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 19 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 The pucker is the flat ring's softest motion by a factor of a hundred. Every internal curvature of the flat pentagonal bipyramid under a 1/r repulsion, on a logarithmic axis: the eleven eigenvalues of the energy's second derivatives once the three rigid rotations are removed. The pair at the far left is the ring's pucker, a degenerate pair; the next motion is a hundred times stiffer with equal bonds. IF₇'s longer ring bonds make the pucker three times stiffer and leave it still thirty-four times softer than anything else.

The ring held flat by its long bonds

Given iodine heptafluoride's own composition — five long bonds and two short — the repulsion model puts the short pair on the axis, as the molecule does, and keeps the equatorial ring flat. But the flatness is the softest thing in the arrangement by a factor of a hundred. With equal bonds, a pucker of the ring costs nothing at all to second order under an inverse-square law and lowers the energy under any steeper one. What holds IF₇'s ring flat is the four per cent by which its ring bonds are longer, and that holds only under laws softer than about the 4.4th power.

shape · VSEPR A few thousandths of the energy in a contact term puckers IF₇'s ring. For each contact law added to the Coulomb repulsion of IF₇'s bond domains, the share of the Coulomb energy the contact term must carry before the flat ring stops being a minimum, on a logarithmic axis. Steep powers need very little: 2.8 per cent at 1/r⁶, 0.16 per cent at 1/r¹². An exponential contact needs 0.09 to 2.5 per cent while its range is short — and from a range of 0.35 Å it needs an infinite amount, because it holds the ring flatter instead. So does 1/r⁴. Fluorine's own contact — the square of its 2p overlap, with Slater's exponent for the atom and for the anion — needs 0.11 and 0.16 per cent. Open circles mark the share needed with equal bond lengths, for comparison.

The fluorines' range decides the ring

IF₇'s flat ring was held by its bond lengths under every repulsion softer than about 1/r^4.4, with the fluorines left out. Put them back as a second set of repelling points and under the same law they change nothing at all — they sit on the same rays, and the model cannot tell where along a ray a point sits. They matter only as a contact repulsion of a different shape. Then a steep power puckers the ring with a few thousandths of the energy. An exponential contact puckers it only if its range is under 0.346 ångström, and a longer one holds it flatter. Fluorine's own orbital gives a range near 0.16. And the ring that puckers does not settle: its envelope and its twist differ by a ten-thousandth of the pucker's depth, so the pucker runs round the ring.

shape · VSEPR Xenon hexafluoride's fluorines move its lone pair off the axis and through a face. At a lone-pair weight of 1.5, the energies of the capped octahedron and the edge-capped octahedron relative to the pentagonal pyramid, against the six fluorines' Born–Mayer contact repulsion as a share of the domain energy; filled dots are the lowest structure found from any nudged start. With no contact the pyramid, lone pair on the axis, is lowest by 0.063. The capped octahedron reaches it at a share of 2.3% and is the lowest from 3 per cent; in between the lowest structure has left the axis and keeps only a mirror plane.

The lone pair goes through a face

Xenon hexafluoride is the other famously floppy seven-domain molecule, and its seventh domain is a lone pair: a domain with no atom at its end, which a contact repulsion between fluorines cannot see. With the domains alone the model puts the lone pair on the axis of a pentagonal bipyramid at every lone-pair weight — a pentagonal pyramid of fluorines. Add the six fluorines' contact repulsion and, past a share of the energy that rises with the lone pair's weight, the lowest structure becomes a distorted octahedron with the lone pair through one face. Every contact law tried does this. On the way the lone pair leaves the axis continuously, and past it the molecule's floppiness is the lone pair hopping from face to face over an edge — not a ring's pucker moving round a ring.

shape · VSEPR The capped octahedron joins the regular one as the lone pair's weight falls. How far xenon hexafluoride's six fluorines are from a regular octahedron in the capped-octahedron structure — the spread of the twelve cis F–Xe–F angles — against the lone pair's weight q, with no contact between the fluorines. As q falls to zero the spread falls to zero, at 58.5° per unit of q: a weightless lone pair leaves a regular octahedron with the lone pair over a face. With the domains alone the structure is the lowest of all below q = 0.657 (shaded); it stays a minimum up to 0.821 and is a saddle beyond (dashed).

A lone pair weaker than a bond

The repulsion model put xenon hexafluoride's lone pair on an axis at every weight VSEPR allows, and only a contact between the fluorines could move it into a face. Below the weight VSEPR allows — a lone pair that repels less than a bond, which is what an inactive pair is — it goes into a face on its own. As the weight falls to zero the distorted octahedron becomes a regular one continuously, at 58.5° of angle spread per unit weight; and between the face and the axis there is a region nobody had looked at, where the lone pair sits over an edge.

shape · VSEPR Wherever the face is lowest, the long bonds go opposite the lone pair. For xenon hexafluoride's capped octahedron with three bonds long and three short in the ratio 1.972 : 1.856, the energy of the isomer with the long three round the face the lone pair opens, minus the isomer with them opposite, against the lone pair's weight q. Above zero the long bonds go opposite. With the domains alone the sign changes at q = 0.727, but the face is the lowest structure only below 0.657 (shaded), so wherever the model's face is lowest its long bonds sit away from the lone pair. With a one-per-cent contact between the fluorines the preference for opposite is larger and never changes sign.

The model lengthens the wrong three

Xenon hexafluoride's capped octahedron has its three longer bonds round the face the lone pair opens. Give the repulsion model three long bonds and three short and ask where it puts them: wherever the face is its lowest structure — at every lone-pair weight and every contact share mapped — it puts the long three on the opposite side. The lone pair's push on its neighbours and the fluorines' push on each other nearly cancel, and the lone pair wins only above a weight of 0.727, where the face has already given way to the edge.

shape · VSEPR With the bonds free to stretch, which three are long depends on where the lone pair goes. The near three bonds' mean length over the far three's, minus one, against the lone-pair weight, with every bond on a spring of stiffness 20 and the six bonds stretched by about 8.7%. If the lone pair keeps its place among the bonds the long three are opposite the face below a weight of 0.725, as with lengths fixed by hand. If it stays where it was, or sits on a bond's spring, the long three are round the face across that range instead.

The lone pair has no bond to stretch

Put every one of xenon hexafluoride's bonds on a spring and let the repulsion set the six lengths, and which three come out long is decided by something the model has never had to say: how far from the xenon the lone pair sits once the bonds have moved. Three equally defensible answers give three different verdicts — and none of them produces a split within a factor of twenty of the measured one without stretching every bond beyond recognition.

shape · VSEPR

Named alongside it

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

RepulsionModel limitLone pairMinimisationBond angleBond lengthCoordination numberAxialDegeneracyEquatorialLocal minimumTetrahedral angle

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