Molecular Geometry & Bonding
what the shape decides
Concept

Repulsion — where it appears

The mutual push between electrons, or between the domains a shape model treats as objects. Which one is meant matters: the first is an interaction in a Hamiltonian and the second is a device for predicting angles.

Named by 20 essays across 4 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 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.

Five sites are not alike

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.

shape · VSEPR 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 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 Two orbitals, 4 electrons, S = 0 and S = 0.25. Two interacting orbitals with 4 electrons in them, drawn twice: once with the overlap set to zero and once with it kept at 0.25. Dropping the overlap makes the two shifts equal, which is the picture usually taught; keeping it makes the upper level rise by more than the lower falls, which is why four electrons in two orbitals is a repulsion.

The antibonding level goes up more

The two-level diagram every course draws is symmetric, and the symmetry is an artefact of setting the overlap to zero. Keep it, and the upper level rises further than the lower one falls — which is why helium has no molecule and why closed shells push each other apart.

bonding · Overlap 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 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.

applied · Ligand field The same angles, the same torsions, and not the same molecule. twelve closed conformers of a ring of 10 at a bond angle of 111.5 degrees. Every one has exactly the same bond angles, so an account built from angles and torsions places them all on the horizontal axis alone. The vertical axis is the closest approach of two atoms four or more bonds apart, which no term in that account mentions: two of these differ by 0.08 kilojoules in torsional energy and by 0.75 ångström in how close they come.

The atoms that meet across a ring

Twelve closed conformers of a ten-membered ring at one bond angle, so every one has identical angle strain by construction. Two of them differ by 0.026 kilojoules a mole in torsional energy and by 0.80 ångström in how close two atoms on opposite sides of the ring come — 2.331 against 3.131, where two carbons are in contact at about 3.4. An account built from angles and torsions calls those two structures the same.

shape · Strain Where two He atoms stop, with no contact distance put in. The repulsion between two He atoms, computed from the overlap of their filled valence orbitals — four electrons in a bonding and an antibonding pair, of which the antibonding one rises further — against London's dispersion attraction from the measured polarisability and ionisation energy. The minimum is at 3.14 ångström where the tabulated van der Waals contact is 2.8, and nothing anywhere in the calculation is a length.

The radius that was tabulated

A van der Waals radius is a fitted number that every structural argument in chemistry uses. Computing it instead — a repulsion taken from computed overlap integrals, an attraction taken from two measured scalars, and no length anywhere — puts helium's contact at 3.140 ångström against a tabulated 2.80, neon's at 3.226 against 3.08 and argon's at 3.996 against 3.76. And the surface two atoms actually stop at encloses 99.4 per cent of the density, not ninety.

orbitals · Contour The gap is flat along it everywhere, and the repulsion is flat only where a symmetry says so. At each distorted geometry, how far the sixteen-electron gap and the ligand–ligand repulsion move along the direction the gap is blind to, over 5 degrees. On the symmetric line the repulsion moves by about a tenth of a per cent and downward, and its own null direction is the same one; off it the repulsion climbs by up to 1.30 per cent and its null direction is elsewhere. A blindness a symmetry produces is inherited by every function of the arrangement; one a gradient search produces is inherited by nothing.

A blindness that is inherited

There is a direction the sixteen-electron gap does not move along, and it is tempting to call it the one a complex is softest along without paying for it. That second half is a claim about an energy the gap model has no term for. Put the ligand repulsion on the same family and the answer splits: where a symmetry fixes the blind direction the repulsion is blind to it too, exactly, and slightly downhill — and where no symmetry fixes it, the two are nearly perpendicular.

applied · Electron count Two conditions, and one factor that nearly meets both. The mean error of each group of pairs as the third-row p exponents alone are contracted, with the second-row pairs untouched by construction. One factor has to bring both other groups onto them. The pairs with one third-row ion arrive at 1.376 and potassium chloride at 1.346, 2.2 per cent apart — a test that could have produced two factors nowhere near each other, and did not.

One contraction for two conditions

The ionic model's errors run with the row of the periodic table, and the test proposed for that — stiffen the repulsion and watch for second-row pairs moving out and third-row pairs moving in — produces exactly that pattern from a repulsion that knows nothing about shells. The test that can fail contracts the third-row shells alone and asks one factor to bring two different groups of pairs onto the second-row ones. The two factors needed are 1.35 and 1.38.

orbitals · Contour Three ions with the same shell, and three different contractions. Each pair with one third-row ion, its error plotted against a contraction of that ion's p exponents alone. Sodium chloride's error falls to the second-row pairs' mean when chloride is contracted by 1.302, potassium fluoride's when potassium is contracted by 1.387, and calcium oxide's when calcium is contracted by 1.441. The single factor that contracts every third-row shell at once, 1.376, is drawn faint: it sits between the three, and the three span 10.7 per cent.

Three contractions for one shell

One contraction of the third-row p shells removed the row pattern from an ionic model's errors, with two conditions met by factors 2.2 per cent apart, and left the order of three pairs untouched. Taken ion by ion, chloride needs 1.302, potassium 1.387 and calcium 1.441 — the pairs' own order — and potassium chloride, fitted on nothing, lands among the second-row pairs. The single factor's two conditions agreed because both were averages of these three.

orbitals · Contour 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 strength closes the gap by pulling the pairs apart; a range closes it with them together. The six rock-salt pairs' mean error in the separation against the spread between the largest and smallest, as each of two adjustments is turned up from where the per-ion contractions left them: a mean of −14.1% and a spread of 1.3 points. Multiplying the repulsion's strength reaches a mean of zero at 6.15 with a spread of 8.8 points. Multiplying every ion's exponents by one factor reaches it at 0.870 with a spread of 0.98.

The shortfall was a length, not a strength

With each third-row ion's orbital range set by its own contraction, an ionic model leaves all six rock-salt separations about fourteen per cent short, the same for every pair to 1.3 points. A shortfall common to every pair is what a strength constant is for, and one closes it at a factor of 6.15 — while spreading the six over 8.8 points, because a strength moves each equilibrium by a length, the repulsion's own decay length times the logarithm of the factor, and those lengths are not the same fraction of every separation. Scaling every ion's orbitals by one factor instead closes the mean with the six within a point of each other. It works because it is exactly a small strength change followed by a uniform stretch.

orbitals · Contour 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 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.

Model limitVSEPRMinimisationBond angleBond lengthClosed-shell configurationsAxialCoordination numberEquatorialLone pairOverlap integralTrigonal bipyramid

All concepts