Concept

Zero-point energy — where it appears

The energy a vibration retains at absolute zero, half a quantum in each mode. It puts the atoms away from the minimum of the potential and, because a real well is not symmetric, moves the average bond length outward.

Named by 17 essays across 3 fields — each of them below, with the objects they name alongside it.

$water, drawn with its nuclei the size they are. The molecule with a disc round each nucleus whose radius is the computed root-mean-square displacement of that nucleus in the vibrational ground state. The hydrogens' discs are a substantial fraction of the bond length, and this is at no temperature at all.$

The atoms are not at the points

Every structure in this collection is a set of points, and every bond angle it argues about is a property of that set. Computed from the fitted force fields it already has, water's hydrogens are 0.094 Å from where they are drawn and its bond angle has a spread of 8.9 degrees — larger than the difference between 104.5 and the tetrahedral value that half the essays here are about. This is at no temperature at all.

shape · Approximation

H³⁵Cl: the well, its states and their averages. The Morse potential built from H³⁵Cl's measured vibrational constants, with the lowest four states drawn at their computed energies and the average separation of each marked. Every average lies to the right of the minimum, because the well is not symmetric — and they move outward as the state rises.

The bond length that depends on the isotope

Hydrogen chloride and deuterium chloride have the same potential energy curve, and their measured equilibrium lengths agree to three hundredths of a milliångström. Their average bond lengths differ by 4.34 mÅ — a hundred times more — because a lighter atom explores more of a well that is not symmetric.

wrong · Approximation

What boron trifluoride's bands are strong in, and what they move. Every infrared-active mode of boron trifluoride, with its band strength and the root-mean-square displacement of its atoms in the zero point, each scaled to its own largest. The two do not order the modes the same way — the rank correlation between them is 0.2 — and the strongest band belongs to the mode at 719 cm⁻¹, in which 89.93 per cent of the motion is the lightest atom's. Every mode moves the same weighted amount of mass, exactly, so that is not what separates them either.

The mode that moves least radiates most

Boron trifluoride's strongest infrared band is the one in which the fluorines barely move: ninety per cent of the motion belongs to the boron, which is a fifth of the molecule's mass. The mode that moves the most mass is nine and a half times weaker. Across five molecules the rank correlation between band strength and how far the atoms actually go runs from +1 to −0.66, and every normal mode carries exactly the same weighted motion by construction.

shape · Dipole

A photoelectron band is a filter, and the group chooses the filter. The Huang–Rhys factor of every vibration of five molecules under a change of geometry that lengthens every bond alike — which is what removing an electron from a non-degenerate orbital does. On a logarithmic scale spanning sixteen decades, eight modes carry the whole of it and the rest sit on the floor at arithmetic noise. Which ones is decided by the point group: only a totally symmetric vibration can appear, whatever the size of the change.

A band is a filter on the modes

A photoelectron band's vibrational structure reports the frequencies of a few of the ion's vibrations and is silent about the rest, and which few is decided by the point group before any geometry is known. Under a change of shape that lengthens every bond alike, methane's totally symmetric stretch gets a Huang–Rhys factor of 0.905 and its other eight modes get between 10⁻²⁸ and 10⁻³⁵.

spectra · Photoelectron

Every sign, lost. Formaldehyde in its own principal axes. Open circles are the atoms where they are; filled ones are where Kraitchman's equations put them, from the change in the three moments when each atom in turn is made heavier. The two agree to 7.6e-8 ångström — the equations are an identity for a rigid structure — but they return the square of each coordinate, so the two hydrogens at b = ±0.9348 both come back at +0.9348 and land on the same point.

The coordinate an isotope reports

Kraitchman's equations return an atom's position from the change in the moments when that atom alone is made heavier, and for a rigid structure they are an identity — formaldehyde's four atoms come back to a part in ten million. What they return is the square of each coordinate, so both hydrogens at b = ±0.9348 come back at +0.9348; every out-of-plane coordinate comes back imaginary at a moment error of one part in a hundred thousand; and the famous error cancellation, measured at a factor of thirteen, still leaves the answer two and a half times worse than a direct fit.

spectra · Rotation

The factor of thirteen was generous. An invented error model against the one computed from a force field. Its invented mismatch of five per cent made the correlation between the parent and the substituted species worth a factor of 12; the computed mismatch of 35 per cent makes it worth 2.5. The substitution structure went from being 2.5 times worse than a direct fit to 26, and at the computed size of the correction its worst coordinate is out by 10.7 per cent.

The correction that was invented

A standard error model puts the zero-point error in a rotational constant at a few tenths of a per cent, shared between the three moments by invented weights, with the parent and its deuterated form differing by five. Computed from a force field it is 1.88 per cent, one of the three shares is negative, and the mismatch is 35 — so the cancellation the substitution method rests on is worth a factor of 2.5 and not thirteen.

spectra · Rotation

One expression, four molecules, a factor of eleven. The computed zero-point correction to each molecule's moment of inertia against the expression A²·Σ(1/ν)/I, which has no fitted quantity in it. The dashed line is the mean dimensionless coefficient, 0.6343; the four points lie within 18 per cent of it, on corrections that span a factor of 11.4. The expression is evaluated from a moment of inertia and a list of wavenumbers, which is what a spectroscopist has before doing anything.

An expression for what was a warning

The zero-point correction to a moment of inertia comes out at 1.88 per cent where a few tenths had been assumed, and heavy molecules are safer. Written out, the correction is a mean curvature times the sum of reciprocal wavenumbers over the moment — one line, evaluated from things a spectroscopist has before starting. One coefficient serves four molecules whose corrections span a factor of eleven.

spectra · Rotation

The coefficient each principal axis needs. The dimensionless coefficient the molecule-averaged expression requires, evaluated on each principal axis separately rather than on the mean of the three. Across the twelve axes the positive ones span a factor of 2.33, against the 1.36 the molecule-averaged version spans — and one of them is negative, which no positive constant can be.

The axis that goes the other way

One dimensionless coefficient turned a zero-point correction into an expression a spectroscopist could evaluate, and the three principal axes were averaged over to get it. Split by axis it gets worse, not better — the coefficients span 2.3 where the molecule-averaged ones span 1.36 — and water's smallest moment does not grow at all. It shrinks.

spectra · Rotation

The frame H₂O → HDO turns. H₂O → HDO: the parent's principal axes and the daughter's, drawn on the same nuclei. Two of the three turn by 21.12° and the third does not move at all, because it is the normal to a plane no substitution can tilt. A per-axis comparison between these two species is comparing moments about lines this far apart, which is a rotation of the frame rather than a correction to a number.

Two moments about two different lines

Asked axis by axis, the substitution method's near-cancellation gives numbers as large as 163 per cent. The arithmetic is the smaller half of the answer. A principal axis is an eigenvector of a tensor built from the masses, so one deuterium turns water's frame by 21.12° — and the two moments being compared are not moments about the same line.

spectra · Rotation

The two terms in a vibrationally averaged rotational constant. A rotational constant averages 1/r², not r², so the moment it reports carries +2⟨Δr⟩/rₑ from the anharmonicity and −3⟨Δr²⟩/rₑ² from the harmonic spread. The two have opposite signs in every molecule here, and the anharmonic one — which is exactly zero in any symmetric well and therefore absent from every harmonic force field — is larger by a factor of 1.94 to 2.58.

The term a harmonic field cannot produce

The usual zero-point correction to a moment of inertia comes from a harmonic force field, which contains the mean square displacement and nothing else. A rotational constant does not average that. It averages one over r squared, whose leading correction is the mean displacement — zero in any symmetric well — and which enters with the opposite sign and about twice the size.

spectra · Rotation $A Morse curve's αₑ falls 4 to 15 per cent short of the measurement. The vibration–rotation constant αₑ of each diatomic, averaged over the states of a Morse curve built from its measured ωₑ, ωₑxₑ and rₑ, as a fraction of the tabulated value. H³⁵Cl: 0.27747 against 0.3072 cm⁻¹, ×0.903; D³⁵Cl: 0.10240 against 0.1133 cm⁻¹, ×0.904; ¹²C¹⁶O: 0.01674 against 0.0175 cm⁻¹, ×0.957; H¹⁹F: 0.68164 against 0.798 cm⁻¹, ×0.854. Every one is short, and HCl and DCl — one potential with two masses on it — are short by the same fraction.$

The cubic a Morse curve guesses

The two terms in a vibrationally averaged rotational constant were computed on Morse curves built from measured constants, and the vibration–rotation constant αₑ is the measurement that tests them. In all four molecules the Morse curve's αₑ is short, by four to fifteen per cent. The averaging is not the error — it matches the closed form to four parts in ten thousand. The curve's cubic is, and the measurement asks for more of exactly the term a harmonic field cannot produce.

spectra · Rotation

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

Every prediction of the isotope ratio overshoots, and the mass decides nothing. The ratio of NH₃'s ground inversion splitting to ND₃'s, predicted six ways, against the measured 14.94. At the published barrier the usual mass gives 15.77 and the bond-conserving mass 17.79. With a quartic fitted to NH₃'s splitting they give 19.26 and 19.00; with a well whose shape is fitted to both of NH₃'s lines, 17.62 and 18.48. The bond-conserving mass is nearer the measurement in one of the three pairs and further in two, and the difference within any pair is smaller than the distance of either from the measurement.

Deuterium cannot tell the masses apart

A reduced mass built by holding ammonia's bonds rigid predicts a deuterated molecule differently from any constant mass, and ND₃'s splitting is measured. Run as a test, it cannot choose. Every well and every mass needs a barrier for ND₃ several per cent lower than for NH₃, every prediction of the isotope ratio from a well fitted to NH₃ overshoots by eighteen to twenty-nine per cent, and the two masses differ by less than either misses — in opposite directions in the two wells.

shape · Inversion

The covariant operator is BenDaniel–Duke plus this. The difference between the Laplace–Beltrami operator — the one a one-dimensional manifold with metric μ(x) distinguishes, carried across to the flat measure by the unitary map ψ ↦ μ^(¼)ψ — and the BenDaniel–Duke ordering, divided by the function it was applied to, at forty-one positions inside the molecule's own range. The curve drawn through the marks is the two-function fit every ordering is a combination of, and it passes through them to a part in ten million.

The ordering a manifold picks

A position-dependent mass leaves the kinetic energy with no unique quantum form, and an earlier sweep of the five orderings in use found half a per cent between them. A one-dimensional reduction is a one-dimensional manifold, a manifold has a distinguished Laplacian, and carrying it to the flat measure lands on exactly one of those five — not the one with no extra potential, and not the one anybody reaches for.

shape · Inversion

Two pyramids, and only one of them has ever shown a line. What ammonia and phosphine each bring to the same one-dimensional model. The barrier is six times larger and the apex sits twice as far out, so the action under the barrier is seven times larger and the splitting is twenty-four orders of magnitude smaller. What is not six times anything is the transverse zero-point difference between a molecule and its deuterated twin: the modes that carry it are the ligands' own, the ligands are the same three hydrogens, and the two numbers are within thirty per cent of each other.

The same fraction at six times the barrier

Ammonia's two fitted barriers leave a gap that an isotope-dependent term would have to fill, and the term was sized at five per cent of the transverse zero-point difference. Phosphine can be asked the same question and cannot answer it the same way: its splitting is twenty-four orders of magnitude below anything the solver can represent, and the solver does not fail cleanly — it reports values that rise as the barrier rises. Calibrated where a splitting is still visible, the action carries the question across, and the same fraction moves phosphine's isotope ratio by sixteen per cent against ammonia's twenty-one.

shape · Inversion

The band that cannot supply what it changes

A missing transverse zero-point term was sized at five per cent of the difference between two molecules' frozen frequencies, as though the five frozen modes were one thing. They are three bands carrying 26, 50 and 24 per cent of that difference, so the requirement is not one fraction but three, spanning a factor of two — and the band a flattening pyramid changes most is the one that would have to move furthest. What survives the choice is the barrier: every version implies the same bare electronic value to one and a third per cent, because every band of a pyramidal hydride has the same isotope frequency ratio.

shape · Inversion $Two patterns, one shape, and a slope between them. How far the logarithm of each splitting falls as one hydrogen at a time is replaced by deuterium, for the ground doublet and for the first excited one. Neither falls in equal steps — the first step is half again the last — and the two patterns are the same shape: the excited step is a fixed fraction of the ground step at every stage, within a few per cent. The well is fitted to both of NH₃'s measured lines and nothing is fitted to any deuterated molecule.$

The steps were a square root of a mass

Replacing ammonia's hydrogens one at a time does not lower its splitting in equal steps, and the obvious readings are the potential's shape or the tunnelling arithmetic. Both lines — the ground doublet and the first excited one, which nothing in this argument has ever been fitted to — come out straight in the square root of the reduced mass and bent against the count, by a factor of eight, so the unequal steps are the shape of a square root drawn against the wrong variable. And the excited line is the one test the missing correction cannot pass by construction: it improves it threefold and over-shoots it.

shape · Inversion

Named alongside it

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

Model limit Reduced mass Convention Isotopologue Harmonic approximation Inversion splitting Isotope substitution Moment of inertia Normal mode Approximation Born–Oppenheimer separation Closed form

All concepts