An undergraduate-oriented comment about inverting spectral data to determine the interatomic potential

For nearly fifty years, I started each of my courses preaching about the benefits of self-study ahead of class, referring to the introduction of book printing by Gutenberg as the point in history when the role of the professor should have transformed from a knowledge provider to guide in an adventurous intellectual pursuit. For the majority of my students, these sermons remained futile.

The recent abrupt need to switch to remote teaching/learning allowed a radical transformation of my teacher–student mode of interaction, remotely teaching an elective undergraduate Molecular Spectroscopy course offered to second- and third-year Materials Science and Engineering students who had completed a one-semester quantum chemistry course at the Guangdong-Technion Israel Institute of Technology in Shantou, Guangdong, China. Rather than mimicking traditional lecturing via live (“zoom”) or video-recorded classes, the students were offered a (closed) website (“moodle”) where they could find reading materials, references to appropriate chapters in their course textbook,¹ and exercises, along with a detailed learning guide accompanying each chapter. The students were promised credit for “good” questions and comments, and I made an effort to respond promptly (synchronizing my day with China time—six hours ahead of Israel), loading crudely edited questions and responses on the course website. Indeed, a steady flow of enquiries required my attention, providing me with feedback that I have never before benefitted from about the students' progress and their difficulties.

The Rydberg–Klein–Rees (RKR) procedure for inverting vibration-rotation data to obtain the potential energy function of a diatomic molecule is briefly discussed in the textbook used in the course mentioned above¹ (p. 216). The non-uniqueness of the inversion of vibrational-only data into a potential function intrigued the curiosity of one of the students (see acknowledgements). I quickly offered an appropriate source,² supplemented by an elementary introduction to the WKB (semiclassical) approximation, which the RKR procedure is based on. The purpose of the present comment is to illustrate that, in a particularly simple context, vibrational energy levels are not sufficient to determine the potential. In more fancy language, to illustrate the existence of isospectral potentials (e.g., Ref. 3).

We will only consider potentials, $V(x), −∞<x<∞$⁠, with a single minimum at some x_e, i.e., such that V(x) is monotonically decreasing over $−∞<x< x e$ and monotonically increasing over $x e<x<∞$⁠. For energies that satisfy $E≥V( x e)$⁠, we will refer to the two roots of the equation V(x) = E as the left- and right classical turning points, $x ℓ$ and x_r. If $lim x → ± ∞ V ( x ) = V ± ∞ < ∞$⁠, then we define $x ℓ = − ∞$ for $E ≥ V − ∞$ and $x r=∞$ for $E≥ V ∞$⁠.

The following definitions and facts are provided:

1. The classical periods of oscillation in a one-dimensional harmonic (parabolic) potential do not depend on the energy (or amplitude) of the oscillation. Potentials that satisfy this property are referred to as isochronous.

2. Two potentials, $V ( 1 )(x)$ and $V ( 2 )(x)$⁠, which have the same periods of oscillation at all energies, are referred to as isoperiodic.

3. Two potentials that have the same distance between the right and the left classical turning points $x ℓ(E)− x r(E)$⁠, at all energies, are referred to as shear equivalent. Shear equivalent potentials are isoperiodic.^4,5

4. Potentials that are shear-equivalent to a parabola are isochronous.^6,7 A much broader view of isochronous systems was offered by Calogero.⁸ 

5. A potential, which is continuous with continuous first and second derivatives (a C² potential), is isochronous if and only if it is shear-equivalent to a parabola.^9,10 However, singular isochronous potentials have been reported.¹¹ 

6. Shear-equivalent potentials have equal Bohr–Sommerfeld phase integrals, $∮ pdx$⁠, at all energies. It follows that such potentials have the same EBK (or lowest order WKB) spectra, except in limiting cases in which the Maslow index has to be modified, as illustrated below.

7. Shear equivalent potentials need not share a common quantum spectrum.¹² 

In the RKR procedure (which is based on the semiclassical quantization), the vibrational spectrum provides the distances between the left- and right “classical turning points” (⁠ $R ℓ$ and R_r, respectively) at the energies of the available vibrations. This is not enough to construct the potential. The task of determining the potential is completed by examining the rotational levels associated with each vibrational level. They provide a distinct moment of inertia at each vibrational level, from which a “bond-length” is derived, which is (naively) assumed to be equal to $R ℓ + R r/ 2$⁠.

What about quantum mechanics?

That was easy! But what about the second potential?

Well, semiclassically, it ought to be isospectral. For the harmonic oscillator, the semiclassical treatment yields the same spectrum of equidistant levels as the solution of the Schrodinger equation (except that in the semiclassical treatment, the energy is given in the form of $ℏωn$—the form originally postulated by Planck—rather than $ℏω n + 1 2$⁠, unless we invoke the Maslow correction). So one might guess that the same would hold true for $V ( 2 )(x)$⁠.

So $V ( 2 )(x)$⁠, just like $V ( 1 )(x)$⁠, has a spectrum of equidistant levels, separated by $ℏω$⁠. To make these two potentials fully isospectral, we need to pull $V ( 2 )(x)$ down by $ℏ ω/ 4$⁠, but this does not affect the wavefunctions or anything that depends on them.

We note (without attempting an interpretation) that these electric dipole matrix elements rise with n somewhat more slowly than those for the symmetric harmonic potential, $V ( 1 )$⁠.

The comparison between the harmonic oscillator and the half-oscillator spectra provides an elementary illustration of isospectrality. Examination of the diagonal and the off-diagonal dipole matrix elements suggests that the vibration-rotation spectra as well as the vibrational transition probabilities can be used to distinguish between such isospectral potentials.

The author's “hidden agenda” is to illustrate that the transition to remote teaching/learning imposed by the Covid-19 pandemic can possibly revolutionize the professor's role from an information provider to a guide in an intellectual pursuit. This requires an intensively involved teaching style that university administrators who look for cheaper mass-teaching technologies will not be happy with, as well as curious and committed students.

The author wish to thank Mr. Zhixiang YAO, second year MSE student at GTIIT, for his curiosity, which inspired the present paper. Professor Ed Montgomery kindly read the paper and provided helpful advice.

This appendix is presented as an opportunity to introduce a somewhat richer application of generating function ideas than are usually offered to undergraduates.

The various matrix elements reported above were calculated by explicit integration (over $0≤x<∞$⁠) of appropriate expressions in terms of harmonic oscillator wave functions.

The old-fashioned application of such generating functions involved somewhat tedious expansion of the generating function $π/ 2 exp (2st)(1+erf(s+t))$⁠, a task that a symbolic programming platform such as maple readily performs. We note that the expansion contains no terms with $n 1≠ n 2$ when they are both odd (since they correspond to distinct eigenfunctions of the “half-oscillator” Hamiltonian), but it does contain spurious terms with either n₁ or n₂ or both being even.