Normal mode — where it appears
Named by 41 essays across 4 fields — each of them below, with the objects they name alongside it.
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.
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.
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.
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.
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.
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.
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.
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.
A dipole is not what an infrared spectrum sees
Carbon dioxide has no dipole moment at all and three of its four modes are infrared active. Methane has none and six of nine; boron trifluoride none and five of six. Water, which has the largest dipole of the five, has three modes and three bands — and its dipole predicted neither number.
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.
How far, and along which coordinate
A continuous symmetry measure returns one number: how far a structure is from a shape. Projecting the same displacement onto the twelve symmetry species of benzene's group turns it into a list that sums to 1.000000000000, says which coordinates the structure left along, and shows that the totally symmetric part — 3.4 per cent of this one — moves every atom by 0.0200 ångström and lowers the symmetry measure by nothing at all.
The mode that moves least radiates most
Boron trifluoride's strongest infrared band is the one in which the fluorines barely move: ninety per cent of the motion belongs to the boron, which is a fifth of the molecule's mass. The mode that moves the most mass is nine and a half times weaker. Across five molecules the rank correlation between band strength and how far the atoms actually go runs from +1 to −0.66, and every normal mode carries exactly the same weighted motion by construction.
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.
More coordinates than motions
Methane has ten internal coordinates and nine ways to vibrate, boron trifluoride seven and six, formaldehyde seven and six. The excess is not bookkeeping: it is a combination of coordinates that describes no displacement of any atom, and adding twenty-five units of force constant along it moves every frequency by five parts in a hundred million.
The one intensity symmetry does fix
Symmetry says which bands exist and declines to say how strong they are. It makes exactly one exception, and it is a ratio: every Raman band that is not totally symmetric is depolarised by exactly three quarters, whatever the molecule and whatever the model — and methane's symmetric stretch is polarised by exactly zero, because its group is cubic.
The coordinate it was already soft along
A distorted structure can be resolved into the symmetry species of its ideal group, and that resolution cannot say which coordinate a molecule fell down by itself and which one something outside pushed it along. The force field answers that, and the two halves agree on the case where they must — a unit distortion along a normal mode costs exactly that mode's frequency, to a part in a million, computed from masses and force constants by one side and from coordinates and characters by the other.
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 table that could not have mattered
Charges taken from one of four electronegativity tables invite a worry, because the tables disagree with each other. They do disagree — hydrogen cyanide's dipole runs over a factor of eleven between them — and for the molecules in question the worry could not have applied, because a molecule of two elements has charges that are one number times a fixed pattern and a rank correlation does not notice a rescaling.
Ten directions no frequency can see
A redundant force field has one obvious flat direction. There are ten: methane has fifty-five independent force constants and a ten-dimensional subspace of them that no spectrum can touch. The literature's repair is to project — and the metric a vibrational analysis is naturally written in cannot define the projection at all, because a redundancy is a null vector of it.
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.
A label that prices nothing
Pricing a distortion by its symmetry species works when the species appears once. It appears more than once for every one of seventeen molecules — 71.4 per cent of their vibrations on average belong to a species that is not unique — and for nine of the seventeen nothing forces it: their groups have room to spare and they repeat anyway.
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.
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.
A formula that predicts minus eleven vibrations
A molecule's count of totally symmetric vibrations is often explained as one per orbit of internal coordinates, less one per redundancy, with methane as the worked example. Computed for fifteen molecules it is right for seven and wrong for eight — and for sulfur hexafluoride, benzene and both ferrocenes it returns a negative number. Two independent corrections turn it into an identity.
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 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 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.
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 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.
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.
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.
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.
The two coordinates a square plane was missing
A square-planar centre has nine vibrations and its stretches and in-plane bends span seven of them: no angle in the plane changes to first order when the centre leaves it, or when the ligands pucker. The construction that closes the gap is two coordinates, one per pair of trans ligands, each the signed distance of the centre from its own pair's axis — and two coordinates raise the rank by exactly two. Why nothing measured this for three essays is the more useful half: both molecules with a square-planar centre have bonds longer than the census's default cutoff, so they were dropped before anything looked at them.
Half the distance from a coin
Every essay of this argument that priced a distortion by its symmetry species carried the same caveat: where a species appears more than once, the label cannot say which vibration the distortion goes into. With every coordinate set now complete or incomplete for a stated reason, the caveat can be measured against an actual projection on the five molecules that have force fields. The label carries the answer from 0.500 to 0.761 — about half the distance from a coin to a decision — settling six of twenty-four distortions and leaving ten essentially undecided, the worst of them ammonia's bond angles at 0.516.
The shape was the group and the size was not
A depolarisation probe swept round a circle of distortions gives a reading with a sixty-degree period and a largest-over-smallest of about two, and the residual left in the sum over bands scaled as the amplitude to the 1.248. Run the identical probe on ammonia — the same three stretches, the same three-fold axis, the mirror plane gone — and the period and the ratio come back within three per cent, while the departure is a hundred and sixty-nine times larger, the residual twenty-three times, and the exponent is 0.816. The angular shape is the group's. Nothing else is, and the pyramid's residual is not even a power law.
The third mass that cannot help
Boron trifluoride's E′ block has three force constants and two frequencies, so its exact fits form a closed curve, and a boron-10 substitution cuts that curve to two fields. The obvious remedy is a third isotopologue, and it was named as the first data that could pick one of the two. It is not, if the third substitution keeps the molecule's symmetry: every such isotopologue gives the same spectrum under both fields to a millionth of a wavenumber, because its sum rule is one plane written in two reciprocal masses and its product rule is Teller and Redlich's. Only a substitution that breaks the symmetry adds an equation — and it separates the two fields by half a wavenumber.
Three ways to stretch a bond
A symmetry label was measured at half the distance from a coin to a decision about which vibration a distortion goes into. The distortion was a B-matrix row, which moves a central atom as far as a ligand — and in the metric where the vibrations are orthogonal, three quarters of water's bond-angle distortion turns out to be the molecule moving rather than deforming. Two displacements that realise the same sentence and carry no rigid motion at all double what the label decides, from eleven coordinates of twenty-nine to twenty-two, and move the worst case from ammonia to boron trifluoride. Between them sits an exact identity that found a rounding error in a stored molecule.
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.
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.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
Model limitForce constantVibrational modesIrreducible representationsValence force fieldDegeneracyHarmonic approximationInternal coordinatePoint groupConventionLeast-squaresMode composition