Molecular Geometry & Bonding
what the shape decides
Concept

Electron count — where it appears

The number of electrons in a complex's orbitals below the antibonding set. Eighteen for an octahedron with π-accepting ligands and twelve with π-donating ones, and the sign of one parameter decides which.

Named by 26 essays across 4 fields — each of them below, with the objects they name alongside it.

Eighteen electrons, from a reduction. The ligand σ orbitals of an octahedral complex reduced in Oh (A₁g ⊕ Eg ⊕ T₁u), matched against the metal's nine valence orbitals by species, and counted. 6 bonding and 3 non-bonding orbitals hold 18 electrons.

Eighteen is a count

The eighteen-electron rule is usually justified by adding up an s, three p and five d orbitals. That is a restatement rather than a reason. Reduce the ligand orbitals in the complex's own point group, match them against the metal's by symmetry, and the number that comes out is the count of orbitals lying below a gap — which is eighteen for an octahedron, sixteen for a square plane, and eighteen again for a tetrahedron for a different reason.

applied · Electron count Donation strengthens the C–O bond, back-donation weakens it. Two interactions computed as two-level problems: 0.39 of an electron donated out of the ligand's σ orbital, worth 0.06 on the C–O bond order, and 0.5 donated back into π*, worth -0.4. Beside them, six measured stretching frequencies: five isoelectronic species differing only in charge, and free CO.

Back-bonding is two interactions

A carbon monoxide molecule bound to a metal donates from an orbital that is slightly antibonding and accepts into one that is strongly antibonding, so the two halves of the bonding move its stretching frequency in opposite directions. Five isoelectronic complexes differing only in charge settle which wins — and one of them stretches above free CO.

applied · Electron count One metal orbital, two ligands competing for it. Metal–ligand bond orders in a three-orbital model as the left-hand ligand's interaction is turned up. Its own bond order rises and the bond order to the ligand opposite falls, from 0.62 at equal strengths to 0.42 at the strongest. Nothing else in the model can carry the effect: switch the second bond off and it vanishes exactly.

The trans influence is an overlap argument

Two ligands on opposite sides of a metal both bond through the same metal orbital, and there is only one of it. Strengthen one and the bond order to the other falls — computed exactly on three levels, and measured as a bond length that grows by a tenth of an ångström.

applied · Overlap 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 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 bonds are what is left over. Thirteen carbonyls and clusters, with the total valence electron count, the number of metal–metal bonds that leaves over from eighteen per metal, and the number the crystal structure has. They agree for every cluster up to five metals. At six they disagree by one, in both entries tested — and the skeletal count in the last column, which is the rule boranes are analysed with, comes out at n + 1 for both, meaning a closed deltahedron, which an octahedron is.

The bonds are what is left over

Every metal wants eighteen electrons and a metal–metal bond gives one to each of its partners, so the number of bonds in a cluster is what is left over after the counting. It works for every carbonyl cluster up to five metals and fails at six by exactly one bond — where the other counting rule, the one boranes are analysed with, is right.

applied · Electron count Strong outside, weak inside. The bond orders along each chain. A three-centre system has two equal bonds of 0.707 — not the one half the electron count suggests — and every longer chain alternates, strong at the ends and weak in the middle. The spread grows with the chain: 3:0.000, 5:0.211, 7:0.271, 9:0.296. Nothing here is about iodine.

Hypervalency does not stop at three centres

The three-centre four-electron bond is written up everywhere as an arrangement peculiar to hypervalent molecules. It is the first member of a family — five centres and six electrons, seven and eight — and the family predicts alternating bond strengths that the polyiodide crystal structures have.

beyond · Hypervalency Where two bands lie, as their centres are pulled apart. The σ band and the π band of a two-orbital chain, drawn as the intervals they occupy, against the difference in site energy between the two orbitals. Below a difference of 3 the two intervals overlap and the filled-band count stops deciding anything.

A full band is not an insulator

Two electrons per atom, two orbitals per atom, and the lower set exactly full: the count says insulator, and magnesium is a metal. The count is not wrong about the count. What it assumes is that the two sets of levels occupy separate ranges of energy, and whether they do is a comparison of four numbers that has nothing to do with how many electrons there are.

wrong · Metal Which count closes a shell, ligand by ligand. For each ligand the spectrochemical series has parameters for: its π parameter, the two gaps, and which electron count the deeper one sits above. The π donors close at twelve and the π acceptors at eighteen, and the ligand's charge predicts neither.

The count that is not always eighteen

The eighteen-electron rule is a shell closure, and an octahedral level diagram has two of them — one at twelve electrons and one at eighteen. Which is deeper is decided by the sign of one parameter: with a π acceptor the gap above eighteen is 3.720 and above twelve 2.280, and with a π donor the two swap over exactly.

applied · Electron count Four, however many ligands there are. For each molecule, the number of ligand σ combinations that find a partner among the central atom's four s and p orbitals, against the number of ligands and lone pairs it has. The matched count rises along the diagonal and then stops at four, because there are four orbitals; everything above the ceiling is a pair with nowhere on the central atom to go, and that is what the word hypervalent names.

Four is all that s and p can match

Reduce the ligand σ set of ten molecules in each one's own point group and ask how many of its components transform as one of the central atom's four valence orbitals. The answer is never more than four — not by arrangement, in every geometry from linear to octahedral — and what is left over is n + L − 4, with exactly twice that many electrons in excess of an octet.

beyond · Hypervalency The total is the same in both columns; nothing else is. Ten complexes counted by both conventions. The neutral method gives the metal its group number and every ligand what it brings as a neutral fragment; the ionic method assigns an oxidation state and gives every anionic ligand a pair. The two totals agree in every row. The oxidation state and the d count do not agree wherever the oxidation state is not zero, and the moments the two d counts predict differ by as much as 2.83 Bohr magnetons.

The same count, two oxidation states

Count a complex by the neutral method and by the ionic one and the total is the same integer every time — eighteen for ferrocene, sixteen for tetrachloroplatinate, twenty for hexaaquanickel. The oxidation state and the d count are not: hexaaquairon is d⁶ on one convention and d⁸ on the other, and the two predict spin-only moments of 4.90 and 2.83 against a measured 5.40.

applied · Electron count The metal's charge is a coordinate, and the count is not. The metal's charge in an octahedral d6 complex, against how much of each shared pair the ligand is given. Half each is Mulliken's rule and the whole to the ligand is the assumption an oxidation state makes; the answer runs over 2.06 electrons between them. The oxidation state itself is 0, which is off the end of the range, and the electron count is the same number at every point on it.

An integer nobody measured

The oxidation state of chromium in the hexacarbonyl is zero. Its charge, computed from the same wavefunction, is anywhere between −3.04 and −0.98 depending on how the shared electrons are divided — and the integer sits outside that whole range. The electron count, meanwhile, is eighteen at every point on it.

applied · Electron count However hard the π channel is driven, the counted orbital stays the metal's. The metal's share of the filled T₂g orbital against the π coupling, with the eighteen-electron count drawn beside it. The share falls from 1 to 0.5467 across a coupling range of 80,000 cm⁻¹ and approaches a half from above without reaching it: the lower eigenvector of a two-level problem always carries more of the lower basis function, whatever the coupling. The count is eighteen at every point.

The count that cannot be broken by strength

Back-donation puts electrons into orbitals that are not the metal's, and the eighteen-electron rule counts the metal's nine. Turning the π channel up as far as it will go never breaks it: the counted orbital's metal share falls from 100 per cent to 54.67 and approaches a half from above without reaching it. What does flip it is not strength but order.

applied · Electron count Ten molecules, twenty-six arrangements, three disagreements. Every arrangement of every one of the ten molecules, with the point group recovered from the arrangement's own coordinates and the ligand σ set reduced in it. 27 of them can be worked in a tabulated group; the formula n + L − 4 is right for 23 and wrong for 4. Every failure is a planar arrangement of four or more ligands, and every one of them is a molecule that does not adopt that arrangement.

The square that wastes an orbital

The orphan count that prices hypervalency was treated as a property of a molecule's composition — ligands plus lone pairs minus four. Run on twenty-six arrangements of the same ten molecules it is right for twenty-three and wrong for three, and all three are flat. A planar arrangement gives a main-group centre three usable orbitals rather than four, so the count is a property of the shape.

beyond · Hypervalency The gap that makes sixteen special does not move. The gap above the sixteen-electron closure of a square plane and above the eighteen-electron closure of an octahedron, against the π strength. The octahedron's is 3eσ − 4eπ and moves at every value; the square plane's is exactly 2eσ until the π strength reaches a quarter of the σ one, because the orbital that sets it is d(z²) and a square-planar ligand set has nothing of that symmetry to offer. Past the threshold the two are the same number, which is not a coincidence: beyond it the square plane's gap is set by d(xy) and the expression is the octahedron's.

The orbital a ligand cannot reach

The sixteen-electron count of a square plane is a statement about an energy rather than about symmetry matching, so it was the count that ought to be sensitive to a π channel where the eighteen-electron one is not. It is not sensitive either — and for a sharper reason. The orbital that sets its gap is d(z²), and a square-planar ligand set contains nothing of that symmetry, so the gap is exactly 2eσ until the π strength reaches a quarter of the σ one.

applied · Electron count The sixteen-electron gap, from a plane to a tetrahedron. The gap above eight d electrons as four ligands are folded out of a square plane towards a tetrahedron, at three π strengths. It is 2eσ exactly at the plane whatever the π strength is, and exactly zero at the tetrahedron whatever it is — the upper three levels there are the degenerate t₂ set. In between the three curves separate, and the separation is what the π channel is doing.

The gap that only a tetrahedron closes

The sixteen-electron gap is 2eσ exactly, and a square-planar π set cannot touch it because d(z²) has no partner there. Fold the ligands out of the plane and the gap survives almost intact for fifteen degrees, closes to nothing only at the tetrahedron — and loses its exactness at the very first degree.

applied · Electron count The repulsion against the count, and they do not sort together. Every arrangement of the census by how many ligand σ combinations are left without a central partner and by how far its ligand repulsion sits above the best arrangement of that many points. The three that break the formula are marked. They are not the expensive ones: they sit at 4.2, 6.3, 9.8 per cent while the arrangements the formula gets right run to 26.5.

Expensive is not the same as unadopted

A counting formula right for twenty-three arrangements and wrong for three invites a reading: the failures are the arrangements nothing adopts, so the formula is reliable because chemistry stays away from where it breaks. Put the repulsion energy on the same axis and the reading fails — the most expensive arrangement in the census is one the formula gets right.

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 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 One steps, the other slides — and the step is at the far end. The repulsion energy and the orphan count along the path from a tetrahedron to a square plane. The energy rises smoothly and monotonically, lowest at the tetrahedron and highest at the plane. The count is zero everywhere — including at 89.99° — and becomes one only at 90° exactly. The step is not near the energy's minimum; it is at its maximum, and it is at a single point.

A count that changes at one point

Where does the orphan count step along a distortion, relative to where the energy's minimum sits? A rule that depends on an exact symmetry may have no answer for a real molecule. On the path from a tetrahedron to a square plane the count is the same at every angle up to 89.99° and changes only at 90° exactly — which is the energy's maximum, not its minimum, and a single geometry out of a continuum.

beyond · Hypervalency The energy runs the whole way; the count exists at the two ends. The Coulomb repulsion of six ligands along the Bailar twist, from the trigonal prism at 0° to the octahedron at 60°, with the geometries that have an orphan count marked underneath. The energy is smooth and monotone, lowest at the octahedron. The count is defined at the two ends and at the handful of angles the symmetry finder rounds into them, and nowhere else — not because the geometry is unsymmetrical, but because its group is not tabulated.

The group nobody wrote a table for

The orphan count is predicted to be constant along the Bailar twist, because the twist keeps D3 the whole way. The premise is exactly right — at every angle strictly between the prism and the octahedron the symmetry finder assembles six operations that hold to four parts in 10¹⁶. The conclusion cannot be tested here, because a count is a reduction, and the calculation's nineteen character tables did not include D3.

beyond · Hypervalency The same number, in three different groups, all the way along. The orphan count along the Bailar twist once the D3 character table is written. It is 2 at every one of the 15 geometries the symmetry finder answers for, across D3h at the prism, D3 through the whole interior and Oh at the octahedron. The lower row shows what the same sweep gave before the table existed: two ends and nothing between. This constancy was predicted from the premise that the twist keeps D3 throughout, and the premise was right.

The count the table was hiding

The interior of the Bailar twist looks uncountable — exactly D3 at every angle, and D3 is a point group whose character table is rarely written out. Writing it takes nine lines and settles the prediction made for the path: the orphan count is two at every geometry on the path that can be answered, in three different groups, out of three different decompositions.

beyond · Hypervalency One fold opens it, two close it, and the antisymmetric one does neither. The gap along three directions out of the square plane: folding one pair, folding both equally, and — from a symmetric point ten degrees out — folding one pair further while unfolding the other by as much. The first rises, the second falls, and the third leaves at zero slope. That third direction is the one the question asked for.

The direction the gap cannot see

A two-ligand fold opens the sixteen-electron gap and a four-ligand bend closes it, so a distortion mixing the two must pass through a direction the gap does not move along. It does, and the direction is the antisymmetric fold — one pair of trans ligands up, the other down. Its blindness is exact, because exchanging the two pairs is a symmetry of the arrangement.

applied · Electron count 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 centres, one path, and the leftover changes sides. The count of leftover orbitals along the Bailar twist, for a main-group centre and for a transition metal. At four valence orbitals against six ligand combinations, two combinations are left with no partner and the count is a count of orphans; at nine against six, every combination finds one and three metal orbitals are left instead. Both are constant along the whole path, in three different point groups, out of decompositions that share no species — which is the replacement rule holding in a case where the arithmetic runs the other way.

The leftover changes sides

A main-group centre brings four valence orbitals against six ligand combinations, so two are orphaned. A transition metal brings nine, so the arithmetic inverts and three metal orbitals are left instead — three at every geometry of the Bailar twist, out of decompositions that share no species. Run past the whole arrangement census, exactly one arrangement orphans anything at a metal, and it needs an f orbital to fix.

beyond · Hypervalency

Named alongside it

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

Model limitIrreducible representationsLigand fieldDegeneracyd orbitalsReduction formulaSymmetry operationHypervalencyNon-bonding orbitalsPoint groupCoordination complexEighteen-electron rule

All concepts