Molecular Geometry & Bonding
what the shape decides
The collection

Every essay — page 2

Page 2 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.

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.

5 figures Two humps and a dip, which is not what a trend looks like. The measured enthalpy of hydration of the first transition series, in kilojoules per mole, against a straight line fitted through it. The measurements do not fall on the line: they rise and dip at manganese, rise and dip again at zinc. Both dips are at configurations with no ligand field stabilisation — d⁵ high spin and d¹⁰ — and the line alone accounts for only 73 per cent of the variation.

The double hump and what removes it

The hydration enthalpies of the first transition series do not lie on a line — they rise, dip at manganese, rise and dip again at zinc. Subtract the ligand field stabilisation computed from the same model that describes their spectra and what is left is a line, with the one fitted parameter landing inside the range a spectrum measures.

4 figures The bonds are what is left over. Thirteen carbonyls and clusters, with the total valence electron count, the number of metal–metal bonds that leaves over from eighteen per metal, and the number the crystal structure has. They agree for every cluster up to five metals. At six they disagree by one, in both entries tested — and the skeletal count in the last column, which is the rule boranes are analysed with, comes out at n + 1 for both, meaning a closed deltahedron, which an octahedron is.

The bonds are what is left over

Every metal wants eighteen electrons and a metal–metal bond gives one to each of its partners, so the number of bonds in a cluster is what is left over after the counting. It works for every carbonyl cluster up to five metals and fails at six by exactly one bond — where the other counting rule, the one boranes are analysed with, is right.

5 figures 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.

7 figures The electrons repel each other less inside the complex. Seven chromium(III) complexes. Δ is the first band; B is solved from the second in closed form; β is B against the free ion's 918 cm⁻¹, which is measured on the gaseous ion. Every β is below one — the electrons in a complex repel each other less than the same electrons in the free ion, because they have more room. The two orderings are different: fluoride splits least and reduces the repulsion least, cyanide does both most, and the middle of the two series is not the same middle.

The electrons repel less inside the complex

Two measured bands determine two parameters in closed form, and one of them is the repulsion between the d electrons — which comes out below the free ion's value for every complex, by between two and forty-seven per cent. The ligands that split most are not the ligands that reduce the repulsion most, so a complex is characterised by two numbers rather than one.

7 figures 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.

8 figures Which count closes a shell, ligand by ligand. For each ligand the spectrochemical series has parameters for: its π parameter, the two gaps, and which electron count the deeper one sits above. The π donors close at twelve and the π acceptors at eighteen, and the ligand's charge predicts neither.

The count that is not always eighteen

The eighteen-electron rule is a shell closure, and an octahedral level diagram has two of them — one at twelve electrons and one at eighteen. Which is deeper is decided by the sign of one parameter: with a π acceptor the gap above eighteen is 3.720 and above twelve 2.280, and with a π donor the two swap over exactly.

8 figures The total energy against the distortion, at several gaps. The elastic cost plus the second-order lowering, for five gaps between the ground state and the excited state it mixes with. The critical gap is 1.00: above it the symmetric structure is the minimum, below it the minimum has moved off zero, and nothing about the molecule is degenerate in either case.

A distortion needs two states

A degenerate electronic state cannot survive — that is the Jahn–Teller theorem, and it can be computed. A closed shell can fail to survive too, and the condition is a number: the symmetric structure holds only while the nearest excited state of the right symmetry lies above 2λ²/k, and one of ten symmetry species in an octahedron is the right one.

8 figures 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.

8 figures 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.

5 figures The splitting against the square of one computed overlap. five chromium(III) complexes: the measured ligand-field splitting against the square of the metal–ligand σ overlap, computed from Slater-type functions at the measured bond lengths. The angular overlap model says the splitting is proportional to that square and to no other power, and the line drawn through the origin is that proportionality with nothing fitted but its slope. Across a series in which the splitting doubles, the ratio varies by 30.75 per cent.

The splitting against something structural

The angular overlap model says a ligand field splitting is proportional to the square of one overlap integral and to no other power. Computing that integral from Slater functions at the measured bond lengths, for five chromium complexes whose splittings run from 13,600 to 26,700 wavenumbers, the ratio varies by thirty-one per cent with nothing fitted. And a power law on the donor's effective charge, at an exponent nobody predicted, does slightly better.

5 figures 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.

7 figures