H³⁵Cl: the well, its states and their averages

One of the figures on rotation and vibration: Moments of inertia, normal modes, isotope shifts, and the frequencies a force field does and does not fix.

Five 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.

The bond length that depends on the isotope

Hydrogen chloride’s potential, built from its measured vibrational constants, with its four lowest states at their computed energies and the average separation in each marked. Every average lies to the right of the minimum, and they move outward as the state rises: 15.30 mÅ in the ground state and 113.85 in the third.

Deuterium chloride’s well, which is hydrogen chloride’s well: the same curve with a heavier particle in it. Its levels are closer together and lower, so its averages sit nearer the minimum — the whole isotope effect in one picture.

Four diatomics with the minimum of the potential, the average separation in the ground state, and the length implied by ⟨1/r²⟩ — which is what a rotational constant measures. For the hydrides the three span more than a hundredth of an ångström.

The width of a band is a bond length

The vibrational states of a Morse potential. Every intensity in this essay is an overlap between one of these on one curve and one on another, and the fact that a state has a spread rather than a position is what makes the overlaps non-zero at all.

The term a harmonic field cannot produce

The two contributions to the moment a rotational constant reports, in the ground state, for four diatomics. A harmonic force field can produce the middle bar and nothing else.

Each term, the two-term series, and the exact change in the reported moment, for each molecule’s ground state.

The ratio of the two terms against the anharmonicity constant, as a fraction of the fundamental. It falls as the well gets more anharmonic.

The cubic a Morse curve guesses

The Morse curve’s $\alpha_e$ as a fraction of the measured value for four diatomics. Every bar is below one.

How far the averaged $\alpha_e$ sits from the closed form for the same Morse curve, and how far it sits from the measurement, for each molecule.

The cubic Dunham coefficient each molecule’s measured $\alpha_e$ implies, beside the one its Morse curve implies. The measurement asks for a steeper cubic in every case.

Two coefficients are not a potential

Each molecule’s $\alpha_e$ as a fraction of the measured value, from its Morse curve and from a quartic potential carrying the measured cubic and quartic coefficients.

Each molecule’s Morse curve with its series stopped after the $x^{4}$, $x^{6}$ and $x^{8}$ terms, and complete, with $\alpha_e$ as a fraction of the complete curve’s.

For each molecule, the Morse curve’s shortfall in $\alpha_e$ and the change in its own $\alpha_e$ when its series is stopped after the quartic, both as percentages of the measured $\alpha_e$.