The bond length that depends on the isotope

Every drawn structure is a set of points, and the obvious objection is that the atoms are not at them. They are spread out by the zero-point motion, and water’s hydrogens are nearly a tenth of an ångström from where they are drawn.

The zero-point spread is usually computed from fitted force fields, which are harmonic — and a harmonic well is symmetric, so a harmonic spread is symmetric about the equilibrium value and moves the average nowhere. The atoms are smeared out and the mean stays put.

A real potential is not symmetric. It rises steeply when the atoms are pushed together and levels off as they are pulled apart, so the average separation is longer than the equilibrium one — and by an amount that depends on the mass.

Three lengths, one potential #

The consequence is that a molecule does not have a bond length. It has several, and which one a number refers to depends on how it was obtained.

rₑ is the minimum of the potential. It is the number a quantum chemistry calculation reports, it is the only one that is a property of the potential alone, and no experiment measures it directly.

r₀ is the average separation in the ground vibrational state, ⟨r⟩. It is the nearest thing to what an electron diffraction experiment sees, and it is 15.30 mÅ longer than rₑ for hydrogen chloride.

The rotational length comes from a rotational constant, and a rotational constant is proportional to ⟨1/r²⟩ rather than to ⟨r⟩ — so it gives a third number, 1.28299 Å, between the other two. A bond length out of a spectrum is this quantity, and the essay that derives it is careful to call the result an effective length.

Fifteen milliångström is not a rounding. It is larger than the difference between many pairs of chemically distinct bonds, and it is far larger than the precision either kind of measurement claims.

The isotope, which is the sharp case #

The clean demonstration is the one where nothing about the electrons changes.

Substituting deuterium for hydrogen changes no electronic property whatever: the potential is a function of the nuclear positions and of the electrons’ solution to them, and the mass of a nucleus does not enter it. So the two molecules have the same potential, and the measurement confirms it — the two quoted equilibrium lengths differ by 0.03 mÅ, which is the accuracy of the measurement rather than a difference.

The two molecules do not have the same bond length in any measured sense. The heavier one has a lower zero-point energy, sits further down a well whose walls are not symmetric, and has an average separation 4.34 mÅ shorter.

That difference is real, is measured, and cannot be explained by any argument about bonding. Nothing about deuterium’s bonding differs from hydrogen’s, and its bond is shorter.

What the shift is made of #

The size of the shift follows from the shape of the well, and the scaling can be checked rather than described.

A Morse potential has one parameter controlling how asymmetric it is, and that parameter goes as the square root of the anharmonicity constant ωₑxₑ — which is measured, from the spacing of the vibrational levels. The shift in the average separation goes as the same parameter.

So dividing the anharmonicity by four hundred should divide the shift by twenty, and it does. That is checked directly, and it is worth having because it is a scaling law rather than a limit: a check that the effect is the anharmonicity and not the solver, made without ever setting the anharmonicity to zero.

The mass enters through the zero-point energy. A heavier reduced mass gives a lower ground state, which samples a narrower region of the well, which is less asymmetric — so the shift falls. Carbon monoxide, with a reduced mass seven times hydrogen chloride’s, shifts by 3.95 mÅ against 15.30.

The zero-point spread of water’s coordinates comes out of its fitted harmonic force field, and that spread is symmetric about the equilibrium value by construction — a harmonic potential has no way to be lopsided. The lengths in this essay differ precisely because the real potential is not harmonic, so the spread is not symmetric and the average is not the minimum.

The spread and the shift are different sizes #

One number in the table deserves separating out, because it is the one that connects the mean to the spread and the two are easily confused.

Hydrogen chloride’s zero-point spread — the root mean square width of the ground state — is 76.8 mÅ. Its shift is 15.30. So the bond wanders over a range five times larger than the amount by which its average is displaced.

Those two numbers answer different questions and both are needed.

The spread says how badly a fixed set of coordinates describes the molecule: at any instant the bond is somewhere within about a twentieth of an ångström of anywhere, and no single number describes where the atoms are. That is the argument that the atoms are not at the points and it applies to a harmonic well as much as to a real one.

The shift says how badly the average describes the minimum, and it exists only because the well is anharmonic. A harmonic molecule would have the full spread and no shift at all — smeared out, and centred exactly where it was drawn.

The practical difference is which errors they cause. A spread makes a measurement noisy in a way that is symmetric and averages away over many observations. A shift is systematic: every measurement of every molecule of the sample is displaced the same way, and no amount of averaging removes it.

The heavier isotope in every column #

The isotope comparison holds for every quantity in the table and it is worth reading down the two rows rather than only across.

Deuterium chloride’s average separation is shorter, its spread is smaller — 64.8 mÅ against 76.8 — and its rotational length is nearer its equilibrium value. All three follow from one fact: a heavier reduced mass gives a lower zero-point energy, and a state lower in the well samples less of it.

That produces a rule of thumb worth having: the heavier isotopologue is always the more classical one. It sits deeper, moves less, and its measured structure is closer to the structure a calculation reports. In the limit of infinite mass every one of these differences goes to zero and the molecule sits at the bottom of its well, which is the structure every drawing in this collection is.

So the isotope effect on a bond length is not merely a curiosity that demonstrates the zero-point motion. It is the one experimental handle on how far a measured structure is from the calculated one, available on any molecule with a substitutable hydrogen, and it costs one extra spectrum.

What was computed, and what is quoted #

Quoted: ωₑ, ωₑxₑ and rₑ for each molecule, from spectroscopy, and the isotopic masses. Nothing is fitted here.

Computed: the Morse parameters that reproduce those constants, the potential, the vibrational states by finite differences on a four-hundred-point grid, and the expectation values ⟨r⟩ and ⟨1/r²⟩ in each state.

Checked: the computed levels against the exact Morse spectrum, which is the reason for choosing that potential rather than a more realistic one — a numerical solution needs something to be checked against that is not another numerical solution. The agreement is within two per cent of the level spacing for the four lowest states, and the residue is the finite-difference error.

Also checked: that every average exceeds the equilibrium separation, that the excess grows with the state, that the shift scales as the square root of the anharmonicity, and that the two isotopes’ equilibrium lengths are identical while their averages are not.

What this model cannot do #

One dimension. A diatomic has one internal coordinate and everything above is exact for the potential given. A polyatomic has 3N−6, its anharmonicity is a tensor of third derivatives, and the shift in each coordinate depends on the others — which is why vibrational averaging in a polyatomic is a serious calculation and not a scaling law.

The Morse form is a choice. It has the right shape and an exact spectrum, and it is wrong in detail: real potentials are steeper at short range and have a different long-range form. What survives is the sign of the effect and its rough size, both of which follow from the well being steeper inward than outward.

No rotation–vibration coupling. The rotational constant of a vibrating molecule is not ⟨1/r²⟩ of a non-rotating one, and the correction — the α constant of a spectroscopist — is the same anharmonicity acting through the centrifugal term. The third column above is therefore an illustration of the distinction rather than a prediction of a measured constant.

And the potential is not measured directly. ωₑ and ωₑxₑ are extracted by fitting a Morse-like expression to observed levels, so the constants that build the potential were obtained assuming something close to it. The circularity is mild and it is not nothing.

The same substitution seen in a vibrational spectrum is a different kind of statement: every frequency falls by a factor the masses fix exactly. That shift is arithmetic and this essay’s is not — the frequencies are fixed by the masses and the force field, while the bond length is fixed by where in an anharmonic well the amplitude puts the average.

What follows for reading a structure #

Three practical statements, and they are the reason this belongs among things taught confidently and wrongly.

A calculated length and a measured length are different quantities. A calculation reports rₑ and an experiment reports r₀ or something like it, so a computed bond that is 0.01 Å shorter than the measured one may be exactly right. Comparing them without saying which is which is comparing two things that differ by more than the error being discussed.

An isotope effect on a bond length is not a bonding effect. It is a zero-point effect, and it is the standard way of demonstrating that a measured length is a vibrational average — because there is nothing else it could be.

A structure drawn as points is a set of rₑ values or a set of r₀ values and never both. The atoms are not at the points makes the case about the spread; this essay makes it about the mean, and the mean is the number everybody quotes.

Hydrogen fluoride, and why it is the worst case #

The fourth row of the table is the extreme, and the reason is worth stating because it identifies which molecules the whole difficulty is about.

Hydrogen fluoride’s equilibrium bond is 0.91680 Å and its ground-state average is 0.93143 — a shift of 14.63 mÅ on a bond that is only nine tenths of an ångström long. As a fraction that is 1.6 per cent, against hydrogen chloride’s 1.2 and carbon monoxide’s 0.35.

Two things make a molecule bad in this respect and hydrogen fluoride has both: a light reduced mass, so the zero-point energy is high, and a large anharmonicity, so the well is strongly asymmetric where that energy puts it. The two are related — a light molecule has widely spaced levels and reaches further up its well — which is why the hydrides dominate the top of any list of this kind.

Carbon monoxide is the opposite case and is instructive. Its reduced mass is seven times larger and its anharmonicity constant is a quarter of hydrogen chloride’s, so its shift is a quarter as large and its spread is less than half. A structure of a molecule made only of heavy atoms is nearly the structure at the bottom of its well.

So the rule for reading a table of bond lengths is a rule about hydrogen: a bond to hydrogen is where the equilibrium and the average part company, and a molecule of heavy atoms is where they nearly agree. Which is unfortunate, because bonds to hydrogen are the ones a structure determination is least able to locate in the first place.

Why the effect is not a correction #

A last distinction, because the shift is often described as a correction to be subtracted.

A correction implies a true value being approached. There is one here — the minimum of the potential — and it is not what any experiment measures, so subtracting the shift from a measured length would produce a number that no measurement returns and that a calculation already gives directly.

The useful framing is the other way round. The equilibrium length is a property of the potential; the average is a property of a state of the molecule; and a molecule is always in a state. So the average is the physical quantity and the minimum is the calculated one, and the fifteen milliångström between them is not an error in either.

What follows is a rule for comparison rather than for correction: compare like with like. A calculated structure belongs beside another calculated structure, a measured one beside another measured one obtained the same way, and a comparison across the boundary needs the vibrational average computed — which is exactly the calculation done here.

The fourth length, and the one rule that generates all of them #

Three lengths were computed above because three quantities were evaluated. There is no reason to stop at three, and stopping there hides how simply the whole family is related.

Every experimental length is an average of some power of the separation, and every technique picks its own power because of what it physically responds to. A diffraction experiment scatters off pairs of nuclei and its intensity carries a factor of $1/r$, so what it returns is governed by $\langle 1/r \rangle$; a rotational constant is an inverse moment of inertia and so is governed by $\langle 1/r^2 \rangle$; an average separation is $\langle r \rangle$ itself. Three powers, three numbers, one distribution.

The relation between them needs no new calculation. Expanding each average about the mean of the ground-state distribution, and writing $\sigma$ for the spread computed above,

$$\frac{1}{\langle 1/r \rangle} \approx \langle r \rangle \left(1 - \frac{\sigma^2}{\langle r \rangle^2}\right), \qquad \langle 1/r^2 \rangle^{-1/2} \approx \langle r \rangle \left(1 - \frac{3\sigma^2}{2\langle r \rangle^2}\right).$$

So the three lengths are the same average read through coefficients of $0$, $1$ and $\tfrac{3}{2}$ on one quantity, $\sigma^2/\langle r \rangle$ — and that quantity is the square of the zero-point spread, divided by the bond. Everything separating the experimental lengths from one another comes out of the width of the distribution; only the separation from $r_e$ comes out of its asymmetry.

The numbers close the loop. For hydrogen chloride the spread is 76.8 mÅ against an average separation of 1.28985 Å, so $\sigma^2/\langle r \rangle$ is 4.57 mÅ. Subtracting it once gives 1.28528 Å, which is the diffraction-weighted length; subtracting one and a half times gives 1.28299 Å, which is the rotational length in the table above, arrived at here from a spread rather than from an expectation value and agreeing to the last digit quoted.

That agreement is worth more than the number. The rotational length was computed by integrating $\langle 1/r^2 \rangle$ over a numerically solved Morse state; the same figure now falls out of two moments and a Taylor expansion, which is an independent route to it. A quantity obtained twice by different means is a quantity, and one obtained once is a solver’s output.

So the sequence of lengths for hydrogen chloride runs $r_e = 1.27455$, then the rotational 1.28299, then the diffraction 1.28528, then $\langle r \rangle = 1.28985$ — four numbers spanning 15.3 mÅ, of which the last three span only 6.9. The gap from the potential’s minimum to any of them is twice the gap between all the measured ones, which is the practical form of this essay’s argument: the disagreements among experiments are small beside the disagreement between every experiment and the calculation.

Still open: polyatomic shapes, and temperature #

The natural open question is the polyatomic case, where the shift is not a number but a pattern: every coordinate’s average moves, and the moves are coupled, so the shape of a molecule changes with its vibrational state as well as its size. A famous consequence is that some molecules are bent at their potential minimum and effectively linear on average, and the two descriptions are both correct about different quantities.

The nearer question is what happens to the family of lengths above once temperature is admitted. Every average here is over the ground state, and a sample at three hundred kelvin has a population in the first excited state whose averages sit further out again — so each of the four numbers becomes a temperature-weighted mean over its own levels, and two techniques run at different temperatures return different lengths for the same quantity. That is why the literature’s count of distinct bond lengths is nearer six than four, and why several of them carry a temperature in their definition.