Molecular rotational energy levels

In addition to the vibrational degrees of freedom, molecules also have rotational degrees of freedom, which lead to discrete energy levels. Consider a molecule which has moments of inertia Ia, Ib, Ic about its three principal axes a, b, c, which are aligned along lines of rotational or reflectional symmetry of the molecule, where the moment of inertia is defined as

By convention the axes are ordered such that Ia < Ib < Ic. For the molecule to have a rotational degree of freedom about a certain axis, the moment of inertia must be nonzero. Hence, linear molecules, for which the moment of inertia is zero along one axis, have two degrees of rotational freedom, while nonlinear molecules, for which the moment of inertia is nonzero along all three axes, have three degrees of rotational energy. Depending on their moments of inertia, molecules may be categorized as linear rotors, symmetric rotors (oblate or prolate), asymmetric rotors, and spherical tops, as defined in Table 6.1 and discussed below. From quantum mechanics, the square of the total angular momentum is quantized as

where J is an integer quantum number and the allowed components of angular momentum along a symmetry axis are given by

where K is a quantum number that satisfies |K| < J.

Table 6.1. Symmetry classifications of molecules relevant to giant planets.

Name

Definition

Examples

Linear rotor

Ia = 0, Ib = Ic

CO2, C2H2 (acetylene)

Symmetric rotor or top

Prolate: Ia = Ia, Ib = Ic = Ib Oblate: Ia = Ic, Ia = Ib = Ib

CH3C2H (methyl acetylene),

NH3, AsH3, ph3

Asymmetric rotor or top

Ia < Ib < Ic

H2O, O3, C3H8 (propane), H2S

Spherical top

Ia = Ib = Ic

CH4, GeH4

NB: Diatomic molecules are by definition linear rotors.

L2

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