Molecular Geometry & Bonding
what the shape decides
Series

Normal mode — the series

14 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · spectra
  2. 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.

    part 2 · wrong
  3. 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.

    part 3 · spectra
  4. 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.

    part 4 · wrong
  5. 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.

    part 5 · spectra
  6. 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.

    part 6 · spectra
  7. H₂O with 1→D: what each mode is made of. H₂O with 1→D. 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. 3 of 3 here are.

    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.

    part 7 · spectra
  8. 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.

    part 8 · spectra
  9. 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.

    part 9 · symmetry
  10. 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.

    part 10 · symmetry
  11. 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.

    part 11 · spectra
  12. 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.

    part 12 · spectra
  13. 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.

    part 13 · spectra
  14. 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.

    part 14 · spectra

All series