The collection

Every essay — page 33

Page 33 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 a spectrum settles

A spectrum is a list of positions, and how many there can be is decided by the shape before any of them is measured. Counting them, and reading a structure back out.

$Three molecules the family cannot reach, and one it reaches twice. Along a family of potentials that all hold the measured ωₑ and ωₑxₑ, the vibration–rotation constant αₑ as a fraction of the measured one, against the parameter λ that bends the Morse curve's range; λ = 0 is the Morse curve. Hydrogen chloride and deuterium chloride rise to 0.957 and 0.957 and fall again; hydrogen fluoride to 0.909. None reaches one. Carbon monoxide crosses one twice.$

Three constants, two potentials

A Morse curve falls short of the measured vibration–rotation constant, and a quartic built from the two measured coefficients overshoots it. A Morse curve whose range bends with the separation has one more shape parameter, and fitted to ωₑ, ωₑxₑ and αₑ together it should say whether 'the shortfall is the cubic' was ever a well-posed statement. For carbon monoxide it is: two members of the family fit all three constants, and both carry the leading-order cubic to five figures. For hydrogen chloride, deuterium chloride and hydrogen fluoride no member fits at all — and every member that gets close has stopped being a curve that dissociates.

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Ammonia's direction goes at a narrower width than boron trifluoride's. How much the probe's reading varies round the circle of distortions — largest over smallest — against the spectrum's line width, for the pyramid and the plane, both distorted by 0.02 Å. Both start near two. The plane's falls to half its excess at 3.48 cm⁻¹, set by its bending pair's 2.82 cm⁻¹ splitting; the pyramid's at 2.56 cm⁻¹, set by its stretching pair's 2.12 — although both the pyramid's pairs are split more than either of the plane's.

The pyramid trades its pairs

Ammonia's depolarisation probe was expected to survive a wider line than boron trifluoride's, because both of its degenerate pairs are split more by the same distortion. It survives a narrower one. On the plane the bending pair carries the reading; on the pyramid the stretching pair does, thirty times over, and its splitting is the smaller of ammonia's two — so the direction halves at 2.6 cm⁻¹ instead of 3.5. What the pyramid keeps is the size. The contrast a measurement has to resolve is 178 times the plane's, and at a ratio precision of one per cent ammonia's direction becomes readable from 0.046 Å of distortion, where boron trifluoride's is unreadable at 0.16.

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Carbon monoxide's fourth constant picks the fit that turns over. Along the Morse curves with a linearly bent range that hold carbon monoxide's measured ωₑ and ωₑxₑ, the ωₑyₑ read from the lowest four levels, against the bend λ. The two members that also have the measured αₑ are marked: the one at λ = −0.221 dissociates and predicts 0.0182 cm⁻¹; the one at λ = −0.453 turns over at 2.10 bond lengths and predicts 0.0110. The measured value, read from the published level expansion the same way, is 0.0110 — the second, to 0.0 per cent.

The fourth constant chose a curve, not a depth

Read from its lowest four levels the way every potential here is read, carbon monoxide's measured ωₑyₑ is 0.01101 cm⁻¹ — and of the two bent-range Morse curves that fit its first three constants, the one that predicts 0.01100 is the one that never dissociates. Let the range bend and stay bounded and every molecule, the hydrides included, is fitted to all four constants by a curve that does dissociate. The depths those curves carry are 6 to 31 per cent too deep, further from the measurement than the plain Morse curve's on all four, and hydrogen chloride and deuterium chloride — one potential — are fitted by two different ones.

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Around boron trifluoride's loop of exact fits, ζ₃ takes every value — each one twice. The Coriolis constant ζ₃ of ¹¹BF₃'s E′ stretch on every field of the closed loop that reproduces its frequencies exactly, against the B–F stretch constant. It runs over its whole possible range, −1 to +1. The level of the fitted field, 0.791, meets the loop at exactly two points — B–F constants 7.425 and 3.966 — which are the fitted field and the partner field a ¹⁰B spectrum cannot tell from it.

The Coriolis constant was an isotope shift

Boron trifluoride's E′ block has a closed loop of force fields that fit its spectrum exactly, a ¹⁰B spectrum cuts the loop to two, and no third isotopologue that keeps the symmetry can choose between them. The Coriolis constant depends on mode shapes rather than frequencies, so it looked like the measurement that could. Around the loop it takes every value from −1 to +1 — and under the two fields it is the same to nine decimals, because for a planar top it is exactly the boron's share of the motion, which is what a boron isotope shift measures.

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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.

ammonia — C3v. 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.

Point groups from coordinates

A molecule's symmetry is not a label to be looked up. It is decidable from the atom positions by searching for the operations that permute them, and the search either finds an operation or it does not.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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