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Harmonic Vibrational Motion This Schrödinger equation forms the basis for our thinking about bond stretching and angle bending vibrations as well as collective phonon motions in solids The radial motion of a diatomic molecule in its lowest (J=0) rotational level can be described by the following Schrödinger equation: - h2/2µ r-2∂/∂r (r2∂/∂r) ψ +V(r) ψ = E ψ, where µ is the reduced mass µ = m1m2/(m1+m2) of the two atoms. By substituting ψ= F(r)/r into this equation, one obtains an equation for F(r) in which the differential operators appear to be less complicated: - h2/2µ d2F/dr2 + V(r) F = E F.

The coupling between rotational and vibrational motion gives rise to rotational B constants that depend on vibrational state as well as dynamical couplings,called centrifugal distortions, that cause the total ro-vibrational energy of the molecule to depend on rotational and vibrational quantum numbers in a non-separable manner. 4. Harmonic Vibrational Motion This Schrödinger equation forms the basis for our thinking about bond stretching and angle bending vibrations as well as collective phonon motions in solids The radial motion of a diatomic molecule in its lowest (J=0) rotational level can be described by the following Schrödinger equation: - h2/2µ r-2∂/∂r (r2∂/∂r) ψ +V(r) ψ = E ψ, where µ is the reduced mass µ = m1m2/(m1+m2) of the two atoms.

Although He is indeed a function of the R-variables, it is not a differential operator involving them. Because He is a Hermitian operator in r-space, its eigenfunctions Ψ i (r|R) obey He Ψ i (r|R) = Ei (R) Ψ i (r|R) for any values of the R-variables, and form a complete set of functions of r for any values of R. These eigenfunctions and their eigenvalues Ei (R) depend on R only because the potentials appearing in He depend on R. The Ψ i and Ei are the electronic wavefunctions and electronic energies whose evaluations are treated in the next three Chapters.

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