Molecular Geometry & Bonding
what the shape decides
The collection

Every essay — page 30

Page 30 of 36, 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.

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 One distortion, resolved into the species of the group it left. An arbitrary displacement of benzene, projected onto each symmetry species of D6h. The weights add to 1, which is the check that the projectors resolve the whole of it: the largest is E2g at 35.0 per cent, and the totally symmetric part — 3.4 per cent here — is the part that changes every distance and no symmetry at all.

How far, and along which coordinate

A continuous symmetry measure returns one number: how far a structure is from a shape. Projecting the same displacement onto the twelve symmetry species of benzene's group turns it into a list that sums to 1.000000000000, says which coordinates the structure left along, and shows that the totally symmetric part — 3.4 per cent of this one — moves every atom by 0.0200 ångström and lowers the symmetry measure by nothing at all.

8 figures Two kinds of answer to a flux, and only one of them is a curve. The π binding of three rings against the magnetic flux through them, in units of beta and of the flux quantum, measured from each ring's own value at zero flux. Beta is negative, so a binding that FALLS is an energy that rises: benzene's does, which is what a diamagnetic ring current is. Cyclobutadiene's rises in both directions from a corner — its energy has no second derivative at zero field at all, and the two one-sided slopes differ by 12.57.

Two rules that share no arithmetic

Hückel's rule is a statement about a gap. Put a magnetic flux through the same rings instead and ask which of them push the field out, and the answer is the same set — eighty cases, no exceptions — although the second calculation counts nothing and has no shells in it. And the rings the rule excludes turn out to have no magnetic susceptibility at all: their energy has a corner at zero field.

9 figures How many coordinates, and how many motions. For each of six molecules, one square per internal coordinate the valence set carries — bond stretches, angle bends and an out-of-plane wag where there is one — with a rule drawn at the number of vibrational degrees of freedom. three of them have more coordinates than motions, and which ones is decided by shape rather than by size: ammonia's three angles are independent and boron trifluoride's are not, and the only difference is that one is flat.

More coordinates than motions

Methane has ten internal coordinates and nine ways to vibrate, boron trifluoride seven and six, formaldehyde seven and six. The excess is not bookkeeping: it is a combination of coordinates that describes no displacement of any atom, and adding twenty-five units of force constant along it moves every frequency by five parts in a hundred million.

8 figures Three quarters, exactly, for every mode that is not totally symmetric. The depolarisation ratio of every Raman-active mode of five molecules, computed from a bond-polarisability model. 18 of the 26 sit on three quarters to better than a millionth, and they are exactly the modes that are not totally symmetric: the mean polarisability derivative is a trace, a trace is invariant, and an invariant has no derivative along any other species. The polarised ones below the line are the totally symmetric modes, and where they sit is a property of the model rather than of the group.

The one intensity symmetry does fix

Symmetry says which bands exist and declines to say how strong they are. It makes exactly one exception, and it is a ratio: every Raman band that is not totally symmetric is depolarised by exactly three quarters, whatever the molecule and whatever the model — and methane's symmetric stretch is polarised by exactly zero, because its group is cubic.

8 figures The shallowest slope, and the steepest. For three molecules, the softest and stiffest vibrations, the species each belongs to, and what a unit distortion along each costs. The cost is not taken from the frequency: it is computed by resolving the distortion onto the normal coordinates in the mass-weighted metric and adding up ω²q². That it comes back as the frequency is the identity the whole comparison rests on — the species decomposition uses only the coordinates and the group, and the cost uses only the masses and the force constants, and the two have to agree on this case before they can be asked to disagree on any other.

The coordinate it was already soft along

A distorted structure can be resolved into the symmetry species of its ideal group, and that resolution cannot say which coordinate a molecule fell down by itself and which one something outside pushed it along. The force field answers that, and the two halves agree on the case where they must — a unit distortion along a normal mode costs exactly that mode's frequency, to a part in a million, computed from masses and force constants by one side and from coordinates and characters by the other.

5 figures