Molecular Geometry & Bonding
what the shape decides
The collection

Every essay — page 34

Page 34 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

What symmetry decides

A molecule's point group follows from its coordinates, and it settles whether the molecule can be polar or chiral with no reference to bonding.

carbon dioxide: D∞h worked in D2h. The vibrations of carbon dioxide, computed in D2h because D∞h has infinitely many operations and the reduction formula divides by the order of the group. Each constructed operation is checked against the molecule before use. The count must come out at 3N−5 rather than 3N−6 — 4 for 3 atoms — because rotation about the molecular axis moves no atom and is not a motion of the molecule at all.

An infinite group, worked in a finite one

A linear molecule has infinitely many symmetry operations, and every formula in character theory divides by the number of them. The standard device is to work in a finite subgroup — and it is worth computing what that trade costs rather than putting it in a footnote.

5 figures F s on sulfur hexafluoride: a₁g ⊕ eg ⊕ t₁u. The character of the basis under each class of operations, which is a count of what did not move, and the multiplicities that come out of the reduction formula. The multiplicities must be whole numbers, and that is the check.

Six bonds and four orbitals

The six fluorine σ functions of sulfur hexafluoride span a₁g ⊕ eg ⊕ t₁u. Sulfur's 3s and 3p supply a₁g and t₁u and nothing else, so four bonding orbitals hold twelve electrons across six bonds — a bond order of two thirds, computed from characters with no energy anywhere in it.

7 figures How many groups a molecule can fall to. For each molecule, every subgroup of its point group, found by closing subsets of the operations recovered from its atom positions — beside the number the corresponding abstract group is known to have. The two agree in all 4 cases. The last column is how many of those subgroups this site holds a character table for, which is a minority in every row but the first.

Every group a molecule can fall to

A distortion takes a molecule's symmetry away, and what is left is not a free choice — it has to be a group. Closing subsets of methane's twenty-four operations returns thirty of them, which is exactly the number the symmetric group on four letters has.

5 figures Why a character table has the rows it has. All 20 groups this site defines, with the two counts that fix the shape of every character table: the number of representations equals the number of classes, and the sum of the squares of their dimensions equals the order of the group. Neither column pair differs anywhere in the table, and there is no room for another row in any of them.

Why a character table stops where it stops

A character table is square, and both of its dimensions are forced. The number of representations equals the number of classes, the squares of their dimensions sum to the order of the group, and between them there is no room for another row.

7 figures The group of benzene, against how much error is forgiven. Four distortions of benzene, none larger than six hundredths of an ångström, and the point group the search reports for each at nine tolerances. The exact structure is D6h at every one of them, so the staircases below belong to the distortions rather than to the search. The cell marked in warning colour is a step at which the order of the group named at the tighter tolerance does not divide the order of the one named at the looser: C6h of order 12, so the sequence is not a chain of subgroups.

The tolerance is a decision

A measured structure is never exactly symmetric, so assigning it a point group means deciding how much error to forgive. Sweep that decision from a thousandth of an ångström to a third of one and benzene, bent by a hundredth, is assigned five different groups — and at one step the group named is not even a supergroup of the one named before it.

6 figures Two answers from one projector: E1g. The E1g projection operator of benzene, applied to the pz function on one atom and then to the one on its neighbour. Both results belong to the same two-dimensional representation and span the same subspace; neither is more correct than the other; and they are different pictures, overlapping by 0.500. The circle areas are the coefficients and the two colours are their signs.

The projector is unique, the basis is not

A reduction says how many times each representation appears. It cannot say which combinations of orbitals they are — that takes a projection operator, and for a degenerate representation the operator returns a different pair of orbitals depending on which function it is handed first. Both pairs are correct, they span the same space, and every energy computed from either is identical.

6 figures Three counts for each molecule, and the barrier between them. Four molecules, with the number of operations available to each: the point group, which is what a rigid rotation or reflection can do; the feasible group, which adds whatever a crossable barrier makes available; and the whole permutation-inversion group, which is every rearrangement of identical nuclei whether a molecule can reach it or not. The middle column is the one that describes a real spectrum, and it equals the first only when nothing moves.

The group of a molecule that will not hold still

A point group is a group of rotations of space, and it presupposes a structure for them to act on. Ethane has no one structure — its methyl groups turn billions of times a second — and the group that describes its spectrum has thirty-six elements where the point group has twelve, out of two thousand eight hundred and eighty conceivable.

7 figures Every set of point groups here that shares a table. Found by comparing character matrices rather than named: five sets among the tabulated groups, at orders 2, 4, 6, 8, 20. The smallest is the most startling — a mirror plane, a centre of inversion and a twofold axis all have the two-row table with entries 1, 1 and 1, −1, and one of the three describes a chiral molecule. The largest is the eclipsed and staggered conformers of ferrocene, which are one molecule at two temperatures.

One table, three groups

A mirror plane, a centre of inversion and a twofold axis have the same character table — two rows, entries 1, 1 and 1, −1 — and one of the three describes a chiral molecule. A character table is a property of an abstract group; a point group is that plus an action on space, and the action is what a molecule has.

8 figures The characters of C6, and the levels they are. The 6 representations of the ring's rotation group, drawn as points on the unit circle at 2πk/6. Each level is twice the horizontal coordinate: a representation and its complex conjugate have the same real part, so they are degenerate, and the levels pair up automatically. Only k = 0 — and k = n/2 when n is even — lands on the real axis, so one level is unpaired at the bottom and the shell closes at 4n + 2. Nothing here has been diagonalised.

The ring's levels are its group's characters

The Hückel energies of a cyclic system are twice the real part of the characters of its own rotation group, so the level pattern — one level, then pairs, then one more if the ring is even — is a group theorem rather than a calculation. Hückel's 4n+2 rule is a statement about the representations of a cyclic group and about nothing else.

8 figures Two moments of the n = 2 shell, and only one of them agrees. ⟨1/r⟩ and ⟨1/r²⟩ for each orbital of the n = 2 shell of hydrogen, computed from the radial functions and checked against their closed forms. The first is the same number for every member — which is why they share an energy — and the second differs by a factor of 3 across the shell.

The degeneracy no group predicts

How many orbitals can share an energy is decided by a group before any energy is computed, and the rotation group of a central potential permits one, three, five and seven. Hydrogen's second shell has four orbitals at one energy and its third has nine. The extra degeneracy is not an accident of the arithmetic — it is the signature of a symmetry that has not been named, and it survives only for a potential that goes exactly as one over r.

5 figures A field splits the n = 2 shell into whole numbers. The eigenvalues of z inside the shell, which are the shifts a uniform field produces to first order. There are three distinct ones and each is a whole number times (3/2)n, so the splitting is proportional to the field itself rather than to its square — which is what no other atom does.

The symmetry that is not a rotation

Hydrogen's n = 2 shell holds four states at one energy and its rotation group accounts for at most three. The operator that accounts for the fourth is built here out of computed integrals: three matrices whose commutators close into the rotations, whose product with the angular momentum vanishes, and whose Casimir comes out at exactly n² − 1.

8 figures How far benzene is from its own group, against how far it has been pushed. The measure — the distance to the nearest structure with the ideal group — against the size of the distortion, for two ways of distorting the same molecule. Each curve is smooth and quadratic; the markers on it are where the point-group search changes its verdict, which happens at one step and says nothing about the steps either side of it.

How much symmetry is left

A point group is a verdict and every real structure fails it. Two distortions of benzene that are two per cent apart in how far they sit from D6h need tolerances 1.88 times apart before the search will call either of them D6h — and one tolerance, applied to six molecules, admits amounts of asymmetry differing by a factor of fifty-six.

8 figures