Molecular Geometry & Bonding
what the shape decides
Series

Electron count — the series

14 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · applied
  2. 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.

    part 2 · applied
  3. 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.

    part 3 · applied
  4. 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.

    part 4 · applied
  5. 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.

    part 5 · applied
  6. 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.

    part 6 · applied
  7. 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.

    part 7 · applied
  8. 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.

    part 8 · applied
  9. 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.

    part 9 · applied
  10. 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.

    part 10 · applied
  11. 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.

    part 11 · applied
  12. 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.

    part 12 · applied
  13. 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.

    part 13 · applied
  14. 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.

    part 14 · applied

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