5: The Rigid Rotor and Rotational Spectroscopy
- Page ID
- 419502
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)One of the most powerful tools for elucidating molecular structure is the analysis of rotationally resolved molecular spectra. These can be observed in the microwave, infrared, and visible/ultraviolet regions of the spectrum. The rigid rotor (or rigid rotator) problem provides the idealized model that chemists use to describe the rotational motion of a molecule. In this chapter, we will explore the quantum mechanical model of a rotating body, and apply the results to lay the foundation for an understanding of the rotational structure in molecular spectra. We’ll look at the shortcomings of the model when applying it to real molecules (which as we saw in the previous chapter, do not have rigid bonds!) and apply these results to the interpretation of pure rotational spectra (generally found in the microwave region of the spectrum) and rotationvibration spectra (accounting for the rotational structure that is observed in infrared spectra of molecules.)
- 5.1: Spherical Polar Coordinates
- The description of a rotating molecule in Cartesian coordinates would be very cumbersome. The problem is actually much easier to solve in spherical polar coordinates.
- 5.2: Potential Energy and the Hamiltonian
- Since there is no energy barrier to rotation, there is no potential energy involved in the rotation of a molecule. All of the energy is kinetic energy.
- 5.3: Solution to the Schrödinger Equation
- The time-independent Schrödinger equation can be written as follows.
- 5.4: Spherical Harmonics
- The solutions to rigid rotor Hamiltonian are very important in a number of areas in chemistry and physics. The eigenfunctions are known as the spherical harmonics and they appear in every problem that has spherical symmetry.
- 5.5: Angular Momentum
- The Spherical Harmonics are involved in a number of problems where angular momentum is important (including the Rigid Rotor problem, the H-atom problem and anything else where spherical symmetry is involved.)
- 5.6: Application to the Rotation of Real Molecules
- While the spherical harmonics are the wavefunctions that describe the rotational motion of a rigid rotator, the names of the quantum numbers are changed to reflect the type of angular momentum encountered in the problem.
- 5.7: Spectroscopy
- The experimental determination of spectroscopic rotational constants provides a very precise set of data describing molecular structure. To see how experimental measurements inform the determination of molecular structure, let’s examine what is to be expected in the pure rotational spectrum of a molecule first.
Thumbnail: The rigid rotor model for a diatomic molecule. (CC BY-SA 3.0 Unported; Mysterioso via Wikipedia)