Anharmonic vibrations and Morse potentials

Let $x=R-R_e$ and choose the model potential $V(x)=D_e(1-e^{-ax})^2$, with positive depth parameter $D_e$ and inverse-length parameter a. The minimum is zero at x = 0 and the dissociation limit is $D_e$ as x tends to positive infinity. Expanding the exponential near zero gives leading potential $D_ea^2x^2$, so the local spring constant is $k=2D_ea^2$. The vibrational kinetic energy uses nuclear reduced mass, not total molecular mass.

The standard analytic solution treats this stretch coordinate on the full real line. A physical radial coordinate instead has R positive and an appropriate short-distance boundary; the extended Morse model is useful when the relevant bound states do not probe that artificial boundary. It is a model potential, not an exact statement about every diatomic electronic surface. In particular, its large-distance exponential approach does not reproduce every inverse-power molecular tail or coupled dissociation channel.

Spectroscopists often divide energy by h c and report a term value in inverse centimetres. Define $\widetilde\omega_e=\omega/(2\pi c)$ and $\widetilde\omega_ex_e=(\hbar\omega)^2/(4D_ehc)$, using c in centimetres per second for these wavenumbers. The symbol called omega e in a table can therefore be a wavenumber rather than an angular frequency. Unit labels determine which quantity is meant; inserting it directly into a time-domain oscillator equation would be incorrect.

Subtracting neighboring terms gives spacing $\widetilde\omega_e-2\widetilde\omega_ex_e(v+1)$. A positive anharmonic constant makes this spacing decrease with excitation. Only a finite number of normalizable Morse bound states exist, with $v+1/2<2D_e/(\hbar\omega)$; equality at the threshold is excluded. Extending the quadratic term-value formula to arbitrarily large v would eventually turn the sequence downward and create spurious states, so the bound-state condition belongs alongside the energy formula.

If isotope substitution retains the same leading electronic potential, D e and a stay fixed while angular frequency scales as the inverse square root of reduced mass. The anharmonic energy coefficient scales as inverse reduced mass. The zero-point energy therefore decreases for heavier isotopes, increasing D zero even though the electronic well depth has not changed. This distinction is essential when comparing a computed potential minimum with an experimental dissociation threshold.

Real spectra include additional terms, rotation-vibration coupling, and perturbations from other surfaces. A fitted pair of Morse-like constants can summarize low-lying levels without faithfully predicting all near-threshold states. Conversely, an accurate empirical potential may use many parameters rather than the simple Morse form. We will use a deliberately hypothetical set of wavenumbers below so the arithmetic illustrates the model and its consistency conditions, rather than implying measured constants for a particular molecule.