Molecular Geometry & Bonding
what the shape decides
The collection

Every essay — page 10

Page 10 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.

The exact σ energy makes the contact's price fall with the coupling. The contact share at which a 1/r²⁰ contact makes the square pyramid win, against the coupling β, with the σ energy exact (lines) and to fourth order (dashes, which do not depend on β in this normalisation). Weak coupling reproduces the fourth-order shares; by β = 1 the share at q = 2 is 9.3 against 233, 25 times smaller, and at β = 1.5 the four q run 0.77, 3.1, 6.4, 15.

Priced against the whole bond

A steep contact between ligands makes a main-group square pyramid only past the fifteenth power, and its price was quoted as a hundred to two thousand times the bipyramid's fourth-order σ cost. With the σ energy exact the threshold and the pyramid do not move, and the price falls by up to two hundred times as the coupling grows — but measured against the whole σ bond energy it settles instead at a fraction set by the contact's steepness alone: a seventh of the bond under the thirtieth power, a third under the twentieth, more than the bond itself under the sixteenth, and the same for every main-group centre.

5 figures Six carbons on each angular ring, and six different answers. Rings three to eight of the twelve-ring chain with a kink of two angular rings at five and six. Inside each of the two angular rings, beside each of its six carbons, is the response of that ring with the heteroatom on that carbon, as a fraction of the straight chain's with the heteroatom on the carbon of the same kind and sublattice. On ring five the six run from 0.81 to 1.45; on ring six from 0.87 to 1.24. The two carbons the published comparison used are circled: both are ring junctions, not CH carbons.

On an angular ring the carbon decides

An interior ring of a fused chain has two CH carbons, not four, and every interior placement of the heteroatom so far sat on a ring junction — including the one that lost a fifth on the kink, which is a bay junction at the short side rather than a carbon beside the long run. Matched carbon for carbon, an angular ring's six positions give its own ring anything from 0.78 to 1.45 of the straight chain's response. On its two CH carbons the ring itself barely moves, and the rest of the chain receives about half.

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Bonding models

Valence bond, molecular orbital, and hybrids — three descriptions of one thing, related by transformations that change no observable.

1s with 1s at 2.8 bohr. The two orbitals in the plane containing both nuclei, with the regions where their product is positive and negative shown faintly. The overlap integral is the signed volume of that product, and where symmetry makes the two regions mirror images it comes out exactly zero. Contours drawn: 1s at 50% of its density, |ψ| = 1.48e-1; 1s at 50% of its density, |ψ| = 1.48e-1.

Overlap decides

Two orbitals interact in proportion to how much they overlap, and the sign of the overlap decides which of the two combinations is the lower in energy. It is one integral, and almost everything about bonding follows from it.

7 figures 1s with 2px at 2.8 bohr. The two orbitals in the plane containing both nuclei, with the regions where their product is positive and negative shown faintly. The overlap integral is the signed volume of that product, and where symmetry makes the two regions mirror images it comes out exactly zero. Contours drawn: 1s at 50% of its density, |ψ| = 1.48e-1; 2px at 50% of its density, |ψ| = 3.16e-2.

Exactly zero

Where symmetry forbids an interaction the overlap is not small. It is zero — and computing it and finding arithmetic noise is a different kind of statement from computing it and finding a small number.

8 figures sp3 hybrids. The directions the hybrids point, with the angle between them computed from the coefficients rather than quoted. The set is an orthogonal transformation of the atomic orbitals, so it describes the same space in different coordinates.

Hybrids are a basis

An sp³ hybrid set is an orthogonal matrix applied to the atomic orbitals. Rotating a basis changes no observable, so asking whether the electrons are really in hybrids is asking which coordinate system nature prefers.

7 figures Splitting goes with overlap. For each pair, the atomic levels on the outside and the combinations they form in the middle, with the splitting drawn in proportion to the computed overlap. A pair that symmetry forbids does not split at all, because its overlap is exactly zero.

Molecular orbital and valence bond

Two frameworks, taught as rivals, describing the same molecules. One starts from delocalised orbitals and localises; the other starts from localised bonds and delocalises. Pushed far enough they meet.

5 figures Hückel levels of benzene. 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.

Hückel theory and what it gets right

A conjugated system's orbital energies are the eigenvalues of a matrix of ones and zeroes. Nothing about carbon enters, no geometry enters, and the results that survive are exactly the ones that depend on neither.

7 figures Which rings close a shell. Each ring is filled with its own number of pi electrons and asked whether the highest occupied shell came out full. Of the rings drawn here, C6 close — at 6 electrons — which is Hückel's 4n+2, produced here rather than recalled.

Aromaticity as a computed shell closure

Hückel's 4n+2 rule is not a rule. It is what comes out when every ring size is filled and asked whether its highest occupied shell came out full — and the calculation that answers does not know the phrase.

7 figures π bond orders in naphthalene. 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.

Bond order from the eigenvectors

Having diagonalised the matrix, the coefficients are already there. Two sums over them give every bond's order and every atom's charge, and naphthalene's three kinds of bond come out in the order the measurements find them.

7 figures Four bonds, or one a₁ and three t₂. The same four occupied orbitals written in two bases. On the left each orbital sits on one bond; on the right one is shared over all four hydrogens and three follow the Cartesian directions. The transformation between them is orthogonal, so the density is unchanged.

The localisation transformation, demonstrated

Four equivalent bonds or one a₁ and three t₂ — two descriptions of methane's bonding electrons that disagree about everything except the electron density, which they agree about to the last bit a double can hold.

5 figures 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.

7 figures pyridine against benzene. The Hückel levels of pyridine beside those of benzene, 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.

Hückel with a heteroatom

A hydrocarbon's Hückel matrix contains no fitted number at all. Put one nitrogen in the ring and two arrive at once, the levels stop being symmetric about α, and the delocalisation energy stops being quotable — all from changing three entries in a matrix.

5 figures