Molecular Geometry & Bonding
what the shape decides
Concept

Internal coordinate — where it appears

A bond length, angle or torsion, as against a Cartesian displacement. A mode's composition is shared out among a chosen set of them, and the shares depend on which set was chosen and how angles were made commensurate with lengths.

Named by 18 essays across 3 fields — each of them below, with the objects they name alongside it.

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.

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 The same angles, the same torsions, and not the same molecule. twelve closed conformers of a ring of 10 at a bond angle of 111.5 degrees. Every one has exactly the same bond angles, so an account built from angles and torsions places them all on the horizontal axis alone. The vertical axis is the closest approach of two atoms four or more bonds apart, which no term in that account mentions: two of these differ by 0.08 kilojoules in torsional energy and by 0.75 ångström in how close they come.

The atoms that meet across a ring

Twelve closed conformers of a ten-membered ring at one bond angle, so every one has identical angle strain by construction. Two of them differ by 0.026 kilojoules a mole in torsional energy and by 0.80 ångström in how close two atoms on opposite sides of the ring come — 2.331 against 3.131, where two carbons are in contact at about 3.4. An account built from angles and torsions calls those two structures the same.

shape · Strain 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 ceiling rises and the measurements fall, so they cross. The largest acceleration the rotamer account can produce, against the ring being closed, with the measured gem-dimethyl accelerations on the same axis. Closing a bigger ring means freezing more rotations, so the ceiling rises steeply; the measurements go the other way. The five-membered ring's 250-fold acceleration is above its own ceiling of 36.5 and the six-membered ring's tenfold one is far below its 121 — so the account is refused at one size and sufficient at the next.

A ceiling that rises where the measurements fall

The rotamer account of the gem-dimethyl effect has a largest possible acceleration, which looks like a limitation. It is a prediction: the ceiling is a closed form in the number of rotations a closure freezes, it rises steeply with ring size, and the measurements fall — so the account is refuted for the five-membered ring and more than sufficient for the six.

shape · Strain 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 How far the rotor count would have to be wrong. The ceiling against the number of rotations a closure freezes, with the two measured accelerations drawn across it. The five-membered closure freezes three and its ceiling is 36.46; the ceiling does not reach the measured 250 until 5 rotors, so the count would have to be wrong by 2 on a ring that has three rotations to freeze. The six-membered closure freezes four at a ceiling of 120.88, and stays above its measured 10 down to 2 — so the refusal is airtight and the sufficiency is comfortable.

An estimate that can be wrong by two

The ceiling on the gem-dimethyl effect is exponential in the number of rotations a closure freezes, and that number was taken as n − 2 without counting — which left the refutation at five rings probable rather than airtight. It is airtight. The ceiling does not reach the measured 250 until five rotors, on a ring that has three, and no hindering of the tether can raise it.

shape · Strain 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 A species label is ambiguous for every one of 17. For each molecule, the share of its vibrations belonging to a symmetry species that appears more than once — the distortions a species label cannot price, because the label picks a space rather than a mode. Every molecule in the census has at least one such species, the share averages 71.4 per cent, and for 9 of 17 the repetition is not forced by the group's capacity — those molecules have fewer vibrations than their group could hold without repeating, and repeat anyway.

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.

symmetry · Point group 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.

spectra · Normal mode Where the explanation gives a negative number of vibrations. The usual formula against the answer, for the 15 molecules with internal coordinates here. It is right for 7 of them and wrong for 8, and for sulfur hexafluoride, benzene and both ferrocenes it predicts a negative number of totally symmetric vibrations — which is the clearest possible sign that the quantity being subtracted is not the one that should be.

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.

symmetry · Point group Which torsions a molecule's own operations turn backwards. Every torsion orbit of the molecules here that have torsions: how many operations carry the torsion onto itself, how many of those are proper, and which operations reverse it. 7 of 9 orbits are reversed, and in every one the operation doing it is improper — the plane of a planar molecule, a mirror bisecting the torsion's bond, or a centre of inversion at that bond. Proper operations fix torsions too, benzene's twofold axes and hydrogen peroxide's among them, and never reverse one. Staggered ferrocene gives the same rows as eclipsed.

Five coordinates for six vibrations

A torsion is reversed by every improper operation that carries it onto itself and by no proper one, so adding torsions tests the orbit rule on a second kind of signed coordinate — and the rule holds on every molecule. It also moves hydrogen peroxide's count of totally symmetric vibrations from three to four, which a property of a molecule cannot do. Its five coordinates never spanned its six vibrations, and three of fifteen coordinate sets had been counting vibrations they did not describe.

symmetry · Point group 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 Water's two hydrogens are closer than platinum's chlorine. For each molecule, the longest pair that is a bond and the shortest pair that is not, on a logarithmic length axis. A cutoff on the length has to sit to the right of every filled mark and to the left of every open one, and it cannot: the longest bond in the collection is 2.3200 ångström and the shortest non-bond is 1.5144. The two populations overlap by a factor of 1.53, so the rule in use is not a rule with a badly chosen number in it — it is a rule with no number that works.

No length separates them

A bond list here is one distance cutoff with a clause about hydrogen, added when peroxide came back with five bonds instead of three. The clause repaired one molecule. Across the twenty-three molecules drawn here, the longest bond is 2.32 ångström and the shortest pair that is not a bond is 1.51 — so no cutoff can work at all, and five molecules currently come back with no bonds.

symmetry · Point group Three constructions of one number, spanning a factor of 5.6. The reduced mass of NH₃'s umbrella coordinate under each construction, against position along the coordinate. Two of the three are constants and the third is not: if the bonds are held at their measured length, the ligands must slide outward as the apex descends, and their radial motion adds to the mass. It runs from 2.4866 at the plane to 2.9874 at the pyramid — 20 per cent — and the coordinate itself stops existing at one bond length, which is where the curve ends.

The mass nobody chose

Every one-dimensional treatment of ammonia's inversion needs a mass, and the measurement does not supply one. Three constructions are defensible and they give 1.35075, 0.94420 and 0.0000055 wavenumbers. The honest one is not a constant at all, and it moves the answer thirty per cent towards the measurement — which means the usual choice is the wrong one.

shape · Inversion The orbit identity holds on all nineteen, and the formula it replaced on eight. For each molecule in the census under the radius rule: its number of totally symmetric vibrations, the corrected count — symmetric orbits less symmetric redundancies — and the usual formula, orbits less redundancies. The corrected count lands on the molecule's own count every time. The usual formula is right for 8 of 19, and for ferrocene under the new bond list it predicts minus sixty-six.

The census a bond rule was hiding

Every internal coordinate, redundancy and totally symmetric count here is built on a bond list, and the bond list came from a length cutoff that gave five molecules no bonds. Rebuilt on the radius rule, the census reaches nineteen molecules instead of fifteen, the orbit identity holds on every newcomer, ferrocene's coordinates finally span all its vibrations — and a different gap appears: no bond rule can give a square-planar centre its two out-of-plane vibrations.

symmetry · Point group The term two of the four molecules have and two do not. The radial coefficient of the bond-conserving reduced mass for the four isotopologues: the sum of the ligand masses, and what is left after the asymmetric correction. Three ligands at a hundred and twenty degrees on a circle whose radius changes as the apex descends move their own horizontal centre of mass outward — unless the three masses are equal. A frame that does not translate has to subtract that motion, and the amount is half the sum of the squared mass differences over the total mass. It is zero at both ends of the series and the same number in the middle.

The two that are not on the line

Ammonia and its fully deuterated twin are two points, and two points cannot show a curve. Putting the partly deuterated molecules between them needs a term neither symmetric one has — three ligands of unequal mass move their own centre of mass sideways as the apex descends — and it moves the prediction by half a per cent, which is what a whole change of mass construction was worth.

shape · Inversion

Named alongside it

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

Model limitConventionNormal modeForce constantDegeneracyLeast-squaresVibrational modesUnderdeterminationValence force fieldIrreducible representationsSymmetry operationOrbit (group theory)

All concepts