The spectrum that weighs a bond

Every minimum is a parabola makes a promise and a disclaimer in the same breath. The promise is that any smooth well looks like a parabola near its bottom, so every small vibration — a pendulum’s, a chemical bond’s, a crystal’s — has a frequency set by the curvature there. The disclaimer is that the curvature is all such a vibration measures: two wells with the same curvature and very different depths vibrate at the same frequency, so a spectrum of small vibrations cannot say how strongly a molecule is held together.

That is true of the lowest line in the spectrum and false of the spectrum as a whole. A quantum oscillator in a well has a stack of levels, and in a parabola they are evenly spaced for ever. In a real bond they are not. The levels crowd together as they climb, because the well widens and softens towards the top, and at the top — the energy at which the atoms come apart — the gap between neighbours falls to nothing. Measure enough levels and the spectrum contains the depth. Raymond Birge and Hertha Sponer turned that into a method in 1926, and the method, its famous flaw and the correction to it make a short history of what the parabola leaves out.

The same bottom, and a different number of levels #

The two wells are the standard models of a chemical bond. Philip Morse’s, of 1929, is built to be solvable: an exponential wall that rises steeply as the atoms are pushed together and levels off as they are pulled apart, with a closed-form set of levels. The Lennard-Jones well, of 1924, is built to have the right long-range tail: two neutral atoms far apart attract with an energy falling as $1/r^6$, the dispersion attraction that the attraction that needs no charge traces to fluctuating dipoles, and the Lennard-Jones form has that tail built in. Matched in depth and curvature at the bottom, the two wells are almost indistinguishable in the drawing’s lower half.

The levels come from the quantisation condition of the old quantum theory, refined: the action round one cycle of the classical motion must be $(v + \tfrac12)h$, where the half is the quarter cycle each turning point costs. All the physics of the drawing is in one dimensionless number, $\Lambda = \sigma\sqrt{2\mu D}/\hbar$, which compares the well’s size and depth with the quantum scale set by the mass. The drawings use $\Lambda = 100$, which gives a few dozen levels, the right number for a light diatomic molecule.

Near the bottom the two stacks agree level for level, because near the bottom both wells are the same parabola: the lowest level, whose height above the floor is the zero-point energy that cannot be removed, sits at the same place in both to four figures. Near the top they part completely. The Morse well’s exponential closes in fast and holds nineteen levels. The Lennard-Jones well’s tail reaches out a long way at very small energy, and in that long shallow region it fits eight more levels, packed into the last few per cent of the depth. The same curvature and the same depth give different numbers of states.

Why the gaps have to close #

That the gaps shrink to zero at the top is not a detail of either model. It follows from the correspondence between a quantum stack of levels and the classical motion in the same well, which the quantum picture hands back to the old one wherever the levels are many: the gap between neighbouring levels is Planck’s constant divided by the period of the classical oscillation at that energy. A gap is a frequency in disguise.

So the question becomes how long a classical oscillation takes, and the answer at the top of any well is: longer and longer without limit. An atom pulled almost free climbs slowly out along the flat tail of the potential, turns round very far out where it is barely moving, and takes a long time to come back. At exactly the dissociation energy it never comes back, and the period is infinite. The period that depends on the swing finds the same thing for a pendulum, whose period grows without bound as its swing approaches the inverted position; the top of a molecular well plays the part of the pendulum balanced upside down. An infinite period means a zero gap, so every stack of levels in a well of finite depth must close up at the top.

How the period diverges depends on the shape of the top of the well, and that is where the two models part. A Morse well’s exponential tail is short, and its period grows only as one over the square root of the remaining binding energy. A $1/r^6$ tail is long, and an atom climbing it spends much longer far out, so its period grows faster as the top approaches and its gaps close up faster — which is why the Lennard-Jones well packs so many levels into so little energy.

The area under the spacings is the bond #

The Birge–Sponer construction is a piece of bookkeeping. The energy needed to lift the molecule from its lowest level to the dissociation limit is the sum of all the gaps between successive levels on the way up. Plot the gaps against level number and the sum is the area under the plot. If only the first few gaps have been measured, extrapolate.

For the Morse well the extrapolation is exact. Its levels are

$$E_v = \hbar\omega\left(v+\tfrac12\right) - \frac{(\hbar\omega)^2}{4D}\left(v+\tfrac12\right)^2,$$

so the gaps fall in a perfectly straight line, from nearly $\hbar\omega$ at the bottom to zero at the top, and the line through any two of them hits zero at the last level and encloses exactly the dissociation energy. Measure the fundamental and the first overtone, and the whole well is known. That is the method’s appeal, and it is why the coefficient of the squared term — the anharmonicity constant, $\omega_e x_e$ in a spectroscopist’s notation — is quoted for every diatomic molecule: with $\omega_e$ it fixes a Morse well, and the Morse well fixes the depth as $\omega_e^2/4\omega_e x_e$.

For the Lennard-Jones well the extrapolation is wrong, and the drawing shows why. Its lowest gaps fall a little faster than the Morse well’s, so a straight line through them heads for zero sooner, around level fifteen. The true gaps do not; they bend over and trail away in a long tail of tiny steps, and the area under that tail — invisible to anyone who has measured only the first five gaps — is the sixth of the bond energy the line misses. The error has the opposite sign from the one a reader of the oscillator that answers at three times the question might expect from its leaning resonance: it is not that the well is softer than a parabola but that it is softer than a Morse well, in exactly the part of it nobody has measured.

How many levels a straight line needs #

The practical question is how much of a spectrum is enough. The drawing answers it for this well by pretending to be a spectroscopist who has measured the lowest $k$ gaps and fits a line to them. With only the first three, the estimate is eighteen per cent low. It improves slowly: ten gaps still leave a tenth of the bond unaccounted for, and even twenty — three quarters of all the levels there are — leave one and a half per cent. Only when the fitted levels reach into the crowded region near the top does the estimate converge.

Real spectra are usually worse placed than this. Absorption and emission from a molecule’s ground state reach the lowest levels easily and the highest ones with difficulty, because the transitions to them are weak. Birge–Sponer estimates made from the first handful of levels were the main source of dissociation energies for decades, and they were in error by amounts of this size. Iodine, whose visible absorption spectrum generations of physics and chemistry students have fitted with a Birge–Sponer plot, is the standard classroom case, and the manuals for that experiment warn that the line misjudges the top of the well its data do not reach.

The error need not have the sign the model shows. Hydrogen chloride’s infrared spectrum has a fundamental at 2,886 reciprocal centimetres and a first overtone at 5,668, a little short of twice the fundamental; the shortfall gives the anharmonicity constant, about 52, and with the harmonic frequency it fixes a Morse well whose depth is about 5.3 electronvolts. The measured depth is 4.6. For hydrogen chloride the straight line overshoots by about a seventh, because its real well narrows faster near the top than the Morse well matched to its bottom, where the Lennard-Jones well in the drawing widens. Which way a straight line errs is a statement about the part of the well it cannot see, and the only way to know is to see it.

The right variable makes the top straight #

In 1970 Robert LeRoy and Richard Bernstein showed what shape the top of the stack must have, and the answer depends only on the tail. The levels near dissociation are bound very weakly, and a weakly bound molecule spends almost all its time far out, where the potential is only its long-range tail $-C_n/r^n$. Applying the quantisation condition to that tail alone gives

$$D - E_v \propto (v_D - v)^{2n/(n-2)},$$

where $v_D$ is an effective, generally non-integer, level number at which the binding would reach zero. For two neutral atoms, $n = 6$, the exponent is three: the cube root of the binding energy falls in a straight line near the top of the stack. The drawing confirms it. The Lennard-Jones well’s cube-root binding energies are straight at the top, and a line through the top four levels predicts the last level’s binding to within one per cent.

For the Morse well the corresponding variable is the square root, and it is straight everywhere, because an exponential tail behaves in this respect like a tail with $n \to \infty$. The comparison carries the lesson. The shape of the top of the stack is set by the physics at large distances — the dispersion attraction, the $1/r^3$ interaction of two atoms that share an excitation, the $1/r$ of two ions — and not by the chemistry at the bottom. The straight line Birge and Sponer drew was in the wrong variable for every molecule whose tail is not exponential.

This is now how the tops of such stacks are used. Photoassociation spectroscopy of ultracold atoms, since the 1990s, measures precisely the highest few levels of molecules as they are formed from pairs of free atoms; fitted with the LeRoy–Bernstein form, those levels give the dispersion coefficients $C_6$ and the dissociation energy with an accuracy that no extrapolation from the bottom could reach, and they give the atoms’ scattering length, which decides whether an ultracold gas of them is stable.

One well, three stacks of levels #

Isotopes give a check that no single spectrum can. The electrons that make a chemical bond neither know nor care which isotope of each nucleus they are holding, so $\mathrm{H_2}$, HD and $\mathrm{D_2}$ share one potential well to very high accuracy. Their levels differ, because the spacing near the bottom goes as one over the square root of the reduced mass: the heavier molecule has lower levels, more of them, and a lowest level sitting deeper in the well. The energy needed to break the bond from its lowest level is therefore larger for the heavier isotope, although the well is identical.

For hydrogen itself the numbers are measured. $\mathrm{H_2}$ has fifteen bound vibrational levels in its ground state and $\mathrm{D_2}$ twenty-one, and their dissociation energies from the lowest level are 4.478 and 4.556 electronvolts; the difference, 0.078 electronvolts, is almost entirely the difference in zero-point energy, which is the same fact why heating a perfect spring changes nothing relies on when it says that the harmonic part of a well holds no information about its asymmetry. Hydrogen’s dissociation energy is now one of the most precisely known quantities in molecular physics, measured to about one part in a billion, and at that precision it is a test of quantum electrodynamics in a molecule rather than of any model of the bond — every correction, down to the size of the proton, has to be included for theory and experiment to agree.

Each level is also longer than the last #

The crowding of the levels has a companion that a spectrum also records, and it connects them to a property of bulk matter. A molecule on a higher level spends more of its time on the soft, outer side of an asymmetric well, so its average length grows as it climbs. Rotational spectroscopy measures that length directly. Each vibrational level has its own set of rotational levels, spaced in proportion to one over the molecule’s moment of inertia, and the spacing shrinks steadily from one vibrational level to the next. For hydrogen chloride it falls by about three per cent per vibrational level; for hydrogen, whose vibration is large compared with its length, by about five.

That is thermal expansion, one molecule at a time. Why heating a perfect spring changes nothing shows that a solid expands when heated only because its wells are asymmetric, and that a parabola, however vigorously it vibrates, keeps its centre where it was. A molecule’s rotational constants, read level by level, show the same asymmetry at work — the average length growing with every level, by an amount the parabola says should be zero — and the same asymmetry is what closes the gaps at the top. The spacing of the vibrational levels and the stretching of the rotational ones are two readings of one departure from the parabola, made in two different parts of the same spectrum.

Where the picture of levels stops #

A single well and a rigid rotor. Real diatomic levels also rotate, and rotation adds a centrifugal term that raises the effective well and makes the highest levels quasi-bound, trapped behind a barrier through which they can tunnel. A spectrum of a rotating molecule has to be separated into vibration and rotation before any of this applies.

WKB levels. The drawn levels come from the semiclassical quantisation condition. It is exact for the Morse well, which is checked, and accurate to a small fraction of a gap for the Lennard-Jones well except for the very highest level, whose wavefunction reaches into the region where the tail is changing too slowly for the approximation to hold. The fitted numbers in the drawings are statements about WKB levels.

Model wells. Neither well is a real molecule’s. Both are stand-ins, one with the right tail and one with the right solvability, and the size of the Birge–Sponer error depends on which: a well with an ionic $1/r$ tail holds infinitely many levels and defeats any straight-line extrapolation entirely, while one with an exponential tail is extrapolated exactly.

What the stacked levels cannot show #

The drawings show energies and nothing about the molecule at each level. Near the bottom the two atoms vibrate a few per cent of their separation; near the top of the Lennard-Jones well the last few levels are enormous, weakly bound objects in which the atoms spend most of their time many times their equilibrium distance apart — the “halo” states that ultracold experiments create. A picture of the levels as lines drawn across a well gives no sense of that: the lines near the top are as long as the well is wide, and the molecule on them is mostly not where the well is deep.

Still open: how weakly a molecule can be bound #

The last level in a well can be bound by almost nothing. The helium dimer, two helium atoms held by the dispersion attraction alone, has a single bound level whose binding energy is about a millikelvin, a ten-thousandth of the well’s depth, and its average separation is about fifty ångströms, twenty times the distance at the bottom of the well. It was detected in 1994 and imaged directly in 2015. Whether such an object is best described as a molecule at all, and how universal the properties of barely bound states are — for three atoms, weakly bound states arrange themselves in an infinite geometric sequence predicted by Vitaly Efimov, and whether real systems show more than two or three members of that sequence is still being established experimentally — is a live area where the top of the stack has become more interesting than the bottom.

The habit worth carrying away is to ask which part of a system a measurement samples. The bottom of a well decides the frequency and the top decides the depth, and a spectrum carries both only if it reaches the top — and the top of every stack is shaped by the forces at long range, which the bottom never feels.