Molecular Geometry & Bonding
what the shape decides
Concept

Magnetic moment — where it appears

The magnetic dipole a molecule carries, measured through its susceptibility. For a well-isolated ground state it counts unpaired electrons; where two states are close together it is a thermal average and counts nothing.

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 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 Why d⁸ is square planar and nothing else is. The ligand field energy a square plane buys over a tetrahedron, for every d^n, each geometry in its own ground state. The maximum is at d⁸ (1.73 in units of eσ), it is exactly zero for an empty shell and a full one, and d⁸ is one of only two fillings where the plane is diamagnetic and the tetrahedron is not.

VSEPR does not reach a transition metal

Four ligands minimising their repulsion give a tetrahedron, whatever the metal. Half the four-coordinate complexes of the platinum group are square planar, which has larger repulsion, and the term that overrules it is largest at d⁸ and exactly zero at d⁰ and d¹⁰ — which is where the repulsion rule works again.

wrong · VSEPR Which way the line moves counts the electrons. Three ions, their computed g-values and their measured ones. The shift is −2λ times a sum of squared matrix elements over energy denominators; the matrix elements are exactly two for Lz between dx²−y² and dxy and exactly one for Lx between dx²−y² and dyz, which is why the shift along the axis is four times the shift across it. The last column is what is left over — the orbital reduction factor, which is below one when the electron spends part of its time on the ligands and is a covalency measured with a magnet.

The g-value is the orbital coming back

A ligand field quenches the orbital angular momentum of a d electron, and spin-orbit coupling gives some of it back — upward for a shell more than half full and downward for one less than half full. Which way a resonance line moves counts the electrons, and the size of the move comes from two matrix elements and one optical splitting.

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 The total is the same in both columns; nothing else is. Ten complexes counted by both conventions. The neutral method gives the metal its group number and every ligand what it brings as a neutral fragment; the ionic method assigns an oxidation state and gives every anionic ligand a pair. The two totals agree in every row. The oxidation state and the d count do not agree wherever the oxidation state is not zero, and the moments the two d counts predict differ by as much as 2.83 Bohr magnetons.

The same count, two oxidation states

Count a complex by the neutral method and by the ionic one and the total is the same integer every time — eighteen for ferrocene, sixteen for tetrachloroplatinate, twenty for hexaaquanickel. The oxidation state and the d count are not: hexaaquairon is d⁶ on one convention and d⁸ on the other, and the two predict spin-only moments of 4.90 and 2.83 against a measured 5.40.

applied · Electron count 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 The four things a susceptibility curve measures, in order. The four directions in the parameter space, best fixed first, each written as the product of powers it is. The Jacobian is logarithmic, so a direction is a set of exponents and a combination is a product — which is why the answer can be printed. The best-determined is g · J^-0.33, fixed to 0.08 per cent by a curve measured to one per cent; the worst is tip · rho^-0.40, fixed to 40. Neither is one parameter's own axis.

The product a curve measures

A susceptibility curve's fourth parameter is undetermined, and the question is which combination the free direction actually is. It is a product of powers, because the Jacobian is logarithmic — and at the usual window it is the temperature-independent term divided by the 0.40 power of the monomer fraction, fixed to forty per cent, while the product one place up is fixed to 6.7. A paper could print that instead of four numbers.

applied · Magnetism Two instruments, and they do not agree. Each of this collection's four predictors under both tests. The exact-tie instrument reports the share of the variation a predictor demonstrably cannot account for, and it ranks VSEPR worst and has nothing at all to say about the ring-strain control. The near-tie instrument reports how much steeper any model must be somewhere than the set is overall, and it ranks the control worst and cannot speak about VSEPR, whose predictor is a label rather than a number. Neither instrument is the general one, and a predictor that passes one has not been audited.

The pair that is not a tie

An exact tie bounds every model of a form with no fitting, and it is available only for predictors that read integers. A near tie bounds the model's derivative instead, and a pair the predictor orders the wrong way round refuses monotonicity outright. Run on four standard predictors, the new instrument ranks them in a different order from the old one — and its worst case is the control the old one could not see.

bonding · Models

Named alongside it

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

Model limitUnpaired electronsConventionLigand fieldd orbitalsLeast-squaresDegeneracyExchange couplingSpin stateTemperatureUnderdeterminationApproximation

All concepts