at 300 K. The reading predicted about 240 at a 200 K top and 475 at 400 K. Re-read under nine windows, the forty-one clusters give 243 and 463, and the raising gap collapses them equally well under every one. But the constant is not quite a multiple of kT at the top — it drifts from 1.81 to 1.64 as the top rises — and the drift is the window's width: windows that are scaled copies of one another give the same constant to a per cent, because the model has no energy scale of its own.

applied · Magnetism How the temperatures are spaced moves the constant more than how wide the window is. The median of the upper separation times the spin-raising gap, in units of kT at the window's top, for windows of span 7.5, 15 and 25 with forty temperatures spaced three ways: evenly in the temperature, in its logarithm and in its inverse. At a span of fifteen they give 1.70, 1.49, 1.36. From span 7.5 to 25 even spacing falls by 9.5 per cent and logarithmic by 14.0: the logarithmic design leans more, not less.

The spacing sets the constant

A susceptibility fit's upper separation, in units of kT at the window's top, was found to depend on the window's span — 1.81 at a span of 7.5, 1.64 at 25 — and every window read had forty temperatures evenly spaced, so a wider window was also a sparser one at its cold end. Holding the span and changing the design separates the two. The number of temperatures barely matters: twenty, forty and eighty converge on one value from above, and a design with the census's own step reproduces the forty-point constant at both ends of the range. How the temperatures are spaced matters a great deal. Spaced evenly in their logarithm, as many measurements are, the constant is twelve per cent lower and leans more, not less; spaced evenly in their inverse, lower again.

applied · Magnetism The on-site repulsion allyl needs rises with the neighbour's. The on-site repulsion U at which the three-orbital radical's middle spin matches the allyl radical's, against the repulsion V₁ between each end and the middle, with the end–end repulsion in its Mataga–Nishimoto proportion. The band is the range McConnell's Q allows and the line the value the coupling ratio fixes without Q. With no neighbour repulsion U is 6.1 eV; it rises about nine tenths of an electronvolt for each one of V₁. Where V₁ is the Mataga–Nishimoto value computed from U itself, U is 10.89 eV — against Pariser's valence-state 11.13 for carbon. Ohno's formula overshoots, to 13.8.

Six electronvolts was a difference

A three-orbital radical solved with an on-site repulsion reproduced the allyl radical's negative central spin at a repulsion of about six electronvolts — six tenths of what a carbon atom's own charges cost. Add the repulsion between neighbouring sites and the repulsion allyl needs rises almost one for one with it. With the neighbour values π-electron theory uses, it lands within a quarter of an electronvolt of Pariser's 11.13 for carbon. And the band McConnell's constant left can be closed without that constant at all.

bonding · Overlap

Named alongside it

The objects these essays reach for when they reach for this one.

Model limitExchange couplingUnderdeterminationConditioningExact diagonalisationMagnetic susceptibilityTemperatureMagnetic momentUnpaired electronsBoltzmannConventionSusceptibility

All concepts