Energy Levels of a Morse Oscillator

The Morse function V(R)=(Subscript[D, e])((e^(-2a(R-(Subscript[R, e]))))-2(e^(-a(R-(Subscript[R, e]))))), where R is the internuclear distance, provides a useful approximation for the potential energy of a diatomic molecule. It is superior to the harmonic oscillator model in that it can account for anharmonicity and bond dissociation. The relevant experimental parameters are the dissociation energy (Subscript[D, e]) and the fundamental vibrational frequency (Subscript[ω, e]), both conventionally expressed in wavenumbers ((cm^(-1))), the equilibrium internuclear distance (Subscript[R, e]) in Angstrom units (Å), and the reduced mass μ=(Subscript[m, 1])(Subscript[m, 2])/((Subscript[m, 1])+(Subscript[m, 2])) in atomic mass units (amu). The exponential parameter is given by a=(Subscript[ω, e])(Sqrt[μ/2(Subscript[D, e])]) in appropriate units. The Schrödinger equation for the Morse oscillator is exactly solvable, giving the vibrational eigenvalues (Subscript[ϵ, v])=(Subscript[ω, e])(v+(1/2))-(((Subscript[ω, e])^2)/(4(Subscript[D, e])))(((v+(1/2)))^2), for v=0,1,2,...,(Subscript[v, max]). Unlike the harmonic oscillator, the Morse potential has a finite number of bound vibrational levels with (Subscript[v, max])≈2(Subscript[D, e])/(Subscript[ω, e]).

THINGS TO TRY

MouseOver

DETAILS

Snapshot 1: vibrational states of (Subscript[H, 2]) molecule

Snapshot 2: HCl molecule

Snapshot 3: HI molecule

Reference: P. M. Morse, "Diatomic Molecules According to the Wave Mechanics. II. Vibrational Levels," Phys. Rev., 34(1), 1929 pp. 57–64.

RELATED LINKS

Contributed by: S. M. Blinder

© The Wolfram Demonstrations Project & Contributors   |   Terms of Use