Three ways to stretch a bond

The measurement that put a symmetry label at half the distance from a coin turned every internal coordinate of five molecules into a distortion and asked how much the label decided about where that distortion went. It gave 0.761 against a coin’s 0.500, settled six of twenty-four and left ten essentially undecided, the worst of them ammonia’s bond angles at 0.516.

Every one of those numbers rests on a sentence that sounds as if it had one meaning: stretch this bond. It has at least three, and the one the measurement used is the one of the three that is not a distortion.

Under either of the other two, the label decides twice as much.

A distortion that moves the molecule #

An internal coordinate is a function of the atom positions — a bond length, an angle — and its B-matrix row is that function’s gradient: the direction in the space of all atomic displacements along which the coordinate grows fastest. Normalised, it is the obvious way to turn a coordinate into a displacement, and it is what the earlier measurement did.

The row has one property worth stating before anything else. For a bond between atoms A and B it moves A and B by equal amounts in opposite directions along the bond, whatever the two masses are. For an angle it moves the two outer atoms perpendicular to their bonds and the central atom by the negative of their sum. It is a statement about geometry, and geometry does not know that carbon is twelve times heavier than hydrogen.

The projection onto normal modes does know. The modes are orthogonal only in the metric where each Cartesian displacement is weighted by the square root of its atom’s mass, and that is where the earlier work was careful to make the projection, because doing it without the masses double-counts the overlap between modes. The two decisions — a geometric displacement, projected in a mass-weighted metric — are each correct and do not fit together.

The B row is orthogonal to translation and rotation in plain Cartesian coordinates: a rigid motion changes no bond length and no angle, so the gradient of a bond length has no component along one. In the mass-weighted metric that orthogonality is lost, because the weights are different for different atoms. What survives is a displacement whose heavy atom dominates the weighted length, and a heavy central atom moving is mostly the molecule moving.

The size of it is not a detail. For water’s bond angle, 74.2 per cent of the B-row displacement, in the metric the modes live in, is rigid motion. For ammonia’s bonds and angles it is 69 per cent, for methane’s 58 and 59. Averaged over the twenty-four distortions the earlier census measured it is 46.0 per cent. Only where the central atom and its ligands have similar masses is it small: 9.3 per cent for sulfur dioxide’s angle and 3 to 4 for boron trifluoride, where boron is eleven and fluorine nineteen.

The concentration the earlier measurement reported was taken over the vibrational part alone, so the rigid part did not enter as a number. It entered as a shape. What remains of a displacement after three quarters of it has been discarded as translation and rotation is not the bond-angle distortion a chemist means; it is the residue of the oxygen moving, and the projection faithfully reported where that residue goes.

Two displacements that do not #

There are two natural repairs and they are not the same.

The first is to ask for the displacement that changes the coordinate by a unit at the least kinetic cost — the least mass-weighted length. That is the B row with each atom’s component divided by its mass, $M^{-1}B^\mathsf{T}$ in matrix form, and it moves light atoms far and heavy ones hardly at all: to open water’s angle it swings the hydrogens and leaves the oxygen nearly still. It carries no rigid motion in the mass-weighted metric, exactly, because the same identity that makes the B row blind to rigid motion in plain coordinates makes this one blind in weighted ones.

It has a second meaning that makes it easy to check. Its overlap with a normal mode is exactly the amount the coordinate itself changes in that mode — a row of what spectroscopists call the L matrix. Computed that way instead, from the mode shapes and the B matrix with no projection at all, the shares agree with the projection to the ninth decimal on every coordinate of every molecule. So the least-cost displacement answers a question with an older name: in which vibrations does this coordinate move?

The second repair asks for the displacement that changes this coordinate and no other. Stretching one of ammonia’s bonds by the B row also opens its two neighbouring angles slightly; the least-cost displacement does too. The displacement that holds every other bond and angle fixed is $M^{-1}B^\mathsf{T}G^{+}e_i$, where $G$ is Wilson’s kinetic matrix and the plus is its pseudo-inverse, and it answers a different question: if only this coordinate were changed, which vibrations would be excited?

It cannot always be built. Where the coordinates are redundant there is no displacement that changes one and leaves the rest alone — methane’s six angles have nine independent directions between ten coordinates, and the sum of the six is fixed to first order. The nearest displacement opens the chosen angle by 0.83 and closes each of the other five by 0.17. Boron trifluoride’s three in-plane angles sum to 360 degrees and do the same with two thirds and a third. Where the coordinates are independent — water, sulfur dioxide, ammonia — the displacement changes its own coordinate by exactly one and every other by zero to the eighth decimal, which is the statement that it has been built right.

What the label decides under each #

The same census, three times.

Ammonia’s bond angle — the case the earlier essay called a coin to two decimals, on the commonest distortion there is — goes from 0.516 to 0.883 under the least-cost displacement and to 0.926 under the one that changes the angle alone. It was never at a coin. It was at a coin when a nitrogen fourteen times heavier than each hydrogen was made to move as far as the hydrogens did.

Methane’s bond goes from 0.583 to 0.983 and to 1.000. Water’s angle goes from 0.876 to 1.000 and 0.996. Sulfur dioxide’s angle, the case with the least rigid motion to lose, moves least under the first repair — 0.770 to 0.865 — and furthest under the second, to 0.994.

Counted over every coordinate rather than only the ambiguous ones, the B row lets the label decide eleven of twenty-nine: five outright, because their species appears once, and six by concentration above nine tenths. It leaves ten undecided below six tenths. The least-cost displacement lets it decide twenty-two — nine outright and thirteen settled — and the one-coordinate displacement twenty-one, one outright and twenty settled. Both leave three undecided.

That is the correction to carry. Half the distance from a coin was a true statement about the B row. About the question it was asked to answer — how much a symmetry label says about which vibration a distortion excites — the answer under either clean displacement is most of the way to a decision, on three quarters of the coordinates.

Even the species moves #

The table hides a quieter fact that changes what the coarse test even means. The species a label names is itself a property of the displacement.

Water’s O–H bond, under the B row, is dominated by $B_2$ with 56 per cent of its weight — the antisymmetric species, which appears once, so the label decides it outright. Under both clean displacements it is dominated by $A_1$, which appears twice, and the label has to share it between the bend and the symmetric stretch. Methane’s bond angle goes the other way: under the B row it is mostly $T_2$ and ambiguous; under the least-cost displacement 64 per cent of it is $E$, a species methane has once, and the label decides it without a projection.

So the earlier census’s twenty-four of twenty-nine are ambiguous was also a statement about the B row. The least-cost displacement has twenty ambiguous coordinates and the one-coordinate displacement twenty-eight. The coarse half of the comparison needs no force field, but it does need a definition of the distortion, and the definition was carrying weight nobody had assigned it.

The worst case changes molecule #

Under both clean displacements the coordinates the label cannot resolve are boron trifluoride’s, at 0.573 — the three angles under the least-cost displacement, and the three bonds under the one-coordinate displacement.

The second of those is worth pausing on. Change one B–F bond and hold every angle and both other bonds fixed, and the displacement puts 57.3 per cent of its $E'$ share on the 480 cm⁻¹ bending vibration and 42.7 on the 1454 cm⁻¹ stretch. A pure bond stretch excites the bend more than the stretch.

It is not a paradox, and the reason is not the obvious one. The obvious one would be that holding the angles fixed forces boron off the centre of its fluorines, and that the bend is the vibration in which boron moves. It is the other way round: boron carries 92 per cent of the Cartesian amplitude of the 1454 cm⁻¹ stretch and 38 per cent of the 480 cm⁻¹ bend.

The reason comes from an identity two sections below, and it reads more simply than any mechanism. The share a bond displaced alone puts on each vibration equals the share the angle displaced at least cost puts on the other — and boron trifluoride’s F–B–F angle moves more in the stretching vibration than in the bending one, which is why the least-cost angle sits at 0.573 on 1454. A mode labelled stretch in which the angles move a great deal is the same fact as a bond stretch that excites the bend. Ammonia’s floor has not disappeared so much as moved to the molecule whose $E'$ vibrations are the most kinematically entangled.

The walk between the two #

The B row and the least-cost displacement are the two ends of a family, $M^{-t}B^\mathsf{T}$, in which the power $t$ says how much a heavy atom is spared. Walking it shows that the correction is not a jump between two arbitrary choices but a steady change with one exception.

Ammonia’s angle climbs monotonically from 0.516 to 0.883. Methane’s bond climbs from 0.583 to 0.983 and water’s angle from 0.876 to one. Ammonia’s bond rises to 0.9997 at $t = 0.2$ — at that power it puts all but a trace of its $E$ share on the stretching vibration — and falls back to 0.926. Methane’s angle changes species halfway along, at $t = 0.5$, from $T_2$ to $E$.

The exception is boron trifluoride’s angle. It starts at 0.599, falls to 0.508 at $t = 0.6$ and recovers only to 0.573. For every other coordinate, sparing the heavy atom helps the label; for this one it barely matters, because boron and fluorine are within a factor of two in mass and there was little rigid motion to remove. That is the same fact from the other side: the B row’s error was proportional to the mass contrast between a centre and its ligands, and where there is none, there was no error to correct and nothing for the label to gain.

An exact swap #

Looking down the table, two columns echo each other. Ammonia’s bond under the one-coordinate displacement reads 0.883; its angle under the least-cost displacement reads 0.883. Its angle alone reads 0.926; its bond at least cost reads 0.926. Boron trifluoride’s 0.573 and 0.997 appear once in each column, crosswise.

It is an identity, and the reason is two lines of algebra. The least-cost displacement’s shares are a row of the L matrix — how much each coordinate moves in each vibration. The one-coordinate displacement’s shares are a column of its inverse. Inside a species that appears twice, with one bond-type and one angle-type symmetry coordinate, the relevant block of L is two by two, and the inverse of a two-by-two matrix is its adjugate divided by its determinant: the entries swap across the diagonal and two change sign. Squared and normalised, the column of the inverse belonging to the bond is the row of L belonging to the angle, with its two entries exchanged — and exchanging which occurrence is larger does not change the largest share.

So the two clean displacements answer questions that look different — which vibrations does this coordinate move in and which vibrations does moving only this coordinate excite — and in a two-occurrence species each is the other asked about the partner coordinate. Nothing in the force field enters. The identity holds for any force field whatever, and here it holds in six species of five molecules: water and sulfur dioxide’s $A_1$, methane’s $T_2$, boron trifluoride’s $E'$ and ammonia’s $A_1$ and $E$.

It is the surprising connection in this essay, and it was found by reading a table rather than derived first. That order is worth admitting, because it is also the order in which it found something else.

The swap found a rounding error #

Four of the six species obey the swap to 2 × 10⁻⁹ or better, which is the precision of a Hessian computed by finite differences. Ammonia’s two miss by 1.2 × 10⁻⁵ and 3.7 × 10⁻⁵ — thousands of times more — and the miss does not shrink when the finite-difference step is made three or ten times smaller. It is not numerical noise in the modes.

It is the molecule. The stored ammonia’s three N–H bonds are 1.0159924, 1.0160378 and 1.0160378 ångström, because its nitrogen sits 3.3 × 10⁻⁵ ångström off the threefold axis — the rounding of coordinates written to four decimals. Every symmetry search forgives that, as the tolerance any assignment has to choose requires it to; the group comes out $C_{3v}$ and the modes come out in $A_1$ and $E$ pairs whose frequencies agree to the displayed digit.

An exact identity forgives nothing. With the molecule rebuilt about the axis through its hydrogens’ centroid — same mean bond length, same angles, hydrogens at exactly 120 degrees — the swap holds to 1.1 × 10⁻⁸ or better in both species, the same floor as the other four. So the identity is exact, and the stored geometry is symmetric to five figures rather than to every figure.

The stored molecule is left as it is. Every structure here carries its own tolerance, and a rounding of three hundred-thousandths of an ångström changes no frequency, no species and no concentration past the fourth decimal. What changes is the status of a statement: an identity that should hold exactly and holds only to five figures is information about the input, and in essays that state results to four decimals that is exactly the precision at which an input’s rounding stops being invisible.

Which one the question meant #

The earlier measurement asked how much a label says about which vibration a distortion goes into. A displacement that is three quarters translation is not a distortion, so the B row is out. That leaves two.

The one-coordinate displacement is the closer reading of the sentence. Stretch this bond means lengthen it, not lengthen it and open two angles, and the question which vibration does this excite is a question about the normal coordinates of a static displacement — what resolving a measured structure into species has to answer when it asks which coordinate a molecule fell along. Its drawback is redundancy: for methane’s angles and boron trifluoride’s there is no displacement that changes one coordinate alone, and the nearest one drags its neighbours.

The least-cost displacement answers the question the older literature more often asks — where does this coordinate move — and it is built from the kinetic matrix alone, the same matrix an isotope substitution changes and nothing else does. It never runs into a redundancy.

Neither is wrong, the swap says they are the same question about different coordinates, and they agree on everything that matters here: twenty-one or twenty-two coordinates decided, three undecided, boron trifluoride the floor. The disagreement that mattered was between both of them and the B row.

A mixing that was never there #

The earlier essay explained its best and worst cases by mixing: the label works where the two occurrences of a species are far apart in frequency, so one of them is much more nearly a pure stretch than the other, and it fails where a molecule’s vibrations mix a stretch and a bend in comparable amounts.

Methane refutes that on its own numbers. Its upper $T_2$ vibration is a stretch to three decimals and its lower one carries under one per cent of stretching — as unmixed as two vibrations of one species can be — and its bond distortion sat at 0.583 under the B row, among the worst. Ammonia’s $E$ bend carries three and a half per cent of stretching and its angle sat at a coin. The low numbers were not a property of the vibrations. They were the carbon and the nitrogen, moved as far as their hydrogens and then weighted by their mass.

That is repaired in the earlier essay rather than left for a reader to reconcile, and its headline number stands with its convention named: 0.761 is what the B row gives.

What this rests on #

The same five molecules and the same fitted force fields. Every projection here inherits the fits, and the interaction constants — the terms coupling a bond to its neighbouring angle — are the part of a fitted field least constrained by frequencies, as an earlier measurement of what a fit leaves free found. A concentration is a statement about mode shapes, and mode shapes are what those constants move. Nothing here says how far.

The threshold of nine tenths for settled and six tenths for undecided are the earlier essay’s, kept so that the counts compare. Moved, the counts change and the ordering does not: at 0.85 the B row decides 18 against 26 for both others, at 0.95 it decides 8 against 19 and 18, and for every threshold from 0.6 upward it decides the fewest. Its ten undecided stay ten for every cutoff from 0.6 to 0.7, against three.

The species decomposition of a displacement is made from the group alone, after removing rigid motion in unweighted coordinates, exactly as before. Only the displacement fed to it has changed. That the species can change with the displacement is a finding, not an inconsistency, because the three displacements genuinely have different symmetry content.

The refusal. Each construction is checked against the case it must get right: asked to change the coordinates exactly as a normal mode changes them, the one-coordinate construction must return that mode, and it does, for all twenty-seven modes of the five molecules, to the ninth decimal.

Who wrote down which displacement #

The B matrix is Wilson’s, from 1939, and the kinetic matrix $G = BM^{-1}B^\mathsf{T}$ with it; Wilson, Decius and Cross set out the whole apparatus in 1955, including the L matrix whose rows say how much each coordinate moves in each mode and whose inverse turns coordinate changes into normal coordinates. Eckart’s conditions of 1935 are what separates a displacement’s vibrational part from its rigid part, and they are stated in the mass-weighted metric for the reason this essay turns on.

None of that is new, and neither is the observation that a Cartesian gradient carries rigid motion once masses are applied. What is computed here is the size of the effect on a specific published measurement, the census under two clean displacements, the one-parameter family between the B row and the kinetic displacement, and the swap identity between them, with the one input it caught.

Still open: the constants the fit leaves loose, and a third occurrence #

The obvious open question is the force field. Every concentration above is a statement about mode shapes, and the shapes are set by interaction constants that fitting to frequencies constrains weakly. Ammonia’s stretch–bend interaction constant is large and negative in its fit; holding it at other values and refitting the rest would say how far the concentrations move across force fields that reproduce the observed frequencies almost equally well — and whether a label’s verdict on a coordinate is a property of the molecule or of the least-determined number in its fit.

The nearer question is the swap beyond two occurrences. The identity is the adjugate of a two-by-two block, and a species appearing three times has a three-by-three block whose inverse mixes all nine entries. Whether any relation survives there between a coordinate displaced alone and the others displaced at least cost — a sum rule rather than a swap — needs a molecule with a species that repeats three times, and none of the five has one.