Incident 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.
The illustration at left shows some perspective about the nature of rotational transitions. The diagram shows a portion of the potential diagram for a stable electronic state of a diatomic molecule. That electronic state will have several vibrational states associated with it, so that vibrational spectra can be observed. Most commonly, rotational transitions which are associated with the ground vibrational state are observed.
The 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 Ix represents the moment of inertia about the x-axis, 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.
IndexFor 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.
Determining the rotational constant B
enables you to calculate the bond length R. The allowed transitions for the diatomic molecule are regularly spaced at interval 2B. The measurement and identification of one spectral line allows one to calculate the moment of inertia and then the bond length.
For a rigid rotor diatomic molecule, the selection rules for rotational transitions are ΔJ = +/-1, ΔMJ = 0 .
The moment of inertia about the center of mass is
From the center of mass definition
and substituting to eliminate r1 and r2 gives where μ is called the "reduced mass."