Molecular Geometry & Bonding
what the shape decides
The collection

Every essay — page 29

Page 29 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.

What the group settles. For each molecule, the point group found from its coordinates and the two properties that follow from the group alone. Neither column required knowing anything about the bonds.

Symmetry forbids a dipole

Whether a molecule can have a dipole moment follows from its point group alone. The usual argument — adding up bond vectors — gets the right answer for easy cases by a route that does not generalise.

7 figures bromochlorofluoromethane — C1. The molecule with the point group found from its coordinates rather than looked up: every candidate operation was applied and kept when it permuted the atoms among themselves.

Chirality is a symmetry statement

A molecule is chiral when its group contains no improper operation at all. The four-different-groups rule is a useful special case that misses molecules with no stereocentre and wrongly condemns some that have several.

6 figures The Td character table. The irreducible representations of the molecule's point group. The class headings carry the number of operations found by generating the group from the coordinates, and those counts were produced before this table was opened.

Character tables and reduction

A molecule's point group is a list of matrices, not a label. Generating them, sorting them into classes, and dividing by the group order turns any set of orbitals into a statement about how many energies there can be.

7 figures Dipole selection rules in Td. For every pair of symmetry species, whether an electric dipole transition between them is allowed and along which polarisation. An entry is allowed exactly when the triple product of representations contains the totally symmetric one.

Selection rules are one theorem

An integral over all space vanishes unless the integrand is totally symmetric. Every selection rule in spectroscopy is that sentence with a different integrand — and the rule of mutual exclusion falls out rather than being remembered.

7 figures phosphorus pentafluoride — D3h. The molecule with the point group found from its coordinates rather than looked up: every candidate operation was applied and kept when it permuted the atoms among themselves.

Site symmetry, and what it constrains

A molecule's group is not the only group in the problem. Each atom sits at a position with a symmetry of its own, and what that local group permits decides how many kinds of environment a structure really has.

6 figures The Td character table. The irreducible representations of the molecule's point group. The class headings carry the number of operations found by generating the group from the coordinates, and those counts were produced before this table was opened.

Degeneracy is a group theorem

How many orbitals can share an energy is decided before any energy is computed. The dimensions of a group's irreducible representations are the only degeneracies it permits, and a molecule with no representation larger than one cannot have a degenerate level at all.

7 figures Oh descending to D4h. Every irreducible representation of Oh restricted to D4h and reduced there — A1g, A2g, Eg, T1g, T2g, A1u, A2u, Eu, T1u, T2u. A representation that arrives in more than one piece is a degeneracy the lower symmetry cannot hold, so a level carrying it must split when the molecule distorts.

Descent in symmetry

Lower a molecule's symmetry and its labels stop being available. Which of them survive, which split, and into what, is decided by restricting characters to the operations that are left — arithmetic, not a table to be looked up.

4 figures 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