Molecular Geometry & Bonding
what the shape decides
Concept

Valence force field — where it appears

A set of force constants keyed to bonds and angles rather than to atoms. Every constant in one is fitted to observed frequencies, so a frequency it reproduces is not evidence for the model that produced it.

Named by 20 essays across 3 fields — each of them below, with the objects they name alongside it.

H₂O: 3 distinct modes. The displacement of every atom in 3 normal modes of H₂O, drawn from the eigenvectors of the mass-weighted Hessian. Under each is the internal coordinate with the largest share of the motion and how large that share is; where no coordinate holds nine tenths, the mode is not a motion of one bond or one angle and is labelled so.

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.

spectra · Normal mode ¹¹BF₃: what each mode is made of. ¹¹BF₃. Each row is one distinct frequency and each column one internal coordinate; the bar is the share of the motion in that coordinate. A mode whose largest share reaches nine tenths is a motion of one bond or one angle and is named for it. 1 of 4 here are.

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.

spectra · Normal mode 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.

wrong · Normal mode 6 fitted force fields. Every valence force field fitted here, ordered by the size of its bond stretching constant, with the stretching frequencies of the molecule beside it. The two orders are not the same, which is the whole of what separates a force constant from a frequency. The last two columns say how many constants were fitted to how many observed frequencies, and a field with as many of the first as the molecule has distinct frequencies fits exactly and reports nothing.

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.

wrong · Normal mode H₂O: 3 distinct modes. The displacement of every atom in 3 normal modes of H₂O, drawn from the eigenvectors of the mass-weighted Hessian. Under each is the internal coordinate with the largest share of the motion and how large that share is; where no coordinate holds nine tenths, the mode is not a motion of one bond or one angle and is labelled so.

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.

spectra · Normal mode boron trifluoride at 1454 cm⁻¹, shared out four ways. One mode of boron trifluoride, with each internal coordinate's share of it computed by four conventions. Each convention is a bar in every group; the groups are the coordinates. A number quoted for this band without saying which bar it is has not said much.

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.

spectra · Normal mode How many coordinates, and how many motions. For each of six molecules, one square per internal coordinate the valence set carries — bond stretches, angle bends and an out-of-plane wag where there is one — with a rule drawn at the number of vibrational degrees of freedom. three of them have more coordinates than motions, and which ones is decided by shape rather than by size: ammonia's three angles are independent and boron trifluoride's are not, and the only difference is that one is flat.

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.

symmetry · Normal mode The shallowest slope, and the steepest. For three molecules, the softest and stiffest vibrations, the species each belongs to, and what a unit distortion along each costs. The cost is not taken from the frequency: it is computed by resolving the distortion onto the normal coordinates in the mass-weighted metric and adding up ω²q². That it comes back as the frequency is the identity the whole comparison rests on — the species decomposition uses only the coordinates and the group, and the cost uses only the masses and the force constants, and the two have to agree on this case before they can be asked to disagree on any other.

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.

symmetry · Point group 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.

spectra · Spectrum 10 of 55 force constants that no spectrum can see. The map from methane's 55 independent force constants to its Cartesian Hessian, as a spectrum: 45 directions the frequencies respond to and 10 they do not, out of 55. The null block is exactly the size the redundancy count predicts — 10 for 1 redundancy on 10 coordinates — and the two are computed by different routes, one a rank and one a closed form. A force field quoted to four figures is quoted along 45 directions that were measured and 10 that were chosen.

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.

symmetry · Normal mode The two molecules with a direction no spectrum can see. Methane's fifty-five independent force constants and boron trifluoride's twenty-eight, split into the combinations a spectrum determines and the ones it cannot touch. Both molecules have a redundant coordinate set — methane's six angles at a tetrahedral centre are five coordinates' worth, and boron trifluoride's three angles at a planar centre are two — so both have a flat space, and the smaller molecule's is the larger share of its field.

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.

spectra · Normal mode 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.

spectra · Normal mode 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.

spectra · Normal mode Half the distance from a coin to a decision. Every distortion whose symmetry species appears more than once, sorted by how concentrated the projection is across the occurrences of that species. The dashed line at a half is what a label that said nothing would give, since every one of these species appears exactly twice. The mean is 0.761 — so the label carries the answer about half the distance from a coin to a decision. Six of the twenty-four are above nine tenths and settled; ten are below six tenths and are not.

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.

symmetry · Point group The same three stretches, the same axis, one mirror plane apart. Boron trifluoride and ammonia put through the identical probe: a circle of distortions in the plane of stretch combinations that sum to zero, with the largest departure from three quarters taken over the depolarised bands at each direction. The two molecules share a three-fold axis and differ by a horizontal mirror. The first four rows are the same for both and the last three are not.

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.

spectra · Spectrum No substitution that keeps the symmetry can tell the two fields apart. Ten isotopologues of boron trifluoride, real and hypothetical, each computed under the fitted force field and under the second field a boron-10 substitution cannot exclude, with the largest difference between their frequencies on a logarithmic axis. The six that keep all three fluorines alike give the same spectrum under both fields to a millionth of a wavenumber, the precision of the calculation. The four that replace one or two fluorines differ by four to seven tenths of a wavenumber.

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.

spectra · Normal mode One sentence, three displacements, three answers. For each symmetry-distinct internal coordinate of the five molecules, how concentrated the projection is across the occurrences of the species the label names — under the B row the published measurement used, under the displacement that changes the coordinate at the least kinetic cost, and under the one that changes this coordinate and no other. Ammonia's bond angle goes from 0.516, a coin, to 0.883 and 0.926. Boron trifluoride's angles are the worst case under one clean displacement and its bonds under the other, both at 0.573.

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.

symmetry · Point group 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.

symmetry · Point group 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.

symmetry · Point group 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.

spectra · Normal mode

Named alongside it

The objects these essays reach for when they reach for this one.

Normal modeForce constantIrreducible representationsDegeneracyVibrational modesInternal coordinateLeast-squaresModel limitMode compositionUnderdeterminationConventionIsotope substitution

All concepts