Molecular Geometry & Bonding
what the shape decides
Concept

Umbrella mode — where it appears

The out-of-plane motion of a pyramidal centre against its three ligands, in which the molecule flattens and inverts. It is a large-amplitude coordinate rather than a small vibration, so its reduced mass is a construction rather than a formula and its potential has two minima.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

NH₃: four states in a well the molecule does not sit at the bottom of. The umbrella coordinate of NH₃ — the signed distance of the N atom from the plane of its three H atoms — with the quartic well that has its minima at the measured 0.3816 ångström and its barrier at the quoted 2020 wavenumbers. The lowest 4 states are drawn at their computed energies. The lowest sits 587 wavenumbers above the bottom, which is 29.1 per cent of the way up the barrier, so the state is far from the harmonic bottom that a drawing of a pyramid implies.

A barrier is not what a splitting measures

Ammonia's inversion barrier is quoted everywhere as 2020 wavenumbers. Put that number into the simplest double well its own measured geometry allows and the ground-state splitting comes out at 1.3508 against a measured 0.7935, and the excited one at 68.37 against 35.81. Both are too large because a splitting is an area under a barrier and a height is only one of its two dimensions.

shape · Inversion One per cent on the barrier is 3.6 per cent on the splitting. The ground inversion splitting of NH₃'s quartic well against the barrier height, both logarithmic, with the geometry and the reduced mass held at their measured values. The curve is visibly bent: its local slope is -3.56 at the published barrier and steepens either side, so a power law is a tangent to it rather than a description of it. The measured 0.7935 wavenumbers is reached at 2330, which is 15.3 per cent above the quoted 2020 — so a splitting wrong by a factor of 1.70 is a barrier wrong by a sixth. The same derivative read the other way is what makes a barrier quoted to ten per cent useless for predicting a splitting.

The exponent that runs both ways

How hard does a splitting depend on a barrier? Locally, as the power −3.5628 — and the local slope runs from −2.53 to −6.15 across the same sweep, so there is no power law. What is exact is stranger: rescaling the equation forces the mass exponent to be one below the barrier's and the geometry exponent to be twice the mass's, so the model's three sensitivities are one number and the arithmetic reproduces both identities to six decimals.

shape · Inversion The mass is worth a factor of 1.6, and the other two 1e+4 and 9e+4. Ammonia's umbrella well, with each of phosphine's three differences substituted into it one at a time and then all together. The reduced mass is 11 per cent larger and costs a factor of 1.61. The pyramid is 2.01 times taller and costs 9.6e+3; the barrier is 6.1 times higher and costs 9.4e+4. Phosphine's own splitting is below what the arithmetic resolves, so it is drawn at that bound.

It was never the mass

Phosphine does not invert, and the reason given is that phosphorus is heavier than nitrogen. Three things about phosphine differ from ammonia. Substituting each into ammonia's own well one at a time, the reduced mass costs a factor of 1.61, the pyramid height costs 9,600 and the barrier 94,000 — and the mass is the smallest of the three by four orders of magnitude.

shape · Inversion 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 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.

Double wellInversion splittingModel limitReduced massTunnellingConventionInternal coordinateIsotopologueBond angleError propagationNumerical precisionWavenumber

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