14.S: Nuclear Magnetic Resonance Spectroscopy (Summary)
- Page ID
- 550569
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Up until this point, we’ve been looking at spectroscopic transitions that can occur when the internal states of a molecule interact with the electric component of electromagnetic radiation. In general, these are the strongest and most readily observed spectroscopic transitions for a molecule in free space, but there are a number of other types of spectroscopic transitions that are observable. In this section, we will look at nuclear magnetic resonance spectroscopy: nuclear spin transitions that are observable when a molecule is placed in a magnetic field. Nuclear spin angular momentum is perfectly analogous to electron spin angular momentum: some nuclei possess intrinsic quantum angular momentum that cannot be explained classically; the eigenstates for a nucleus with angular momentum \(I\) are \(\ket{I,m_I}\), where \(m_I\), the projection of the angular momentum onto an axis, has values ranging from \(I, I-1, \ldots, -I\). In free space, these states are all degenerate, but in the presence of a strong magnetic field, each state adopts a different energy. NMR requires first placing a molecule in a strong magnetic field to lift the nuclear spin degeneracies, then uses the magnetic component of light to induce nuclear spin flips, resulting in absorption of the radiation.
NMR spectroscopy, and particularly NMR, is widely used as a tool for determining the structures of organic molecules. The reasons for this are threefold:
- Any molecule possessing hydrogen atoms will have an NMR signature, and the magnetogyric ratio of a proton is one of the largest for any nucleus.
- The exact absorption frequency for a proton depends on the electronic structure near the atom, meaning that the frequencies are sensitive nearby functional groups.
- The spins of protons bound to adjacent carbon atoms couple, leading to spectral patterns that give structural information about how the hydrogen atoms in a molecule are distributed.
From a physical chemistry standpoint, NMR spectroscopy is the quintessential spectroscopic technique that involves magnetism, and the energy levels and spectral splittings observed in NMR spectra can be explained rather easily using simple quantum mechanical tools.
By the end of this chapter, you should know the following:
- How quantum mechanical spin is different than normal angular momentum.
- The commutation relationships among the spin operators \(\op{I}^2\), \(\op{I}_x\), \(\op{I}_y\), and \(\op{I}_z\), and how they act on spin functions
- How the frequency of a NMR spectrometer is related to the magnetic field.
- Chemical shift and why it is used in NMR spectroscopy
- Shielding and how it relates to electron density and chemical shift
- Spin-spin coupling and its effect have on a NMR spectrum
- The \(n+1\) rule
- The effect of chemically equivalent protons on an NMR spectrum
- How and why second-order spectra differ from first-order spectra
Skill goals (what you should be able to do/calculate):
- List the complete set of spin functions for a particle.
- Compute the energy levels for a nucleus in a given magnetic field.
- Convert between raw frequency and chemical shift.
- Use shielding constants to calculate chemical shifts.
- Use first-order perturbation theory to calculate energy levels and frequencies of coupled protons.
- Use the Rayleigh-Ritz variational method to calculate the frequencies in a second-order spectrum.