Molecular Geometry & Bonding
what the shape decides
The collection

Every essay — page 4

Page 4 of 36, continuing through the fields in the same order.

Orbitals Where the atoms go Bonding models What symmetry decides Beyond the octet What a spectrum settles When the molecule does not stop What the shape is for What is taught wrongly Series Named objects Orbitals Refutations Search

What the shape is for

Coordination compounds, where the shape decides a colour, a magnetic moment and a bond length — and where the one-electron picture used everywhere else in the collection stops being enough.

The two overlaps, squared, at the measured bond lengths. For each chromium(III) donor: the σ overlap squared, a metal 3d(z²) against the donor's p(z), and the π overlap squared, a 3d(xz) against its p(x). Everything is computed — the radial functions from Slater's rules, the separation from the measured bond length, the integral by quadrature. Chloride's π overlap is 3.5 times fluoride's, which is the opposite of what the overlap argument required of it.

The overlap the model is not proportional to

Without a computed π overlap, the natural argument reasons about one instead: the denominator over-predicts the halide trend, so the overlap must shrink down the group to cancel part of it. Computed, it grows — 3.5 times from fluoride to chloride. And the fitted parameter changes sign across the series, which no ratio of squared overlaps can do.

6 figures The exponent's sign is the ring's parity. The power the temperature-independent term carries in the free combination, for an open chain and for rings of five to ten spins. Every odd ring is negative and every even ring and the chain is positive — so the free product is ρ·χ_TIP raised to a power whose SIGN changes, which is a different combination rather than a shifted one. Nothing here is a near miss: the closest pair on either side of zero are +0.23 and −0.29.

The sign a frustrated ring changes

There is a sharper question than whether a low-temperature feature buys back a fourth parameter: does it change which combination is free? It does, and by a sign. Every odd ring of spins leaves free the monomer fraction times a negative power of the temperature-independent term, and every even ring a positive power, with no case in between.

6 figures Two charges, and three ligands no charge reaches. The metal effective charge each ligand would need on its own for the model's ratio to equal the fitted one. The band is the range Slater's rules allow chromium. Two ligands have an answer, both far outside it and 2.30 apart from each other. Chloride needs more than the overlap rule can be trusted to compute. Ammonia's fitted parameter is exactly zero and cyanide's is negative, and a quotient of squares is neither.

A contraction that cannot reach three of them

The angular overlap model's own derivation gives a π/σ ratio that disagrees with the fitted parameters by up to sixfold, and the metal's contraction is the obvious candidate to account for it. The whole range Slater's rules allow moves the ratio by a factor of two. Two ligands need charges far outside it, one needs a charge past where the overlap rule can be trusted at all, and two are unreachable at any charge because a quotient of squared overlaps cannot be zero or negative.

6 figures The sign follows the count, on a ring and on a chain alike. The free combination's exponent against the number of spins, for open chains and for rings. Every even count is positive and every odd count is negative, whichever topology it is — and an open chain has no frustration at all. The usual comparison sets frustrated odd rings against an unfrustrated chain of eight, which varies the frustration and the parity together.

It was the count, not the frustration

The free combination's exponent comes out negative on every odd ring and positive on every even ring and on an open chain, and the sign was put down to frustration. The control was a chain of eight. A chain of five is not frustrated in any sense — a chain is bipartite and every bond can be satisfied — and its exponent is −0.792.

6 figures One fold opens it, two close it, and the antisymmetric one does neither. The gap along three directions out of the square plane: folding one pair, folding both equally, and — from a symmetric point ten degrees out — folding one pair further while unfolding the other by as much. The first rises, the second falls, and the third leaves at zero slope. That third direction is the one the question asked for.

The direction the gap cannot see

A two-ligand fold opens the sixteen-electron gap and a four-ligand bend closes it, so a distortion mixing the two must pass through a direction the gap does not move along. It does, and the direction is the antisymmetric fold — one pair of trans ligands up, the other down. Its blindness is exact, because exchanging the two pairs is a symmetry of the arrangement.

6 figures Nine clusters, two candidate rules, and one of them survives. Every cluster's exponent, with whether it is frustrated and whether its count is even. Frustration is decided by whether the coupling graph is bipartite, since an antiferromagnet can satisfy every bond exactly when a two-colouring exists. All four decisive cases come out positive — 0.2136, 0.2084, 0.1924, 0.2048 — so the parity rule survives and the frustration account does not. The gap between the lowest positive exponent and the highest negative one is 1.0497.

The frustrated cluster with an even count

A parity account of the exponent's sign replaced a frustration account and left the two still confounded: every case tested had frustration and odd parity aligned. A tetrahedron of four spins is frustrated and even. Its exponent is +0.2084, and so are those of three more clusters the two accounts disagree about — but the mechanism proposed with the parity rule is refuted along with the account it replaced.

6 figures The gap is flat along it everywhere, and the repulsion is flat only where a symmetry says so. At each distorted geometry, how far the sixteen-electron gap and the ligand–ligand repulsion move along the direction the gap is blind to, over 5 degrees. On the symmetric line the repulsion moves by about a tenth of a per cent and downward, and its own null direction is the same one; off it the repulsion climbs by up to 1.30 per cent and its null direction is elsewhere. A blindness a symmetry produces is inherited by every function of the arrangement; one a gradient search produces is inherited by nothing.

A blindness that is inherited

There is a direction the sixteen-electron gap does not move along, and it is tempting to call it the one a complex is softest along without paying for it. That second half is a claim about an energy the gap model has no term for. Put the ligand repulsion on the same family and the answer splits: where a symmetry fixes the blind direction the repulsion is blind to it too, exactly, and slightly downhill — and where no symmetry fixes it, the two are nearly perpendicular.

5 figures 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.

8 figures Every window sits above the value it was meant to reach. For each of the five chromium(III) complexes, the whole range of π/σ ratios the model can produce as the ligand's donor atom is taken through every oxidation state it has — from its bare nucleus to its closed-shell anion — drawn as a bar, with the fitted parameter marked beneath it. The three ligands whose fitted parameter is positive have bars that begin above it and never come down. The other two have fitted parameters of zero and of a negative number, which a quotient of squared overlaps cannot be at any charge.

The correction that moves three of them backwards

Every ligand radial function in the angular overlap sweeps was a neutral atom's, while three of the five donors carry a formal charge. Giving each one the charge it actually has moves three of the five computed ratios — and moves all three away from the fitted parameter, none towards it. The whole window each donor's own oxidation states allow sits above the value it was meant to reach.

8 figures Two interactions, and they push the metal in opposite directions. Each ligand's filled π and empty π against a metal d level, on one energy scale with the vacuum at zero. The π lies below the metal and pushes it up, which is the only interaction the model's derivation has; the π lies above and pushes it down, which is the one it lacks. Both level positions are measured — an ionisation energy and an attachment energy — and the metal's is the single quantity nothing here measures, drawn at -8.0 electronvolts and swept elsewhere.

The channel that points at the metal

Two ligands in the spectrochemical series carry a fitted π parameter no quotient of squared overlaps can produce, because it is negative. Giving the derivation the second interaction it lacks makes both of them negative at every metal level — and the reason is not the energy denominators, which favour the donor channel in all three cases. It is where each orbital keeps its amplitude.

8 figures 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.

6 figures 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.

6 figures