Molecular Geometry & Bonding
what the shape decides
Figure

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.
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.

One of the figures on where the atoms go: Repulsion minimised on a sphere, the angles that fall out of it, and the arrangements that are not all alike.

Eight essays draw this figure, each at the values its own argument needs rather than at the setting shown above. What each one uses it to show is below, in the words of its own caption.

In the essays

A barrier is not what a splitting measures

The ground doublet on the well it lives in: two levels 1.3508 wavenumbers apart, on a barrier 2020 tall. The gap is seven hundredths of one per cent of the barrier it is a consequence of.

The whole well, with the four states that lie below its barrier. The lowest sits 586.81 wavenumbers up, which is 29.05 per cent of the way to the top.

The two members of the ground doublet, on the well they live in. They differ by a node at the centre and by essentially nothing else, which is why their energies differ by a part in 435.

The exponent that runs both ways

The ground splitting against the barrier, both logarithmic. The local slope is −3.56 where ammonia sits, and the curve steepens either side of it.

The local exponent at four barriers. Quoting one number for it is quoting a tangent as though it were a law.

The same exponent carried in each direction. Predicting a splitting from a barrier multiplies the uncertainty; inferring a barrier from a splitting divides it.

It was never the mass

Ammonia and phosphine to one scale, each from its own measured bond length and bond angle. The height the apex has to be carried through is 0.3816 ångström against 0.7682.

The action under the barrier for each substitution. Ammonia’s is 5.69; phosphine’s mass adds 0.43, its pyramid 8.50 and its barrier 12.39, and all three together add 36.24.

Each of phosphine’s three differences substituted into ammonia’s well on its own, against the splitting each produces. The mass is the shortest bar by four orders of magnitude.

The mass nobody chose

The three constructions across the coordinate. Two are horizontal lines and the third is not; the third also stops, because the coordinate itself only exists while the apex is nearer the plane than one bond length.

The bond-conserving mass across the coordinate, with the ground-state amplitude beneath it. The mass is largest exactly where the molecule spends its time and smallest exactly where the tunnelling happens.

The three splittings, against the measured value. The bond-conserving construction lands nearest and the literal reading of the coordinate’s name lands five orders away.

An ordering worth half a per cent

What each of the four other orderings adds to BenDaniel–Duke across the umbrella coordinate. The largest is a few wavenumbers on a barrier of two thousand.

Each ordering’s change in ammonia’s ground splitting relative to BenDaniel–Duke. Bars are exact; the ticks are first-order theory, and they sit on the ends of the bars.

Fifteen orderings with α and γ between −1 and 0, exact change against first-order prediction. They lie on the identity line.

Deuterium cannot tell the masses apart

The bond-conserving mass of each isotopologue divided by its own value at the flat geometry. ND₃’s grows further across the coordinate.

The barrier fitted to NH₃ and the barrier fitted to ND₃, under each mass and in each well. They should coincide and never do.

The ratio of NH₃’s splitting to ND₃’s predicted by each mass in three settings, against the measured 14.94. Every prediction overshoots.

The two that are not on the line

The radial coefficient of the bond-conserving reduced mass for the four isotopologues, with and without the term only an asymmetric molecule has.

The asymmetric term for each isotopologue, computed from the three ligand masses alone.

The predicted splitting against the number of deuterium atoms, on a logarithmic axis, with the two measured points drawn for scale.

The ordering a manifold picks

The difference between the Laplace–Beltrami operator, carried to the flat measure, and the BenDaniel–Duke ordering, at forty-one positions.

The two fitted coefficients, the sum and product of the von Roos exponents they imply, and the closed forms.

The von Roos plane with each named ordering at its own exponents, the diagonal where the two are equal, and the covariant point marked.

Every figure · Every orbital, by what it encloses · All essays