Molecular Geometry & Bonding
what the shape decides
The collection

Every essay — page 38

Page 38 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.

Ammonia's E verdicts across the fields its spectrum allows. The concentration of each distortion on one occurrence of ammonia's E species — the share of its E projection on the favoured vibration — for every field in the walk, with the stretch–bend constant held and the rest refitted to both isotopologues. The band is where the fit cannot tell the fields apart and the dotted line is the published fit. Across the band the angle under the B row reads 0.502 to 0.579, the angle at least cost 0.840 to 0.923, the bond at least cost 0.890 to 0.957.

The worst case was the loosest constant

A symmetry label's verdict on a distortion is read off mode shapes, and mode shapes are set by a fitted force field. Walked along the direction its own frequencies see least, ammonia's field moves the verdict on its bond angle from 0.840 to 0.923 without the fit getting measurably worse — and boron trifluoride, the census's worst case at 0.573, has a whole curve of fields that reproduce its spectrum exactly, along which the same verdict runs from a coin to 0.862.

7 figures With its measured Ritz series potassium's p level stops being the more polarisable at the thirty-fourth shell. Potassium's α(np₀)/α(ns), every s, p and d level summed, against the principal quantum number, with three readings of the defects. The four-decimal asymptotic defects put the turn below one at n = 36.93; the measured δ₀ alone at 36.99; the measured Ritz series, whose corrections at those shells are a ten-thousandth to a thousandth, at 33.72. The ratio falls by about 0.0074 a shell, which is why corrections that small move it three shells.

Potassium's turn was a fine-structure question

Summed over every level, potassium's p level stops being more polarisable than its s level somewhere past the thirtieth shell, and a thousandth in any quantum defect moved the turn by shells. With the Ritz series measured on ultracold ³⁹K in 2019 the inputs are fixed to a millionth, and the turn lands at the thirty-fourth shell, not the thirty-seventh. But the two fine-structure components of the p series differ by three thousandths: with either one in place of their average the turn is at the twenty-third or the forty-fifth. A sign decided by the spinless sum is decided by an interval a hundred and fiftieth of the s–p gap.

6 figures Ammonia's two Coriolis constants along the family of fields the spectrum cannot tell apart. ζ₃ and ζ₄ of NH₃ on every member of ammonia's family of force fields, each refitted to the frequencies of NH₃ and ND₃ with its stretch–bend constant held, against that constant. The shaded band is where the fit is no worse than its own noise, -1.000 to -0.634. Across it ζ₄ runs from -0.807 to -0.641 and ζ₃ from 0.418 to 0.584, while their sum stays at -0.222 on every member.

Two Coriolis constants are one number

Ammonia's frequencies leave its force field free along a band, and across that band a symmetry label's verdict on the bond angle changes. The measurement proposed to settle it was the Coriolis constant of each E vibration, which reads mode shapes directly — and it would settle it, sixteen times more sharply than the frequencies do. But the two constants are not two measurements. Their sum is fixed by the geometry and the masses, ζ₃ + ζ₄ = I∥/2I⊥ − 1 (the moments of inertia about the axis and across it), on every field there is, so the spectrum's Coriolis constants carry exactly one number about the field between them.

8 figures No state of potassium turns where the turn was reported, and only one turns at all. Potassium's polarisability ratio to ns₁/₂ against the principal quantum number, for the three p states a weak field has: np₁/₂, np₃/₂ with |m| = ½, and np₃/₂ with |m| = 3⁄2. The dashed line is the spinless p state with no angular momentum about the field, which passes one at n = 33.7. Of the real states np₁/₂ stays above one to the hundredth shell, |m| = 3⁄2 climbs away from it, and |m| = ½ passes one at n = 77.2.

The state that turned is not in the atom

Potassium's p level was found to stop being more polarisable than its s level at the thirty-fourth shell, with a warning that the fine structure might move that by eleven shells either way. Given its proper angular factors, no real state of potassium turns there at all. The level that turned has no angular momentum about the field, and a weak field has no such level: spin–orbit coupling blends it with the others, and the blend stays above one. One state crosses, at the seventy-eighth shell. The thirty-fourth keeps a meaning, as the shell where a field strong enough to undo the blending first produces a minimum.

7 figures

What is taught wrongly

The explanations that are confident, memorable and false — stated fairly and then tested against a calculation rather than an opinion.

methane — Td. 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.

Hybridisation does not explain

Methane's photoelectron spectrum has two bands, not one. Four equivalent sp³ bonding orbitals cannot produce that, and the resolution is that hybridisation was never a claim about what a measurement would find.

8 figures Orbitals at the 90 per cent contour. Several orbitals drawn at the same enclosed fraction and, unless stated otherwise, at the same scale — so the sizes on the page are the sizes. Each contour was solved for separately by integrating that orbital's own density. Contours drawn: 1s at 90% of its density, |ψ| = 3.94e-2; 2s at 90% of its density, |ψ| = 7.40e-3; 2pz at 90% of its density, |ψ| = 9.48e-3.

Orbitals are not where the electron is

A many-electron atom has no exact orbitals at all. The orbital picture is a basis for an approximation — an extremely good one — and treating it as a description of reality is the source of most of the confusion in this subject.

7 figures 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.

The dipole is not a sum of bonds

Adding bond dipoles as vectors gets the easy cases right and rests on a quantity with several incompatible definitions. The symmetry argument is exact, needs no electronegativities, and says when the answer must be zero.

6 figures Four scales, four orderings. Each column ranks the elements by one electronegativity scale, most electronegative at the top, with a line joining each element across the columns. Every crossing is a pair of elements that two scales order differently.

Electronegativity is not one quantity

Four scales, four definitions, four sets of units, and a defence — "they correlate well" — that answers a question nobody asked. Two scales can correlate at 0.99 and still put hydrogen on the wrong side of carbon.

5 figures Hückel levels of cyclobutadiene. 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.

Delocalisation is stabilising, and other things that are false in general

Spreading electrons over more centres is supposed to lower their energy. Cyclobutadiene is delocalised in exactly the same sense as benzene, gains exactly nothing by it, and the arithmetic says so before any experiment does.

7 figures Overlap against separation, Z = 3.25. How the overlap integral falls as two atoms are pulled apart, for several pairs of orbitals. Where a closed form exists it is drawn over the computed curve, so the integrator is checked rather than trusted.

A double bond is not two single bonds

Carbon's single bond is 348 kilojoules a mole and its double is 614, which is not twice anything. The two halves are different integrals over different orbitals with different distance dependence, and computing them shows why no arithmetic could have made them add.

6 figures 4s and 3d from K to Zn. The mean radius of the 4s and 3d orbitals across the elements K to Zn, at the nuclear charge each shell actually feels. The screening is Slater's, which is fitted; the radius that follows from it is the closed form for a hydrogen-like orbital, which is not.

The aufbau order is not a property of the atom

Iron's 3d orbital is more than four times smaller than its 4s and, by every one-electron estimate available, far lower in energy. The 4s fills first anyway, and it empties first too — which is not a paradox but a sign that the filling order was never a list of orbital energies.

5 figures Every force field the spectrum permits. 5 force fields for H₂O, each one refitted with rr:O:H,H held at the value on the axis, and every one of them reproducing all of the molecule's own frequencies. The heavy line is the error each makes on the isotopologue it was never fitted to. The spectrum cannot choose between these fields; the isotopologue can.

The force field is not in the spectrum

Water has three vibrational frequencies and its force field has four constants, so the spectrum cannot determine the field. A whole family of quite different fields reproduces all three frequencies exactly, and only a second isotopologue can tell them apart.

4 figures