Molecular Geometry & Bonding
what the shape decides
Concept

VSEPR — where it appears

The rule that electron domains round an atom arrange themselves as far apart as possible. Minimising repulsion on a sphere reproduces its shapes exactly, and the model has nothing to say about a transition metal.

Named by 13 essays across 5 fields — each of them below, with the objects they name alongside it.

4 sites, minimised. The arrangement of 4 points on a sphere that minimises their mutual repulsion. The angles printed were measured off the result rather than quoted, and the shape was not assumed.

VSEPR, computed

The tetrahedral angle is not 109.5 degrees because a textbook says so. It is arccos(−1/3), and it falls out of minimising the repulsion of four points on a sphere without ever being written down.

shape · VSEPR The same d shell in four fields. The five d energies in octahedral, tetrahedral, cubic, square planar coordinations, computed and drawn in units of the octahedral splitting. The dashed line is the barycentre, which the five levels sum to whatever the arrangement.

Two models, one ratio

A tetrahedron splits a d shell by four ninths of what an octahedron does. Two models that share nothing but the ligand directions — an integral over a point-charge potential and a rotated diagonal matrix — both produce that number to eight decimal places, and neither was told it.

applied · Ligand field water — C2v. The molecule with the point group found from its coordinates rather than looked up: every candidate operation was applied and kept when it permuted the atoms among themselves.

Why water is bent

The standard answer is lone pair repulsion, it predicts the right direction, and it cannot predict the magnitude. A better rule can, and the heavier hydrides show where both accounts run out.

shape · VSEPR Eight points: the cube loses. The cube and the minimised arrangement of eight points on a sphere, with the repulsion energy of each computed. The minimum is a square antiprism — the cube twisted by forty-five degrees on one face — and the margin is about one part in three hundred.

The shapes above six coordination

Eight points on a sphere do not arrange themselves in a cube. They twist one face by forty-five degrees, and above six the arrangements stop being the ones anybody would name and start being the ones a minimisation finds.

shape · VSEPR Bond angle against lone-pair weight, 2 lone and 2 bonding. The bond angle a weighted repulsion minimisation gives for 2 lone pairs and 2 bonding pairs, as the lone-pair weight runs from one to 3.2. The marked molecules are water and hydrogen sulfide, each placed at the weight that reproduces its measured angle.

What a lone pair is worth

A lone pair repels more than a bonding pair, says VSEPR, without saying how much more. Put a number on it and fit that number to water, and the same number is wrong for hydrogen sulfide by a factor of two — which means it was never a property of a lone pair.

shape · VSEPR phosphorus pentafluoride: 3 environments. The atoms of phosphorus pentafluoride sorted into orbits under its own symmetry group — two atoms are in the same orbit when an operation of the group carries one onto the other, which makes them indistinguishable by any measurement. There are 2 of F, 1 of P, and a spectrum that resolves environments counts those rather than atoms.

A spectrum counts environments, not atoms

Phosphorus pentafluoride has five fluorines in two inequivalent sets, so its magnetic resonance spectrum should show two signals. It shows one — and the reason is not a symmetry the molecule has but a motion faster than the measurement.

spectra · Spectrum Angles at exponents 1, 2, 3, 6, 12. The distinct angles of the minimised arrangement of 4, 5, 6, 7 points, under a repulsion going as one over r to the power 1, 2, 3, 6, 12. Where the arrangement is the maximally symmetric one the angles do not move at all; where it is not, both the angles and how many of them there are depend on the law assumed.

Which angles are symmetry and which are the model

VSEPR says electron pairs repel and never says by what law. For four, five and six domains it makes no difference whatever — change the exponent by a factor of twelve and not one angle moves. For seven it decides the answer.

shape · VSEPR water: 4 valence bands. The measured valence photoelectron bands of water, each labelled with the symmetry species of the orbital it comes from, and beside them the species the valence basis spans — the central atom's s and p functions and one s on each ligand, reduced in the molecule's own group. A band carrying a species the reduction does not produce would stop this figure being drawn.

Water's lone pairs are not a pair

Every course draws two equivalent lone pairs on water, pointing away from the hydrogens like a pair of ears. Its photoelectron spectrum shows the two bands they would produce at 12.6 and 14.7 electronvolts, two point one apart, in different symmetry species.

bonding · Photoelectron The largest angle each ring size can have. The ceiling on a bond angle in a closed ring of equal bonds, 180°(n−2)/n, which is the interior angle of the regular planar polygon and follows from a closed curve having to turn through a full circle. The line at 109.47° is the tetrahedral angle: rings of 3, 4, 5 atoms cannot reach it at any geometry whatever, and every larger ring can, by leaving the plane.

The angle a ring cannot have

A closed ring of equal bonds has to turn through a full circle, so its bond angles cannot average more than 180°(n−2)/n. Three, four and five atoms are below the tetrahedral angle at every geometry whatever; six is above it, and reaches it only by leaving the plane.

shape · Strain 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 site a long bond takes, and it is not the one the rule says. The energy of the axial placement minus the energy of the equatorial one, against the odd bond's length. Above the line the equatorial placement wins and below it the axial one does. The textbook sentence is that a bulkier group goes equatorial where there is more room; given a longer bond and nothing else, the repulsion model puts it AXIAL, at every length, by a margin that grows. A site pushed further out interacts with everything less, so what is left to decide is the arrangement of the four sites left behind — and three equatorial with one axial is the better set of four.

The sites are not the same size

Every arrangement in this collection puts its sites on one sphere, which is an assumption about bond lengths made silently. Give the repulsion model a bond length and it predicts that the long bond goes axial — the opposite of the rule the model is always cited for.

shape · VSEPR 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 A long bond goes axial at five and equatorial at seven. The energy of the odd site placed axially minus placed equatorially, against the odd bond's length relative to the others, for the trigonal bipyramid and the pentagonal bipyramid. Above zero the equatorial site is preferred. At five a longer bond goes axial and a shorter one equatorial; at seven, from 0.7 to 1.3, every sign is reversed — a shorter bond axial, a longer one equatorial. An open marker is a placement that is not a minimum: at seven the site each bond avoids is one it would slide out of.

The long bond goes to the crowded site

Given one bond longer than the others, the repulsion model puts it axial in a trigonal bipyramid — against the rule it is usually cited for. The reason is a crowding count, and at seven sites the count reverses: the pentagonal bipyramid's crowded site is equatorial, so a long bond goes equatorial and a short one axial. PF₅'s long bonds are axial and IF₇'s are equatorial. And at seven the site a bond avoids is not even a minimum.

shape · VSEPR

Named alongside it

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

Bond angleRepulsionMinimisationCoordination numberLone pairModel limitTetrahedral angleTrigonal bipyramidAxialBent's ruleBond lengthDegeneracy

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