Rotational SpectraIncident electromagnetic waves can excite the rotational levels of molecules provided they have an electric dipole moment. The electromagnetic field exerts a torque on the molecule. The spectra for rotational transitions of molecules is typically in the microwave region of the electromagnetic spectrum. The rotational energies for rigid molecules can be found with the aid of the Shrodinger equation. The diatomic molecule can serve as an example of how the determined moments of inertia can be used to calculate bond lengths.

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Rotational EnergiesThe classical energy of a freely rotating molecule can be expressed as rotational kinetic energy where x, y, and z are the principal axes of rotation and I_{x} represents the moment of inertia about the xaxis, etc. In terms of the angular momenta about the principal axes, the expression becomes The formation of the Hamiltonian for a freely rotating molecule is accomplished by simply replacing the angular momenta with the corresponding quantum mechanical operators.

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Diatomic MoleculesFor a diatomic molecule the rotational energy is obtained from the Schrodinger equation with the Hamiltonian expressed in terms of the angular momentum operator.
where J is the rotational angular momentum quantum number and I is the moment of inertia.

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Rotational Transitions, DiatomicFor a rigid rotor diatomic molecule, the selection rules for rotational transitions are ΔJ = +/1, ΔM_{J} = 0 .

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Moment of Inertia, Diatomic

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