5: Rotational and Vibrational Spectroscopy
- Page ID
- 521300
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 5.1: The Harmonic Oscillator and Infrared Spectra
- This page explains infrared (IR) spectroscopy as a vital tool for identifying molecular structures through absorption patterns. It details the quantum harmonic oscillator model relevant to diatomic molecules, such as hydrogen halides, and the calculation of bond stretching in HCl. The page also covers selection rules for IR transitions that require changes in dipole moment and discusses factors affecting molar absorptivity and transition probabilities.
- 5.2: The Harmonic Oscillator Selection Rule
- This page discusses selection rules in spectroscopy and physical chemistry, which govern the allowed transitions of particles in quantized atomic systems, including electronic, rotational, and vibrational transitions. These rules assist chemists in identifying substances by analyzing light wavelengths that induce transitions, revealing properties like molecular structure and bond strength.
- 5.3: Rotations Accompany Vibrational Transitions
- This page explains the rovibrational spectra of diatomic gas molecules, detailing vibrational and rotational transitions influenced by quantum selection rules and bond length changes. It covers energy quantization, resulting in P- and R-branches, and discusses the rotational constant \(B\) and the Q-branch in spectroscopy.
- 5.4: The Selection Rule for the Rigid Rotor
- This page explains selection rules that govern transition probabilities between quantum levels during the absorption or emission of electromagnetic radiation. It discusses gross and specific selection rules, emphasizing that a molecule must have a permanent dipole moment for rotational spectra and outlines the rule \(\Delta J = \pm 1\) for absorptive transitions. These principles are relevant for both electronic and orbital angular momentum transitions.
- 5.5: Unequal Spacings in Vibration-Rotation Spectra
- This page discusses the differences between real and ideal rovibrational spectra, emphasizing the effects of rotational-vibrational coupling and centrifugal distortion on line spacing in R-branch and P-branch as energy varies. It notes how bond length influences vibrational states and the rotational constant, detailing how the spacing in R-branch decreases with increasing J values, while P-branch spacing increases as J decreases.
- 5.6: Unequal Spacings in Pure Rotational Spectra
- This page discusses the vibrational energy of non-rigid rotators, emphasizing flexible internuclear distances. It explains how centrifugal force necessitates a restoring force, resulting in potential energy. The text derives energy equations, showing differences in energy levels between rigid and non-rigid rotators, and highlights the impact of the centrifugal stretching constant (\(\tilde{D}\)) on energy levels at higher angular momentum, enhancing the accuracy for spectral observations.
- 5.7: Vibrational Overtones
- This page discusses the limitations of the harmonic oscillator model for molecular vibrations, particularly at higher energy levels where anharmonic effects become significant. The anharmonic oscillator model, incorporating higher-order terms, offers more accurate predictions and allows for transitions between various vibrational states, resulting in overtones. Observed frequencies align better with anharmonic models, especially for higher energy levels, leading to weaker intensity lines.
- 5.8: Normal Modes in Polyatomic Molecules
- This page discusses normal modes as independent vibrational motions in molecules, defined by symmetries and their roles in IR and Raman spectroscopy. Molecules have 3N degrees of freedom, with nonlinear and linear molecules having 3N-6 and 3N-5 vibrational degrees, respectively. Each vibrational mode acts as a harmonic oscillator, contributing to energy. Normal coordinates aid in analyzing vibrations by simplifying equations.
- 5.9: Group Theory Determines Infrared Activity
- This page explains how to determine if molecular normal modes are infrared (IR) or Raman active, requiring changes in dipole moment for IR activity and changes in polarizability for Raman activity. It uses water (H2O) as an example to show the application of group theory in identifying active modes. The analysis confirms the vibrational transitions related to different symmetry representations, detailing the IR and Raman active modes along with their energy levels.