# 7: Rotational States

Molecules rotate as well as vibrate. Transitions between rotational energy levels in molecules generally are found in the far infrared and microwave regions of the electromagnetic spectrum.

• 7.1: Introduction to Rotation
Molecules rotate as well as vibrate. Transitions between rotational energy levels in molecules generally are found in the far infrared and microwave regions of the electromagnetic spectrum. We will see that the magnitude of the molecule's moment of inertia causes rotational transitions in these spectral regions. We also will learn why the lines are nearly equally spaced and vary in intensity. Such spectra can be used to determine bond lengths, and even bond angles in polyatomic molecules.
• 7.2: The Hamiltonian Operator for Rotational Motion
Translational motion can be separated from rotational motion if we specify the position of the center of mass by a vector R, and the positions of each atom relative to the center of mass. Since translational motion and rotational motion are separable, i.e. independent, the translational and rotational energies will add, and the total wavefunction will be a product of a translational function and a rotational function.
• 7.3: Solving the Rigid Rotor Schrödinger Equation
To solve the Schrödinger equation for the rigid rotor, we will separate the variables and form single-variable equations that can be solved independently.
• 7.4: Angular Momentum Operators and Eigenvalues
ngular momentum is a key component in the physical descriptions of rotating systems. It is important because angular momentum, just like energy and linear momentum, must be conserved in any process. Consequently angular momentum is used to derive selection rules for spectroscopic transitions, determine which states of atoms and molecules can be affected by various perturbations, and identify possible and forbidden mechanisms in chemical reactions.
• 7.5: Quantum Mechanical Properties of Rotating Diatomic Molecules
In this section we examine the rotational states for a diatomic molecule by comparing the classical interpretation of the angular momentum vector with the probabilistic interpretation of the angular momentum wavefunctions. We want to answer the following types of questions. How do we describe the orientation of a rotating diatomic molecule in space? Is the molecule actually rotating? What properties of the molecule can be physically observed?
• 7.6: Rotational Spectroscopy of Diatomic Molecules
The permanent electric dipole moments of polar molecules couple to the electric field of electromagnetic radiation to induce transitions between the rotational states of the molecules. The energies that are associated with these transitions are detected in the far infrared and microwave regions of the spectrum. The selection rules for the rotational transitions are derived from the transition moment integral by using the spherical harmonic functions and the appropriate dipole moment operator.
• 7.7: Overview of the Rigid Rotor
We found that the rotational wavefunctions are functions called the Spherical Harmonics, and that these functions are products of Associated Legendre Functions and the eimφeimφ function. Two quantum numbers, J and mJ , are associated with the rotational motion of a diatomic molecule. The quantum numbers identify or specify the particular functions that describe particular rotational states.
• 7.E: Rotational States (Exercises)
Exercises for the "Quantum States of Atoms and Molecules" TextMap by Zielinksi et al.