Molecular Geometry & Bonding
what the shape decides
The collection

Every essay — page 6

Page 6 of 42, continuing through the fields in the same order.

Orbitals Where the atoms go Bonding models What symmetry decides Beyond the octet What a spectrum settles When the molecule does not stop What the shape is for What is taught wrongly Series Named objects Orbitals Refutations Search

Beyond the octet

Hypervalency without d orbitals, delocalisation, aromaticity, and the structures the first-year rules quietly cannot describe.

Hückel levels of three-centre four-electron. The orbital energies of the pi system, computed as the eigenvalues of the molecule's adjacency matrix. 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.

Three-centre bonding, computed

Three orbitals in a line and four electrons: a bonding level, a level with exactly zero amplitude on the central atom, and an empty antibonding one. The middle atom never exceeds an octet, and the ligands carry the charge — which is why every molecule that needs this arrangement has electronegative ligands.

5 figures cyclobutadiene: what alternation costs and gains. The π energy of cyclobutadiene as its bonds are alternated by β(1 ± δ), the elastic cost of alternating them at a stated stiffness, and the sum. The π curve falls linearly in δ, which is measured here and is the whole difference between a molecule that distorts and one that does not. The stiffness is a stated parameter, so where the minimum falls is not a claim; which power of δ each curve goes as is.

The vibration that lowers the symmetry

A molecule in a degenerate electronic state distorts until the degeneracy is gone. Which distortion it needs is a direct product; whether it wins is a race between a π energy falling linearly and a σ frame resisting quadratically, and both powers are measured here.

7 figures pyrrole against cyclopentadienyl anion. The Hückel levels of pyrrole beside those of cyclopentadienyl anion, which is the same graph with one diagonal entry and the bonds touching it changed. The parent's levels are symmetric about α because its matrix has nothing on the diagonal; the substituted system's are not, and the asymmetry is the size of the fitted parameter rather than a result.

Six electrons in a ring that is not all carbon

Pyrrole and the cyclopentadienyl anion hold the same six π electrons in the same five-membered ring. The anion spreads them perfectly evenly; pyrrole's nitrogen keeps 1.720 of them and furan's oxygen keeps 1.791 — the more electronegative atom donates less, and the bond orders to it fall with it.

5 figures 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.

8 figures Levels of a ring closed with a half turn in it. The orbital energies of a ring with one resonance integral reversed in sign, which is what half a turn in the ribbon of p orbitals does to it. The levels come in degenerate pairs from the bottom up rather than singly, so the count that closes a shell is 4n rather than 4n + 2. 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.

The ring with a twist in it

Reverse the sign of one resonance integral in a ring and its levels stop being one nodeless orbital above degenerate pairs and become degenerate pairs all the way up. The count that closes a shell changes from 4n+2 to 4n, and every aromatic ring size and antiaromatic ring size exchange places.

6 figures 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.

6 figures How often two electrons are in the same place. Double occupancy per site in the exact ground state of a chain of 4, against the repulsion. It starts at the independent-electron quarter and falls to nearly nothing. The second curve is how the neighbouring spins line up in the same wavefunction, which is already -0.0750 at no repulsion at all — that part is exchange — and deepens as the electrons are kept apart.

The hole that is not repulsion

Two electrons in a bond keep out of each other's way, and the obvious reason is that they repel. Setting the repulsion to zero and computing the spin correlation exactly gives −0.125 rather than nothing, and the number is reproduced to nine decimal places by a determinant with no repulsion in it at all.

5 figures Three orbitals in a line, four electrons in them. The three levels of a linear three-centre system, with the two lowest filled. The lower one is bonding across all three centres; the second has exactly zero amplitude on the middle atom, by symmetry rather than by arithmetic, so the two electrons in it sit entirely on the ends. That is where the charges come from: -0.5 on each end and +1 in the middle. Each of the two bonds has an order of 0.7071, which is 1/√2 and not the half the electron count suggests.

Hypervalency is about the ligands

The three-centre four-electron bond is offered as the reason sulfur hexafluoride needs no d orbitals. Read it forwards instead of backwards and it is a requirement rather than a permission — the arrangement puts half an electron onto each ligand before any electronegativity difference is applied, which is why the hypervalent compounds are fluorides and SH₆ is not a compound.

4 figures benzene: what alternation costs and gains. The π energy of benzene as its bonds are alternated by β(1 ± δ), the elastic cost of alternating them at a stated stiffness, and the sum. The π curve falls quadratically in δ, which is measured here and is the whole difference between a molecule that distorts and one that does not. The stiffness is a stated parameter, so where the minimum falls is not a claim; which power of δ each curve goes as is.

The hexagon is the frame's doing

Benzene's delocalisation energy is quoted as 2β and read as the reason its bonds are equal. Let the bonds alternate and the π energy goes down, not up — at every ring size, for 4n+2 as much as for 4n. The π electrons are not what keeps the hexagon regular; the σ frame is, and the margin between them is uncomfortably thin.

5 figures How far the ground state is from being one determinant. The occupations of the two natural orbitals of the Hubbard dimer against the repulsion. At zero they are two and nothing, which is a single determinant exactly. As the repulsion grows they converge on one and one, which is a state no single determinant has — the failure is in the description rather than in the number. The entropy of the occupations rises to ln 2, one bit: the two determinants of the singlet.

Two kinds of correlation, and only one is small

Correlation energy is defined as a subtraction, and the definition hides that the thing subtracted is not one thing. In the two-site model the local power of the correlation energy in the interaction falls from two to one — and at every interaction strength it equals, exactly, the occupation of the bonding natural orbital.

7 figures One carbon, three lithiums, and no way to make it two. The four localised orbitals of the tetramer, with the share of each pair on each of the eight framework atoms. Each sits on exactly four: one carbon and the three lithiums of the face that carbon caps, which was not put in anywhere. The four shares are not equal — about 59 per cent of the pair is on the carbon — so the participation number comes out near 2.5 rather than at four. A bond over four centres is not a bond divided into four.

Four centres, and the pair that will not localise

The occupied orbitals of a molecule can be mixed freely without changing anything observable, and the freedom is usually spent on making them as local as possible. For the methyllithium tetramer the answer is four centres — one carbon and the three lithiums of the face it caps — and no mixing of the four pairs reduces it.

4 figures 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.

7 figures