Diatomic molecules — Morse potential

The vibrational motion of a diatomic molecule in the Morse potential \(V(r)=D(1-e^{-a(r-r_e)})^2\): an exponential change of variable leads to Kummer's equation, with a finite number of bound states and dissociation. The spectrum \(E_v\) reproduces the real vibrational series (anharmonicity \(\omega_e x_e\)). Finite-difference computation in the Python backend (via gw2py).

Like hydrogen and the harmonic oscillator, Morse lives on the confluent branch \(_1F_1\). The qualitative difference is dissociation: above \(D\) the spectrum is continuous, so the bound states are finite in number.

Internuclear coordinate \(r\); reduced mass and \(\hbar\) set to 1.

Morse potential

\[ V(r)=D\big(1-e^{-a(r-r_e)}\big)^2, \]

anharmonic, with a minimum at \(r_e\) and asymptote \(D\) (dissociation energy).

Reduction to Kummer

With the variable \(z=2\lambda\,e^{-a(r-r_e)}\), \(\lambda=\sqrt{2D}/a\), the equation becomes Kummer's; the bound solutions correspond to truncated \(_1F_1\) (Laguerre polynomials in \(z\)).

Closed-form vibrational spectrum: \[ E_v=\omega\big(v+\tfrac12\big)-\frac{\big[\omega(v+\tfrac12)\big]^2}{4D},\qquad \omega=a\sqrt{2D}, \] with a finite number of bound states, \(v=0,\dots,\lfloor\lambda-\tfrac12\rfloor\).

Spectroscopy

The form \(E_v=\omega_e(v+\tfrac12)-\omega_e x_e(v+\tfrac12)^2\) is exactly the series of real vibrational levels (H\(_2\), HCl, CO): \(\omega_e x_e=\omega^2/4D\) is the anharmonicity constant.

State to represent

State type Vibrational level \(v_1\)

Level \(v_2\)

Level spacing decreases with \(v\) (anharmonicity).

Potential parameters

Well depth \(D\) = 10 Width \(a\) = 0.6 Equilibrium distance \(r_e\) = 3 \(r_{\max}\) = 18

Confluent branch, but finite

Like hydrogen and the harmonic oscillator, Morse lives on the \(_1F_1\) branch. The qualitative difference is dissociation: above \(D\) the spectrum is continuous, so bound states are finite in number — a feature the harmonic oscillator (infinite well) does not have.

Shape invariance

Morse is a shape-invariant potential in supersymmetric quantum mechanics: it is this algebraic property that guarantees its exact solvability, as for hydrogen and Pöschl–Teller.

Computational evidence

With \(D=8,\ a=0.5,\ r_e=3\) the engine reproduces \(E_v\) of the closed form to \(\sim2\times10^{-4}\) (\(E_0=0.9687\) vs \(0.96875\)); the count of bound states matches \(\lfloor\lambda-\tfrac12\rfloor\).

References

P. M. Morse, «Diatomic Molecules According to the Wave Mechanics. II», Phys. Rev. 34, 57 (1929). doi.
NIST DLMF, ch. 13. dlmf.nist.gov/13.
F. Cooper, A. Khare, U. Sukhatme, Phys. Rep. 251, 267 (1995). doi.
G. Herzberg, Molecular Spectra and Molecular Structure I, Van Nostrand, 1950.

WebNIR · CNR-IFAC | demo interface — numerical work is provided by the Python backend (gw2py).

Keywords: Morse, diatomic molecule, vibration, anharmonicity, dissociation, confluent hypergeometric, Kummer, spectroscopy, quantum mechanics