Field

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.

Normal modes are not bond stretches

Water has two stretching frequencies and two O–H bonds, and it is almost irresistible to pair them off. Computed, each mode is exactly half in one bond and half in the other, and neither frequency belongs to a bond at all.

H₂O and its D isotopologue. Every frequency of H₂O joined to the frequency the same force field gives when every H is replaced by D, with the ratio on each join. The force constants were not refitted and could not be: they do not depend on mass. The product of all the ratios is fixed by the masses and the moments of inertia alone, and is checked against that identity while this figure is drawn.

The isotope shift is arithmetic

Replace hydrogen with deuterium and every frequency drops. The usual rule says by a factor of the square root of two — and of water's three modes, not one of them does that. What is exact is a different identity, and the force constants cancel out of it.

Group frequencies, and where they stop

A carbonyl band sits near 1,700 wavenumbers in every ketone anybody looks at, and that regularity is real. Computed for a set of small molecules, four of twenty-one distinct frequencies belong to a single internal coordinate — and the four are exactly the coordinates symmetry leaves alone.

6 molecules, counted. How many vibrations each molecule has, how many symmetry species they fall into, and how many frequencies are infrared active, Raman active, both, or neither. Every column after the first counts frequencies rather than modes, because a degenerate pair is one line in a spectrum. The molecules with a centre of inversion are the ones with nothing in the both column — mutual exclusion as a computed count. Nothing here uses a force constant.

How many frequencies, not how many modes

Benzene has thirty vibrations. It has twenty distinct frequencies, and of those, eleven can be seen — four in the infrared and seven in the Raman, with no band in common. The other nine are invisible to both experiments, exactly.

4 molecules, counted. How many vibrations each molecule has, how many symmetry species they fall into, and how many frequencies are infrared active, Raman active, both, or neither. Every column after the first counts frequencies rather than modes, because a degenerate pair is one line in a spectrum. The molecules with a centre of inversion are the ones with nothing in the both column — mutual exclusion as a computed count. Nothing here uses a force constant.

Two structures, two spectra

A linear XY₂ gives two infrared bands, one Raman band and no band in common. A bent XY₂ gives three of each and three in common. Counting settles the shape, without a force constant, an assignment or a single measured frequency.

benzene: 30 vibrations. The vibrational representation, obtained by subtracting the translations and rotations from the full set of Cartesian displacements, with the infrared and Raman activity of each species read off the same character table.

What an absence proves

A band that symmetry forbids is not weak. Its intensity is zero, exactly, by a theorem — while a band that is merely too faint to see is absent for reasons no theorem covers. The two look identical in a spectrum and support completely different conclusions.

phosphorus pentafluoride: 3 environments. The atoms of phosphorus pentafluoride sorted into orbits under its own symmetry group — two atoms are in the same orbit when an operation of the group carries one onto the other, which makes them indistinguishable by any measurement. There are 2 of F, 1 of P, and a spectrum that resolves environments counts those rather than atoms.

A spectrum counts environments, not atoms

Phosphorus pentafluoride has five fluorines in two inequivalent sets, so its magnetic resonance spectrum should show two signals. It shows one — and the reason is not a symmetry the molecule has but a motion faster than the measurement.

carbonyl sulfide: a linear. The principal axes of carbonyl sulfide drawn at its centre of mass, with the moment of inertia about each. Which kind of top this makes it is decided by comparing three numbers, and the point group forces the same answer independently: an axis of order infinity means a linear molecule, with one moment at zero forbids an asymmetric top.

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 structure out of a spectrum. Two rotational constants and two bond lengths, three times over, from three pairs of carbonyl sulfide isotopologues. Above them, the same inversion run on moments computed from a known structure, which returns it to twelve figures. The measured pairs disagree with one another by 7.4 milliangstrom, which is the difference between a ground-state average and an equilibrium geometry.

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.

Dipole selection rules in Oh. 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.

Why a d–d band is weak

In a centrosymmetric complex the transition between the two halves of a split d shell is forbidden — exactly, by parity, with no small quantity anywhere. What makes it visible at all is that the molecule is never quite centrosymmetric, and the computation says which vibrations do the work.

methane, substituted 5 ways. CH₄, CH₃D, CH₂D₂, CHD₃, CD₄ — the same molecule with the same force constants, differing only in which nuclei are heavy. Every column is a consequence of which of the parent's operations survive the mass pattern: the surviving set is a subgroup, it is named from its own conjugacy classes, and the vibrations are reduced in it. The allowed and computed columns come from group theory and from the mass-weighted Hessian respectively, and they agree in every row.

A spectrum that changes when only a mass does

Methane has four distinct vibrational frequencies and two infrared bands. Replace two of its hydrogens with deuterium and it has nine and eight — with every force constant identical, every nucleus where it was, and the potential energy surface unchanged.

7 molecules, three moments each. The principal moments of inertia of water, sulfur dioxide, formaldehyde, boron trifluoride, ammonia, methane, hydrogen peroxide, in u Å², with the inertial defect and the asymmetry parameter beside them. The defect vanishes exactly for a planar structure and does not for any other, so three numbers computed from the coordinates decide planarity with no model anywhere in the argument. Every classification is checked against the one the molecule's point group forces.

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.

ammonia: 3 valence bands. The measured valence photoelectron bands of ammonia, each labelled with the symmetry species of the orbital it comes from, and beside them the species the valence basis spans — the central atom's s and p functions and one s on each ligand, reduced in the molecule's own group. A band carrying a species the reduction does not produce would stop this figure being drawn.

Fewer bands than electrons

Methane has eight valence electrons and two photoelectron bands. Ammonia has eight and three; water has eight and four. The count is not of electrons, not of bonds and not of orbitals — it is of the symmetry species the occupied orbitals fall into, and it falls as the symmetry rises.

The six motions that are not modes

Three N minus six is quoted in every textbook and the six are almost never computed. Diagonalising a mass-weighted Hessian gives six eigenvalues at arithmetic noise, and projecting their vectors onto the three translations and three rotations written down from the geometry alone accounts for every one of them.

Mutual exclusion across every group here. The 20 groups with tables here, each with its highest rotation order, whether it has a centre of inversion, which of its representations carry a coordinate, which carry a product of coordinates, and whether any carries both. Every group with a centre excludes, which is a theorem. 1 group without a centre excludes as well — D5h — so the rule does not run backwards, and the counterexample needs a fivefold axis.

Mutual exclusion does not prove a centre

A centrosymmetric molecule shows no band in both its infrared and its Raman spectrum. The rule is a theorem and its converse is read off as though it were part of it — but ferrocene in the gas phase has no centre of inversion and no coincidence either, and the reason is that a fivefold axis separates the coordinates from their products where a threefold or fourfold axis cannot.

A microwave constant predicted from an infrared one. For four diatomics: the rotational constant and the stretching frequency, both measured, and the centrifugal distortion constant predicted from them by 4B³/ω² — then the constant a microwave spectroscopist fits to the line positions. The prediction and the fit agree within a few per cent across three orders of magnitude in the quantity, and nothing connects the two measurements except the assumption that the bond stretching under rotation is the same bond that vibrates.

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.

When a mode becomes a bond stretch

Water's two stretching modes are each exactly half in one O–H bond and half in the other, which is why neither of them belongs to a bond. Change one hydrogen to deuterium and the same force field at the same geometry gives two modes that are 99.5 and 99.7 per cent in a single bond each. Nothing about the bonding changed; a mass did.

ammonia's rotational levels, sorted by K. The rigid rotational levels of ammonia up to J = 4, each drawn at its computed energy and grouped by J. Within a group the levels are pushed apart by the second rotational constant, so it is plainly there in the level pattern.

The constant a spectrum cannot see

A symmetric top has two rotational constants and its microwave spectrum reports one of them. Not badly, not with difficulty: ammonia's A of 6.3406 wavenumbers appears in none of its lines at any J and any K, because the term it belongs to cancels exactly out of every transition. The molecule turns about that axis, the energy is real, and the measurement is blind to it.

How much of a band is a bond stretch

Boron trifluoride's 1454 cm⁻¹ band is 36.7 per cent B–F stretch, or 49.9, or 97.9, depending on which of four standard ways of sharing a mode out among internal coordinates is used. All four are defensible, all four sum to one, and the spread between them is sixty-one percentage points on one band of one molecule.

water: every level up to J = 4, from a matrix. The rotational levels of water at κ = -0.4322, each J diagonalised in the symmetric-top basis. A symmetric top would show one level per K with everything above K = 0 doubly degenerate; here every degeneracy is split, and the size of each splitting is what the third constant is measured from.

The top that reports all three

A symmetric top hides one of its two rotational constants in every line of its spectrum. Break the symmetry and the hiding stops: for water, twenty-three of the twenty-five levels up to J = 4 move when A is changed, and the two that do not are the ground state and the one at B + C. There is no formula for any of them.

Three bands of one spectrum, and the bond length behind each. Nitrogen's three photoelectron bands, drawn as the vibrational intensity distributions computed from the measured bond lengths and vibrational constants of the three states of the ion. Each band's lines add to one. The middle band is spread over five lines because the electron removed came out of a strongly bonding orbital and the bond lengthened by 77.22 thousandths of an ångström; the outer two keep 92 and 88 per cent of their strength in a single line.

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.

Three parameters, two numbers, and a curve of answers. seven structures of formaldehyde, every one of which reproduces the measured rotational constants A and B exactly. The C=O length runs from 1 to 1.3 ångström, the C–H length from 1.56 down to 0.95, and the HCH angle from 74.58 to 164.47 degrees. The third constant is not a third number: for a planar molecule it is fixed by the other two, and it comes out at 1.14 for every member.

Three numbers is not a structure

Formaldehyde's rotational spectrum gives three constants, of which a planar molecule's are only two independent numbers, and its structure has three parameters. Seven structures are computed here that reproduce A and B to the last digit the solver carries: the C=O length runs from 1.000 to 1.300 ångström, the C–H length from 1.557 down to 0.952, and the HCH angle from 74.6 degrees to 164.5.

A photoelectron band is a filter, and the group chooses the filter. The Huang–Rhys factor of every vibration of five molecules under a change of geometry that lengthens every bond alike — which is what removing an electron from a non-degenerate orbital does. On a logarithmic scale spanning sixteen decades, eight modes carry the whole of it and the rest sit on the floor at arithmetic noise. Which ones is decided by the point group: only a totally symmetric vibration can appear, whatever the size of the change.

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.

Every sign, lost. Formaldehyde in its own principal axes. Open circles are the atoms where they are; filled ones are where Kraitchman's equations put them, from the change in the three moments when each atom in turn is made heavier. The two agree to 7.6e-8 ångström — the equations are an identity for a rigid structure — but they return the square of each coordinate, so the two hydrogens at b = ±0.9348 both come back at +0.9348 and land on the same point.

The coordinate an isotope reports

Kraitchman's equations return an atom's position from the change in the moments when that atom alone is made heavier, and for a rigid structure they are an identity — formaldehyde's four atoms come back to a part in ten million. What they return is the square of each coordinate, so both hydrogens at b = ±0.9348 come back at +0.9348; every out-of-plane coordinate comes back imaginary at a moment error of one part in a hundred thousand; and the famous error cancellation, measured at a factor of thirteen, still leaves the answer two and a half times worse than a direct fit.

Three lines, then a hundred. The exact removal spectrum of a 6-site Hubbard ring at half filling: every final state of the ion, at the energy it costs to reach and with the intensity the matrix element gives it. With no repulsion there are 3 lines and they are the occupied orbital energies. At U = 8 there are 100, on a molecule with 6 orbitals — so the spectrum cannot be read as a list of orbital energies, because there are more bands in it than there are orbitals to name.

More bands than there are orbitals

A photoelectron spectrum is read as a list of orbital energies, one band per occupied orbital. Computed exactly for a six-orbital ring, it has three bands with no repulsion and a hundred with eight — and by then fifty-three per cent of the intensity is in lines that no orbital corresponds to. The total intensity is three at every repulsion, exactly, because that is a sum rule and not a fit.

The test that works until it does not. How many times stronger the weakest fundamental is than the strongest satellite, against the repulsion, on a half-filled ring of six. It starts at 23.8 and falls to 1.15 — a spectrum whose tallest satellite is as tall as its shortest band. The marked repulsion is where the other test fails as well: satellites start appearing inside the range the fundamentals span, so neither height nor position sorts the spectrum.

A hundred lines and no way to sort them

A spectrum with a hundred lines has six fundamentals in it somewhere. Sorting by height works until the tallest satellite is as tall as the shortest band, and sorting by position works until satellites start arriving between the bands — and on a ring of six both stop working at the same repulsion.

A ratio that measures a distortion, and squares it first. How far ammonia's depolarised bands come off three quarters against how far one of its bonds has been stretched. Undistorted the departure is 3.1e-8, which is the rounding in the stored coordinates rather than a physical effect; at 0.1 Å it is 0.04239, and the slope is 1.999 — the departure goes as the square of the distortion, so a ratio measured to three decimals fixes a length to one and a half.

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.

The factor of thirteen was generous. An invented error model against the one computed from a force field. Its invented mismatch of five per cent made the correlation between the parent and the substituted species worth a factor of 12; the computed mismatch of 35 per cent makes it worth 2.5. The substitution structure went from being 2.5 times worse than a direct fit to 26, and at the computed size of the correction its worst coordinate is out by 10.7 per cent.

The correction that was invented

A standard error model puts the zero-point error in a rotational constant at a few tenths of a per cent, shared between the three moments by invented weights, with the parent and its deuterated form differing by five. Computed from a force field it is 1.88 per cent, one of the three shares is negative, and the mismatch is 35 — so the cancellation the substitution method rests on is worth a factor of 2.5 and not thirteen.

The boundary belongs to the gap

A satellite stops being tellable from a fundamental somewhere, and it can be located on one ring at one filling. Four systems put it at repulsions of 2, 4 and 8 — ordering exactly with each one's own one-electron gap and not with its band width — and the fourth, whose gap is zero, has no boundary at all: its satellites are indistinguishable at every repulsion including none.

$How much of each of methane's force constants the spectrum fixes. Each of methane's 55 force constants, grouped by kind, with the squared length of its projection onto the determined subspace. One means the constant is fixed on its own. All 4 stretching constants are; not one bending constant is, at 0.5357 each. The fractions sum to 45, which is the rank of the map from constants to the Hessian, and that sum is an arithmetic check rather than a result.$

The forty-five that are fixed

Methane's fifty-five-dimensional space of force constants has ten directions no frequency can see, and the useful thing to report is the forty-five that are fixed. It is a table: every stretching constant is fixed on its own, no bending constant is, and projecting a fitted field onto the determined subspace takes four fits that span 0.88 mdyn per ångström down to four that span 0.0027.

One expression, four molecules, a factor of eleven. The computed zero-point correction to each molecule's moment of inertia against the expression A²·Σ(1/ν)/I, which has no fitted quantity in it. The dashed line is the mean dimensionless coefficient, 0.6343; the four points lie within 18 per cent of it, on corrections that span a factor of 11.4. The expression is evaluated from a moment of inertia and a list of wavenumbers, which is what a spectroscopist has before doing anything.

An expression for what was a warning

The zero-point correction to a moment of inertia comes out at 1.88 per cent where a few tenths had been assumed, and heavy molecules are safer. Written out, the correction is a mean curvature times the sum of reciprocal wavenumbers over the moment — one line, evaluated from things a spectroscopist has before starting. One coefficient serves four molecules whose corrections span a factor of eleven.

How fast the three quarters goes, along each coordinate. The coefficient of the square, for every non-symmetric mode of three molecules, against the mode's own frequency. A single measurement takes one of these — along a bond stretch — and reports the exponent rather than the size. The sizes span a factor of thirty within methane alone, and the coordinates a molecule is softest along are not systematically the most sensitive: ammonia's stiff pair is five times more sensitive than its soft one.

One number was one direction

The departure of a depolarisation ratio from three quarters goes as the square of a distortion, and the exponent was first measured along one bond stretch. Computed along every non-symmetric coordinate the exponent is always two and the coefficient is not: it spans a factor of thirty inside methane. And the ratio is not preferentially sensitive to the coordinates a molecule is soft along — in two molecules of three the stiffest coordinate is the most sensitive.

The contrast, out to a repulsion of sixteen thousand. The ratio of the weakest fundamental to the strongest satellite, against the on-site repulsion, for eight systems with two electrons each. Both axes logarithmic. The dashed line at two is the factor the intensity test needs. Every curve flattens above it and none of them crosses, at any repulsion — including a repulsion sixteen thousand times the hopping.

A contrast with a closed form

Below half filling the satellite test flattens instead of failing, and the value it flattens at could be above or below the factor of two the test needs. It is — on all eight systems, by between 1.25 and 3.7 times. And on a ring the limit is (1 + 2cos(π/n))², to six figures, on every ring tried.

The coefficient each principal axis needs. The dimensionless coefficient the molecule-averaged expression requires, evaluated on each principal axis separately rather than on the mean of the three. Across the twelve axes the positive ones span a factor of 2.33, against the 1.36 the molecule-averaged version spans — and one of them is negative, which no positive constant can be.

The axis that goes the other way

One dimensionless coefficient turned a zero-point correction into an expression a spectroscopist could evaluate, and the three principal axes were averaged over to get it. Split by axis it gets worse, not better — the coefficients span 2.3 where the molecule-averaged ones span 1.36 — and water's smallest moment does not grow at all. It shrinks.

The frame H₂O → HDO turns. H₂O → HDO: the parent's principal axes and the daughter's, drawn on the same nuclei. Two of the three turn by 21.12° and the third does not move at all, because it is the normal to a plane no substitution can tilt. A per-axis comparison between these two species is comparing moments about lines this far apart, which is a rotation of the frame rather than a correction to a number.

Two moments about two different lines

Asked axis by axis, the substitution method's near-cancellation gives numbers as large as 163 per cent. The arithmetic is the smaller half of the answer. A principal axis is an eigenvector of a tensor built from the masses, so one deuterium turns water's frame by 21.12° — and the two moments being compared are not moments about the same line.

The two terms in a vibrationally averaged rotational constant. A rotational constant averages 1/r², not r², so the moment it reports carries +2⟨Δr⟩/rₑ from the anharmonicity and −3⟨Δr²⟩/rₑ² from the harmonic spread. The two have opposite signs in every molecule here, and the anharmonic one — which is exactly zero in any symmetric well and therefore absent from every harmonic force field — is larger by a factor of 1.94 to 2.58.

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.

The contrast at three fillings, and the floor two of them reach. The intensity contrast on a ring of 6 against the on-site repulsion, at three fillings. At two electrons it settles on a number well above the factor of two the test needs. At half filling it falls through two and lands on exactly one from U = 64 upward — and every point where it reads exactly one is a point where the cut between fundamental and satellite falls between two lines of identical weight. Those are drawn hollow.

A ratio of exactly one is a tie

Does the intensity contrast fall below two at half filling? It does — it falls to exactly one. But one is the floor of a ratio between two ranked quantities, and it is reached here because the cut between fundamental and satellite lands between two lines of identical weight. The guard installed to catch that case tests the wrong degeneracy, and the guard installed to license the extrapolation cannot tell an exact answer from a divergent one.

Six distortions, and the one the ratio cannot see at all. How far a depolarised band of boron trifluoride comes off three quarters when the bonds are stretched in six different combinations, all at the same size. Five move it. The one that stretches every bond by the same amount moves it by nothing whatever — not a small amount, exactly none — because that distortion keeps every symmetry operation the molecule had, and the band sits at three quarters by its species.

The distortion the ratio cannot see

A depolarisation ratio can be made into a structural probe: break a molecule's symmetry and a band fixed at three quarters comes off it by an amount that depends on the distortion. The question is whether two distortions the ratio can see separately might cancel into one it cannot. They do — and the cancellation is exact, at any size, because the blind direction is the one that keeps the symmetry.

The same three fillings, on a ring and on a chain. The intensity contrast against the on-site repulsion at two, four and six electrons, for both geometries. Two of the ring's three curves flatten onto exactly one and stay there — the hollow marks, where the rank cut falls between two degenerate lines. The chain's corresponding curve approaches the same value from above without reaching it, because a chain of six has no exactly degenerate removal lines at any repulsion at all.

The number the tie got right

On a ring of six the contrast at half filling comes out exactly one, and the one is an artefact — the rank cut had landed between two lines of identical weight, so the ratio was a quantity divided by itself. The chain of six has no such pair anywhere, at any repulsion, at any filling. Its contrast at half filling converges to one anyway.

The reading is a function of direction, with a sixty-degree period. The departure of a depolarised band from three quarters, for distortions of equal magnitude pointing all the way round the plane of stretches that sum to zero. It is not one number: it runs from 8.177e-6 to 1.617e-5, a factor of 1.978. The minima are at one bond against another and the maxima at two bonds against one, and the pattern repeats every sixty degrees.

One number was a direction too

The depolarisation probe is exactly blind to the totally symmetric direction, which suggests it reads the distortion's component outside that species. What is left is a plane, and over a circle in it the reading varies by a factor of 1.98 — smallest for one bond against another, largest for two bonds against one, repeating every sixty degrees.

The sum is flat and the maximum is not. Both readings round the circle, each divided by its own average so the two can be drawn together. The maximum varies by 98 per cent; the sum over all four depolarised bands varies by 0.038. A sum cannot be changed by reordering, so if each band's departure is an isotropic quadratic form the sum must be constant — and to four parts in ten thousand it is.

The sum was flat all along

The depolarisation reading varies by a factor of two round a circle in the non-symmetric plane, and the anisotropy may belong to taking a maximum over four bands rather than to the physics. Summing the four instead gives a reading constant to four parts in ten thousand, against a maximum that varies by ninety-eight per cent.

Reversing the distortion changes the answer, which settles the order. How much the sum changes when the distortion is reversed — the same direction taken backwards — at each angle in the plane. Along a basis direction it changes by two parts in ten million, which is nothing. Halfway between them it changes by 7.36e-4, which is the same size as the isotropy residual itself at 5.24e-4. A response with only even powers of the amplitude cannot do that, so the leading anisotropic term is the cubic one and not the quartic.

The suspect that did not fit

The sum over depolarised bands is flat to four parts in ten thousand rather than exactly, and the quartic term is the obvious suspect. Two tests say otherwise. The residual scales as amplitude to the 1.248, which is neither candidate — and reversing a distortion changes the sum by as much as the residual is, which only an odd power can do. The leading term is the cubic, and the suspect was wrong by one order.

$A Morse curve's αₑ falls 4 to 15 per cent short of the measurement. The vibration–rotation constant αₑ of each diatomic, averaged over the states of a Morse curve built from its measured ωₑ, ωₑxₑ and rₑ, as a fraction of the tabulated value. H³⁵Cl: 0.27747 against 0.3072 cm⁻¹, ×0.903; D³⁵Cl: 0.10240 against 0.1133 cm⁻¹, ×0.904; ¹²C¹⁶O: 0.01674 against 0.0175 cm⁻¹, ×0.957; H¹⁹F: 0.68164 against 0.798 cm⁻¹, ×0.854. Every one is short, and HCl and DCl — one potential with two masses on it — are short by the same fraction.$

The cubic a Morse curve guesses

The two terms in a vibrationally averaged rotational constant were computed on Morse curves built from measured constants, and the vibration–rotation constant αₑ is the measurement that tests them. In all four molecules the Morse curve's αₑ is short, by four to fifteen per cent. The averaging is not the error — it matches the closed form to four parts in ten thousand. The curve's cubic is, and the measurement asks for more of exactly the term a harmonic field cannot produce.

The second molecule with a blind spot

Methane's fifty-five force constants have ten directions no spectrum can touch, and the forty-five that are fixed were worth a table. Boron trifluoride is the only other molecule here with a redundant coordinate set, and it is not the same case: eight kinds of constant rather than five, a quarter of its field invisible rather than a fifth, and an out-of-plane coordinate the redundancy cannot reach.

One isotopologue converges and two run away. Methane's force field refitted with the bend–bend constant restored, to the light molecule alone and to both isotopologues. The first converges to an ordinary field with a C–H constant of 5.42. The second walks along the direction no frequency can see until two constants reach minus ninety-seven thousand, equal to four figures and opposite in effect. Adding data made it worse.

Adding data made it worse

A projection leaves a residual spread across four starting points, and an earlier essay attributed it to the eigenvalue problem rather than to the coordinate set — an attribution rather than a measurement. Refitting with two isotopologues was the proposed test. Run, it produces a runaway to minus ninety-seven thousand that one isotopologue alone does not, and the reason is that the direction it walks along is flat at every mass.

Every exact fit lies on one closed curve. The one-parameter family of force fields that reproduce all six of boron trifluoride's frequencies exactly, drawn in the B–F stretch constant and the stretch–bend coupling. It is a closed curve: the stretch constant runs from 3.94 to 12.11 millidyne per ångström and the coupling from -0.36 to 6.71. The four fits from four starting points sit together in one small patch of it.

The residual was a loop

Projecting away boron trifluoride's flat direction left four fits disagreeing by a tenth in the B–F stretch constant, and the disagreement was put down to the search or to the arithmetic. It is neither. Two frequencies cannot fix three constants in one symmetry block, so the exact fits form a closed curve along which the stretch constant runs from 3.9 to 12.1, and the four fits are four points on a short arc of it.

$The measured cubic and quartic overshoot what Morse fell short of. Each molecule's αₑ as a fraction of the measured value, from its Morse curve and from a quartic potential built with the cubic and quartic coefficients the measured αₑ and ωₑxₑ imply. Every Morse curve is short, by 4 to 15 per cent. Every quartic is over, by 1 to 5 per cent: the repair moves αₑ past the measurement rather than onto it.$

Two coefficients are not a potential

A Morse curve built from measured constants gets the vibration–rotation constant αₑ short by four to fifteen per cent, and the measured αₑ and anharmonicity imply the cubic and quartic a real potential should have. Built with exactly those two coefficients and solved, the potential overshoots instead. The reason is that the Morse curve's own series, cut after its quartic, moves αₑ by a sixth to a half of the error being repaired — so a quartic cannot tell whether the shortfall was the cubic.

Each half has an integer exponent and the mixture does not. On logarithmic axes against the amplitude: the combined residual, the even half's anisotropy and the odd half relative to the mean. The even half is a straight line of slope 2.005. The odd half has slope 0.981 below 0.04 and then bends over. The combined residual, fitted as one power, gives 1.248 — an average of two integers weighted by where the sweep happens to sit, with the halves crossing at 0.084.

Two integers made one exponent

A sum over boron trifluoride's depolarised bands is nearly isotropic, and its residual scaled as the amplitude to the 1.248 — no integer, and a two-term fit left a pattern it could not remove. Averaging each distortion with its reverse splits the residual exactly into an even half that scales as the square, to 2.005, and an odd half that scales as the first power until a fifth-order term turns it over. And the sum was a stand-in: what an unresolved pair of bands would actually show is twenty times less flat.

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