Bond length — where it appears
Named by 31 essays across 7 fields — each of them below, with the objects they name alongside it.
Copper is never quite octahedral
A d⁹ ion in an octahedral field has three electrons in a doubly degenerate pair, which cannot be shared evenly. The energy it gains by distorting is linear in the distortion and the elastic cost is quadratic, so no stiffness holds the symmetric structure — and the four configurations with an even occupation gain exactly nothing.
The rotational spectrum is a moment of inertia
Every line in a microwave spectrum sits at a multiple of one number, and that number is a conversion constant divided by a sum of mass times distance squared. No bonding argument appears anywhere in it.
A bond length out of a spectrum
A linear triatomic has two bond lengths and one moment of inertia, so one measurement cannot determine it. Substituting an isotope gives a second measurement on the same structure, and two equations in two unknowns have a solution — which comes out differently depending on which isotope is used.
The frequency is not the bond strength
Sulfur dioxide's S–O force constant is larger than water's O–H constant, and its stretching bands sit at a third of the frequency. A vibrational frequency carries a mass as well as a force, and the two cannot be separated by looking at a spectrum.
The angle a ring cannot have
A closed ring of equal bonds has to turn through a full circle, so its bond angles cannot average more than 180°(n−2)/n. Three, four and five atoms are below the tetrahedral angle at every geometry whatever; six is above it, and reaches it only by leaving the plane.
The atom does not bring its own orbital
Build a one-electron diatomic from two hydrogen 1s functions and it comes out 25 per cent too long and 37 per cent too weakly bound. Let the molecule choose how large those functions are and the bond length is right to three figures, at an exponent of 1.238 — the orbital contracts by a quarter when the bond forms.
The moment that is the sum of the other two
Three numbers computed from the coordinates and the masses decide whether a molecule is flat. For water, sulfur dioxide, formaldehyde, benzene and every other planar structure here the largest moment of inertia is the sum of the other two exactly; for ammonia it misses by 0.76 and for methane by 3.18.
A contraction is a decision made once
Every published basis set freezes its primitive functions into fixed combinations, on an isolated atom, before any molecule is in sight. Freeing one coefficient recovers three quarters of what that costs — and freeing it at the other end of the basis recovers one per cent.
The sites are not the same size
Every arrangement in this collection puts its sites on one sphere, which is an assumption about bond lengths made silently. Give the repulsion model a bond length and it predicts that the long bond goes axial — the opposite of the rule the model is always cited for.
The double bond a ring cannot hold
A trans double bond needs a ring of nine carbons to sit flat, and the eight-ring will hold one twisted by 40.7 degrees — against 136 degrees measured in trans-cyclooctene. The same eight-ring threshold, applied by counting ring sizes, sorts nine bridgehead alkenes correctly with no bridgehead anywhere in the argument.
The rotor that stretches
A rigid rotor's lines are evenly spaced, and a real molecule's are not — it pulls itself apart as it spins. How much is not a fitting parameter: it follows from the stretching frequency by one relation, and the prediction agrees with the measured constant to a few per cent across four molecules spanning three orders of magnitude.
The atoms are not at the points
Every structure in this collection is a set of points, and every bond angle it argues about is a property of that set. Computed from the fitted force fields it already has, water's hydrogens are 0.094 Å from where they are drawn and its bond angle has a spread of 8.9 degrees — larger than the difference between 104.5 and the tetrahedral value that half the essays here are about. This is at no temperature at all.
The bond length that depends on the isotope
Hydrogen chloride and deuterium chloride have the same potential energy curve, and their measured equilibrium lengths agree to three hundredths of a milliångström. Their average bond lengths differ by 4.34 mÅ — a hundred times more — because a lighter atom explores more of a well that is not symmetric.
The width of a band is a bond length
Nitrogen's three photoelectron bands are one sharp line, a progression of five, and a line with a shoulder. Computed from the measured bond lengths of the three states of the ion, the intensities come out at 0.917, 0.263 and 0.880 in the first line of each — because removing a weakly bonding electron lengthens the bond by 18.7 thousandths of an ångström, a strongly bonding one by 77.2, and an antibonding one shortens it by 23.7.
A band is a filter on the modes
A photoelectron band's vibrational structure reports the frequencies of a few of the ion's vibrations and is silent about the rest, and which few is decided by the point group before any geometry is known. Under a change of shape that lengthens every bond alike, methane's totally symmetric stretch gets a Huang–Rhys factor of 0.905 and its other eight modes get between 10⁻²⁸ and 10⁻³⁵.
The splitting against something structural
The angular overlap model says a ligand field splitting is proportional to the square of one overlap integral and to no other power. Computing that integral from Slater functions at the measured bond lengths, for five chromium complexes whose splittings run from 13,600 to 26,700 wavenumbers, the ratio varies by thirty-one per cent with nothing fitted. And a power law on the donor's effective charge, at an exponent nobody predicted, does slightly better.
The frame that was allowed to relax
Four changes of frame were tested on nine quantities and none of them could move a bond order, because a bond order is a property of the eigenvectors and every change left the eigenvectors alone. Letting the geometry answer back does move them — butadiene's central bond falls from 0.4472 to 0.3676 — and it moves naphthalene's the other way, because its weakest bond is the one two rings share and relaxation strengthens it.
The basis the other atom lent
Two atoms in a molecule are described in each other's functions and the separated atoms are not, so the molecule is treated better than the pieces and the binding comes out too large. That is the basis set superposition error, it is removed by a standard correction, and for H₂⁺ in four Gaussians a centre it is six tenths of a microhartree against an incompleteness error of twelve millihartree — a factor of eighteen thousand the other way.
A correction computed at one length
The counterpoise correction is expensive, so it is evaluated once at a reference geometry and subtracted across a whole potential surface. A constant does not move a minimum — so a frozen correction returns the uncorrected bond length exactly, at every reference geometry and in every basis, and everything the correction does to a structure is the part that has just been thrown away.
A ratio that squares what it measures
A depolarised Raman band sits at exactly three quarters because symmetry says its mean polarisability derivative is zero. Distort the molecule and it comes off — by 4.5 × 10⁻⁴ for a hundredth of an ångström and 0.042 for a tenth, going as the square of the distortion, which makes a ratio measured to three decimals a length known to one and a half.
An anomaly that is not the first of a series
Naphthalene's shared bond behaves differently when the geometry is allowed to answer back, and the obvious reading is that two rings feeding one bond is what does it. Run it up the acenes and the effect falls at every step and changes sign: +0.02036 for two rings, +0.00678 for three, −0.00029 for four, −0.00603 for five. Two rings feeding a bond is not the beginning of anything.
A correction that is two functions
A symmetric pair's counterpoise correction splits exactly in half, at every separation, to the last digit — which is why one frozen number describes it. Give the two atoms different charges and the split runs from 0.03 per cent to 67, changing places at 6.29 bohr: the correction a single number was standing in for is two functions of different shapes.
Four tables and one molecule to disagree about
A molecule of two elements is provably safe from the choice of electronegativity table, and a series down a group should land where the tables do disagree. It does. All four agree that hydrogen iodide is the exception — and they disagree about what its dipole is by 1.245 debye, which is two and a half times the 0.448 that was measured.
The term a harmonic field cannot produce
The usual zero-point correction to a moment of inertia comes from a harmonic force field, which contains the mean square displacement and nothing else. A rotational constant does not average that. It averages one over r squared, whose leading correction is the mean displacement — zero in any symmetric well — and which enters with the opposite sign and about twice the size.
No length separates them
A bond list here is one distance cutoff with a clause about hydrogen, added when peroxide came back with five bonds instead of three. The clause repaired one molecule. Across the twenty-three molecules drawn here, the longest bond is 2.32 ångström and the shortest pair that is not a bond is 1.51 — so no cutoff can work at all, and five molecules currently come back with no bonds.
Oblate in the picture nobody draws
Every drawing of a σ bond shows a density stretched along the bond, and the position-space calculation agrees: the second moment along the axis is twice the one across it. In momentum the same orbital is flattened in the same direction, because the interference between the two centres cuts the distribution off at π over the bond length — and that cut-off is a zero a measurement could find.
The long bond goes to the crowded site
Given one bond longer than the others, the repulsion model puts it axial in a trigonal bipyramid — against the rule it is usually cited for. The reason is a crowding count, and at seven sites the count reverses: the pentagonal bipyramid's crowded site is equatorial, so a long bond goes equatorial and a short one axial. PF₅'s long bonds are axial and IF₇'s are equatorial. And at seven the site a bond avoids is not even a minimum.
The ring held flat by its long bonds
Given iodine heptafluoride's own composition — five long bonds and two short — the repulsion model puts the short pair on the axis, as the molecule does, and keeps the equatorial ring flat. But the flatness is the softest thing in the arrangement by a factor of a hundred. With equal bonds, a pucker of the ring costs nothing at all to second order under an inverse-square law and lowers the energy under any steeper one. What holds IF₇'s ring flat is the four per cent by which its ring bonds are longer, and that holds only under laws softer than about the 4.4th power.
The fluorines' range decides the ring
IF₇'s flat ring was held by its bond lengths under every repulsion softer than about 1/r^4.4, with the fluorines left out. Put them back as a second set of repelling points and under the same law they change nothing at all — they sit on the same rays, and the model cannot tell where along a ray a point sits. They matter only as a contact repulsion of a different shape. Then a steep power puckers the ring with a few thousandths of the energy. An exponential contact puckers it only if its range is under 0.346 ångström, and a longer one holds it flatter. Fluorine's own orbital gives a range near 0.16. And the ring that puckers does not settle: its envelope and its twist differ by a ten-thousandth of the pucker's depth, so the pucker runs round the ring.
The model lengthens the wrong three
Xenon hexafluoride's capped octahedron has its three longer bonds round the face the lone pair opens. Give the repulsion model three long bonds and three short and ask where it puts them: wherever the face is its lowest structure — at every lone-pair weight and every contact share mapped — it puts the long three on the opposite side. The lone pair's push on its neighbours and the fluorines' push on each other nearly cancel, and the lone pair wins only above a weight of 0.727, where the face has already given way to the edge.
The lone pair has no bond to stretch
Put every one of xenon hexafluoride's bonds on a spring and let the repulsion set the six lengths, and which three come out long is decided by something the model has never had to say: how far from the xenon the lone pair sits once the bonds have moved. Three equally defensible answers give three different verdicts — and none of them produces a split within a factor of twenty of the measured one without stretching every bond beyond recognition.
Named alongside it
The objects these essays reach for when they reach for this one.
Model limitApproximationConventionClosed formHarmonic approximationVSEPRExpectation valueMinimisationRepulsionDegeneracyMoment of inertiaOverlap integral