VSEPR — the series · Molecular Geometry & Bonding

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.

part 1 · shape

Five sites are not five of a kind. The minimised arrangement of five points, with the two axial sites marked apart from the three equatorial ones. Their neighbour angles differ, so the two kinds of position are genuinely different places — which the shape's name does not convey.

Every other common arrangement has one or two distinct angles. Five has three, because two of its positions are on an axis and three are round an equator — and a molecule built that way does something about it.

part 2 · shape

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.

part 3 · shape

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.

part 4 · shape

sp3 hybrids. The directions the hybrids point, with the angle between them computed from the coefficients rather than quoted. The set is an orthogonal transformation of the atomic orbitals, so it describes the same space in different coordinates.

Bent's rule, against the substituent series

More electronegative substituents get more p character, so the angle between them shrinks. The rule predicts a trend, the trend is observed, and the quantity it depends on turns out not to be one quantity.

part 5 · shape

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.

part 6 · shape

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.

part 7 · shape

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.

part 8 · wrong

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.

part 9 · shape

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.

part 10 · shape

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.

part 11 · shape

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.

part 12 · shape

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.

part 13 · shape

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.

part 14 · shape

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.

part 15 · shape

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.

part 16 · shape