Vibration and Rotation of a Diatomic Molecule

The Schrodinger equahon for a vibrating and rotating diatomic molecule, or generally for a two-body problem, with p as the reduced mass, is 

With Equation 8.53b, the operator is expressed in spherical polar coordinates and in terms of the angular momentum operator, L, to give 

This differential equation is separable into radial and angular parts, where the angular wavefunctions are the spherical harmonics. That is, if we assume a product form for the eigenfunction, p, 

On substitution of the product form of in Equation 9.20 into the Schrodinger equation, Equation 9.19, we have 

Dividing this equation by the spherical harmonic function, Y , establishes the separability of the Schrodinger equation because the resulting differential equation is only in terms of the variable r. 

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So far we have treated the vibration and rotation of a diatomic molecule as two separate motions. Vibration and rotation occur simultaneously, of course, and transitions between energy levels can involve changes in both vibration and rotation quantum numbers, v and J. The vibration-rotation combination gives rise to observed spectra with not only the expected vibration transitions, but also rotational fine structure around each of the vibrational wavenumbers. Calculation of vibration-rotation... 

Consider the vibration and rotation of a diatomic molecule. Since the molecule is rotating in space, the Hamiltonian is best written in terms of spherical coordinates. The potential V(r) depends only on the separation of the atoms, and it develops from the electrons and the chemical bonding that occurs between the atoms. The Schroedinger equation for a rotating and vibrating diatomic molecule is... 

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