Molecular Geometry & Bonding
what the shape decides
Concept

Unpaired electrons — where it appears

Electrons in singly occupied orbitals, whose number a magnetic moment counts. The count is a whole number and the measured moment need not be, when more than one state is populated.

Named by 14 essays across 3 fields — each of them below, with the objects they name alongside it.

The spin-only count against nine measured moments. Each ion's magnetic moment computed from the number of unpaired electrons alone, √(n(n+2)) Bohr magnetons, beside the measured value. The two agree to a hundredth for the first five and the measurement exceeds the count by up to 0.93 for Co²⁺ — always in the same direction, which is what an omission looks like rather than noise.

A moment counts electrons, not orbitals

A magnetic moment is one of the few chemical measurements that returns an integer. Feed the count of unpaired electrons into √(n(n+2)) and nine first-row ions come back within a hundredth for five of them — and the five that miss all miss the same way, which is what a missing term looks like.

applied · Magnetism 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 coupling that is second order in the hopping. The singlet–triplet splitting of a two-site Hubbard model, and the same quantity multiplied by U. The product settles on −4t² — -4 at U = 64 — which is what makes the coupling a second-order effect rather than a term somebody put in.

What couples two spins

Two magnetic ions a few ångströms apart interact far too strongly to be doing it magnetically — the dipole–dipole energy is about 0.06 wavenumbers and the measured couplings run to hundreds. What couples them is hopping, which the Pauli principle allows for antiparallel spins and forbids for parallel ones, and the exact answer is −4t²/U.

applied · Magnetism How often two electrons are in the same place. Double occupancy per site in the exact ground state of a chain of 4, against the repulsion. It starts at the independent-electron quarter and falls to nearly nothing. The second curve is how the neighbouring spins line up in the same wavefunction, which is already -0.0750 at no repulsion at all — that part is exchange — and deepens as the electrons are kept apart.

The hole that is not repulsion

Two electrons in a bond keep out of each other's way, and the obvious reason is that they repel. Setting the repulsion to zero and computing the spin correlation exactly gives −0.125 rather than nothing, and the number is reproduced to nine decimal places by a determinant with no repulsion in it at all.

beyond · Correlation What the angular momentum operator connects. The five real d functions, with a line between each pair the z component of orbital angular momentum connects and the size of the connection on it. Written in this basis the operator is i times an antisymmetric matrix, so its expectation in any real function is exactly zero — that is the quenching, and it holds before any field is applied. The three t₂g functions are connected among themselves, so as a SET they carry eigenvalues 1, 0, -1; the two eg functions are each connected only to something outside the pair, so as a set they carry nothing.

An orbital carries no angular momentum

The d orbitals every chemist draws carry exactly no orbital angular momentum, and the proof is one line about a matrix being antisymmetric. A set of three of them carries a whole unit, which is why the spin-only formula works for most ions and fails for cobalt by nearly a Bohr magneton.

applied · Magnetism Hückel levels of cyclopropenyl cation. The orbital energies of the pi system, computed as the eigenvalues of the molecule's adjacency matrix. Degenerate levels share a line. The electrons are placed by aufbau with Hund's rule, so a half-filled degenerate shell shows as two unpaired spins.

The same ring, three charges

Three carbons in a ring are aromatic with two π electrons, a doublet with three, and a triplet with a delocalisation energy of exactly zero with four. Nothing about the molecule changed but the count, and adding electrons to a π system can make its π binding energy fall.

bonding · Aromaticity 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 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 What each of these predictors cannot see. Four predictors built elsewhere in this collection, each audited by its ties: the share of the variation in what was measured that the predictor demonstrably cannot account for, because two systems it assigns the same number to were measured to differ by that much. the spin-only moment leaves 23 per cent; the VSEPR angle leaves 92 per cent; the highest occupied Hückel eigenvalue leaves 41 per cent. The control has no ties at all and the instrument returns nothing for it, which is what it must do. No fitting anywhere: a tie is a claim a model of that form cannot escape.

Two systems a model cannot tell apart

Two molecules that share a Hückel eigenvalue but were measured to differ bound a whole family of models at 0.456 eV. That is a reusable instrument, and chemistry has plenty of predictors of exactly the same shape. Turned on themselves: VSEPR cannot account for 92 per cent of the variation in the four angles it predicts, and no fitting is involved anywhere.

bonding · Models Two prices, and which is cheaper is a property of the claim. For each slope floor, what it costs to halve it two ways. The measurement price is a standard error on the measured rise, as a fraction of the measured range — the currency every price in this argument has been quoted in. The model price is the change in the computed run, as a fraction of the predictor's own range. They are different currencies and their ordering differs between claims: the angle strain's floor is three times cheaper to move through its model, and the spin-only moment's cannot be moved through its model at any price at all.

A denominator needs three currencies

Every price in this argument has been a price on a measured rise. A slope floor is a rise over a run, the run is computed rather than measured, and pricing it turns out to need three currencies rather than one — because one predictor's run is a graph eigenvalue, one is a convention, and one is a difference of square roots of integers that nothing defensible moves.

bonding · Models Repulsion puts negative spin on the middle orbital. The spin on the middle orbital of the symmetric three-orbital radical against the on-site repulsion, solved exactly. With no repulsion it is zero: the singly occupied orbital has a node there, and one determinant cannot put negative spin anywhere. With repulsion it is negative at once, and it reaches −0.20 at 8 eV. The shaded band is the allyl radical's central carbon, from its measured proton hyperfine coupling of +4.06 gauss read through McConnell's relation — which the trio meets at a repulsion between 5.5 and 6.9 eV.

Spin below zero

A three-orbital radical read from its singly occupied orbital has no spin on its middle orbital unless its ends are both unequal and coupled — and never negative spin anywhere. Solved exactly with an on-site repulsion, the middle orbital goes negative at once, reaches the allyl radical's measured −0.15 to −0.18 at a repulsion of five and a half to seven electronvolts, and responds to the end–end coupling at first order where the one-determinant leak responds at second. The two have opposite signs, so in a bent, unequal radical the middle orbital's spin falls and then rises, and the unequal ends that let the leak in drain most of the polarisation first.

bonding · Overlap 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 The repulsion allyl needs follows the energy the hopping is built from. With carbon's own 2pπ overlaps at allyl's geometry, the on-site repulsion that reproduces allyl's central spin of −0.164 against the magnitude of the orbital energy the Wolfsberg–Helmholz rule multiplies, for neighbour repulsions from Mataga–Nishimoto and from Ohno. At carbon's −11.4 eV they give 9.49 and 12.68 eV, either side of Pariser's 11.13; Mataga–Nishimoto reaches 11.13 only at -14.4 eV, near hydrogen's −13.6 that the trio used.

The agreement was hydrogen's energy

A three-orbital radical solved exactly with on-site and neighbour repulsion reproduced the allyl radical's central spin at an on-site repulsion of 10.89 eV, a quarter of an electronvolt from Pariser's 11.13 for carbon. Its hopping, 2.91 eV, was never allyl's: it came from a round overlap of 0.25 and hydrogen's energy of −13.6 eV. Computed from carbon, the overlap barely changes — carbon's own 2pπ overlap at 1.39 Å is 0.248 — but the energy does, and with carbon's −11.4 eV the hopping is 2.34 eV, Pariser and Parr's own β. Then the repulsion allyl needs is 9.49 eV with one standard form of the neighbour terms and 12.68 with the other. Pariser's value sits between them, not on either.

bonding · Overlap

Named alongside it

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

Model limitMagnetic momentElectron correlationExact diagonalisationConventionHubbard modelOn-site repulsionSpin stateExchange couplingLeast-squaresUnderdeterminationBoltzmann

All concepts