Molecular Geometry & Bonding
what the shape decides
Concept

Coordination number — where it appears

The number of atoms bonded to a central one. It fixes which arrangements are available and, through the count of electron domains, most of what a repulsion model has to say about the angles.

Named by 25 essays across 7 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 ⟨x²⟩ is the average number of neighbours. For each structure, the mean coordination counted off the edge list beside the mean of x² measured off the eigenvalues. They are equal by an identity about graphs, not by a limit — the full band width beside them obeys no such rule.

The width of a band is a count of neighbours

The mean of the squared level energies equals the average coordination — exactly, for any structure, with no limit taken and no periodicity assumed. It is the one statement in this field that is arithmetic rather than physics, and the usual textbook formula for band width is a special case of something weaker.

solids · Bands in a solid 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 The gap against size: uniform against δ = 0.15. The HOMO–LUMO gap of a half-filled chain plotted against the number of sites, on log axes, for a uniform chain and for one whose bonds alternate. The uniform sequence falls without limit; the alternating one settles at four times the alternation.

The gap is not the band width

Two numbers describe a band and they answer different questions. The width is set by how many neighbours an atom has; the gap is set by how unequal they are. A structure can have a wide band and no gap, a narrow band and a large one, and changing one leaves the other alone.

solids · Peierls distortion benzene — D6h. 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.

Counting electrons in an extended structure

The octet rule, Hückel's 4n + 2 and the 8 − N rule that predicts the structures of the main-group elements are one rule counted three ways. Each says the same thing — close the shell — and each stops being reliable at exactly the point where closing it becomes impossible.

solids · Metal π bond orders in benzene. Each bond drawn with a thickness set by its computed pi bond order, and the number printed beside it. The orders come from the eigenvectors and cost nothing extra once the matrix is diagonalised.

What one pair can hold together

Put a single electron pair into a ring of any size and it supplies a total bond order of exactly two and a π energy of exactly 4β — three atoms, eight atoms or six hundred. Spreading a pair over more centres divides the bonding among them; it neither creates nor destroys any.

beyond · Multicentre The radial set of a 6-vertex cage. The energies of the 6 radial orbitals of a deltahedron — one per vertex, pointing at the centre — as the eigenvalues of the cage's own adjacency matrix. The top level is nodeless and is the only one that is, which is Perron's theorem about a connected graph rather than an observation. Degenerate levels share a line. The electrons are placed by aufbau with Hund's rule, so a half-filled degenerate shell shows as two unpaired spins.

A cage needs one pair more than it has corners

Every closed borane holds n+1 skeletal electron pairs for n vertices, and the extra one is a theorem about connected graphs rather than an observation about boron. A cage's radial orbitals have exactly one nodeless combination, always, whatever its shape.

beyond · Multicentre Sixteen electrons, from a reduction. The ligand σ orbitals of a square planar complex reduced in D4h (A₁g ⊕ B₁g ⊕ Eu), matched against the metal's nine valence orbitals by species, and counted. 4 bonding and 4 non-bonding orbitals hold 16 electrons.

Sixteen is also a count

A transition metal brings nine valence orbitals, and nine filled orbitals is eighteen electrons. A square plane leaves more of those nine unmatched than an octahedron does and still holds fewer electrons, because one of the leftovers is out of reach.

applied · Electron count The state on the end of a chain of 60. The amplitude of one eigenvector at each site of a 60-site chain whose site 1 has its energy raised by 0.6β. The two colours are the two signs. A state belonging to the whole chain would reach across it; this one does not.

The end is the hardest place to bind

A site in the middle of a chain traps a state for any energy difference however small. The site at the end demands a whole β before it traps anything — measured at 1.025, 1.013 and 1.006 on chains of forty, eighty and a hundred and sixty, converging on exactly one. The intuition runs the other way and is wrong.

solids · Defect What holds matter together, per pair. Four kinds of interaction between two units of matter, each computed from the model named beside it, on a logarithmic energy scale. The range from top to bottom is a factor of several hundred, which is the number behind why a molecular solid melts hundreds of degrees below a covalent one.

The lattice sum that depends on the order of adding

An ionic solid's binding is the sum of every pair of charges in it, and the series does not converge absolutely — rearranged, it gives a different answer. That is a genuine mathematical difficulty rather than a technicality, and it is the clearest example of something a real-space, neighbour-by-neighbour method cannot compute at all.

solids · Cohesion ⟨x²⟩ is the average number of neighbours. For each structure, the mean coordination counted off the edge list beside the mean of x² measured off the eigenvalues. They are equal by an identity about graphs, not by a limit — and the binding each bond supplies, in the last column, obeys no such rule and falls as neighbours are added.

The bond that weakens as neighbours multiply

Wrapped structures with two, four and six neighbours per site give a mean of x² of exactly 2, 4 and 6. Their binding per site goes 1.272, 1.611, 1.979 — and their binding per bond falls from 0.636 to 0.330, which is why a metal atom with twelve neighbours has weak bonds and a great many of them.

solids · Cohesion A band becomes a bell curve, and the dimension is the sample size. The normalised fourth moment of a wrapped hypercubic structure's density of states against its dimension, with the closed form 3 − 3/2d drawn through it. They agree to eight decimal places at every dimension from one to 6, and the reason is a limit theorem: a hypercubic spectrum is the sum of d independent one-dimensional ones, so its cumulants fall as powers of d exactly as a sum of independent samples does. The Gaussian value of three is approached and never reached.

A band becomes a bell curve

The normalised fourth moment of a hypercubic structure's density of states is 3 − 3/2d exactly, from one dimension to six, to eight decimal places — because the spectrum is a sum of d independent one-dimensional ones and the central limit theorem is what it is obeying. Reach further in one dimension instead and the shape overshoots a Gaussian rather than approaching it: a chain touching its third neighbour has a cubic structure's coordination and a fourth moment of 3.389 against 2.500.

solids · Bands in a solid The fraction a table encloses is not one number. For each ion, the fraction of its own electron density that lies inside its tabulated radius. Along each isoelectronic series the answer runs over eight percentage points, where three neutral atoms at their contact distances spanned three tenths of one. And it peaks at the neutral rather than falling through it, because a van der Waals radius is fitted to the distance between two atoms that are not bonded and an ionic radius is one term of a sum fitted to the distance between two that are.

The surface a table draws

Three noble gases stop at a surface enclosing between 99.38 and 99.75 per cent of their density — a near-constant, and an argument that a contour is a real boundary. Charge the atoms and it collapses. Across ten electrons the tabulated radius encloses anything from 94.4 to 99.98 per cent, it peaks at the neutral rather than trending through it, and radii built at a fixed enclosure do not add up to a single measured separation.

orbitals · Contour Every arrangement, and the winner is not the one with the most unlike bonds. All 1820 ways of raising 4 of 16 sites on a wrapped square net, at a contrast of 4, each placed by its count of unlike bonds against the binding it gives. The best arrangement has 12 unlike bonds where 16 is available, and it binds at 1.103953 against 1.080031 for the best of those that do have the most. The count and the spectrum are two different orderings.

The arrangement a count cannot pick

Three arrangements of one composition came out ordered by their count of unlike bonds, which looked like a rule. Enumerating every arrangement instead of three shows it is not one: it holds at every composition on a square net at a small contrast, fails at four of them at a large contrast, and fails on a triangular net at any contrast at all.

solids · Cohesion The even sharer is the one the repulsion likes least. trigonal bipyramid: 2 kinds of ligand, spread 0.1137, repulsion 6.4747; square pyramid: 2 kinds of ligand, spread 0.1658, repulsion 6.4844; pentagonal planar: 1 kind of ligand, spread 0.0000, repulsion 6.8819. The planar arrangement gives all five ligands exactly the same charge and costs 6.3 per cent more in repulsion than the bipyramid, which is the arrangement chemistry actually adopts — so the two models disagree about which arrangement is preferred, and about how much charge is moved.

Two models that disagree about the shape

The identity says how much charge a hypervalent molecule's ligands must share and nothing about how. Working out which arrangement shares it most evenly puts the σ model and the repulsion model on one axis for the first time: the even sharer is the pentagonal plane, which is the arrangement the repulsion likes least — and along the interchange chemistry actually uses, one model sees 0.15 per cent of a change and the other sees 54.

beyond · Hypervalency What a fifth ligand does to the gap above eight electrons. The gap between the fourth and fifth d levels as one axial σ donor is brought in, and as two are. It closes exactly linearly — 2eσ less one eσ for each unit of axial σ strength — and a full octahedron has none of it left. The rule of sixteen has a gap to be about only while the axial positions are empty, and how much of it survives is a number rather than a yes or no.

The ligand the rule was waiting for

A sixteen-electron complex is called reactive because it can add a ligand, and the gap that makes it sixteen points straight at where the ligand arrives. Bringing one in closes the gap exactly linearly — and leaves its exactness completely untouched, which is the opposite of what bending the same complex does.

applied · Electron count Seven nets, and the one column that sorts them. Every wrapped net here, with its dimension, coordination, third moment and band shape, beside the exponent the coupled-band measurement returns for it. The 4 nets whose third moment vanishes all give an exponent within 0.03 of −2 and a quotient constant to a fiftieth of a per cent; the 3 that do not all give one near −1 and no constant at all. Dimension does not sort them and neither does coordination — each takes values in both groups.

Seven points that looked like a switch

A constant belongs to a square net and not to a triangular one, which points at the coupling graph. Seven wrapped nets say which property of it: the third moment, and neither the dimension nor the coordination. They also say it is a switch — and made continuous, it is a crossover that every one of the seven sits twenty-five times past.

solids · Bands in a solid Folding two ligands makes the gap bigger before it makes it smaller. The gap above eight electrons as two of the four ligands fold to the same side. It rises first, to 2.0938eσ at 20°, before falling. The four-ligand path only ever closes it, so the direction the gap moves is not a property of bending — it is a property of which ligands bend.

The distortion that opens the gap

Two distortions close the sixteen-electron gap — one by bending all four ligands, one by adding a fifth. Folding two of the four makes it larger, by five per cent, before it makes it smaller. And it costs the exactness at the first degree, while the gap is still growing, so the size of a gap and whether it is exact are not one measurement.

applied · Electron count Nine combinations, and the column that sorts them is not the bands'. Two bands and a coupling, varied separately. The composite graph's third moment splits into triangles that lie inside a band and triangles that use two coupling bonds, and only the second sorts the table: every row with no gap-crossing triangle gives an exponent near −2 and a nearly constant quotient, whatever its bands are made of. Triangular bands carrying an intra-band moment of 7.296 behave exactly like square ones when the coupling is a matching.

The triangles that were never in the bands

A switch in how a gap scales is usually attributed to a band's third moment, and the attribution cannot be tested while the coupling runs along one of the bands. Separated, the bands turn out to decide nothing. Two triangular bands coupled along a matching — which cannot close a triangle across the gap — behave exactly like square ones.

solids · Bands in a solid Water's two hydrogens are closer than platinum's chlorine. For each molecule, the longest pair that is a bond and the shortest pair that is not, on a logarithmic length axis. A cutoff on the length has to sit to the right of every filled mark and to the left of every open one, and it cannot: the longest bond in the collection is 2.3200 ångström and the shortest non-bond is 1.5144. The two populations overlap by a factor of 1.53, so the rule in use is not a rule with a badly chosen number in it — it is a rule with no number that works.

No length separates them

A bond list here is one distance cutoff with a clause about hydrogen, added when peroxide came back with five bonds instead of three. The clause repaired one molecule. Across the twenty-three molecules drawn here, the longest bond is 2.32 ångström and the shortest pair that is not a bond is 1.51 — so no cutoff can work at all, and five molecules currently come back with no bonds.

symmetry · Point group The orbit identity holds on all nineteen, and the formula it replaced on eight. For each molecule in the census under the radius rule: its number of totally symmetric vibrations, the corrected count — symmetric orbits less symmetric redundancies — and the usual formula, orbits less redundancies. The corrected count lands on the molecule's own count every time. The usual formula is right for 8 of 19, and for ferrocene under the new bond list it predicts minus sixty-six.

The census a bond rule was hiding

Every internal coordinate, redundancy and totally symmetric count here is built on a bond list, and the bond list came from a length cutoff that gave five molecules no bonds. Rebuilt on the radius rule, the census reaches nineteen molecules instead of fifteen, the orbit identity holds on every newcomer, ferrocene's coordinates finally span all its vibrations — and a different gap appears: no bond rule can give a square-planar centre its two out-of-plane vibrations.

symmetry · Point group

Named alongside it

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

Model limitBands in a solidConventionTight-binding modelsDegeneracyElectron countRepulsionVSEPRClosed formDensity of statesGraphIrreducible representations

All concepts