The Shapes and Structures of Molecules
- Depending on the nature of a particular chemical compound, there exist many experimental techniques for finding its molecular structure (David Tong would call these experimental techniques scattering which if one views light as a particle, would be an appropriate terminology). Examples include nuclear magnetic resonance (NMR) spectroscopy (where one also has the freedom to select any NMR-active nucleus with \(I\geq 1/2\) such as \(^1\text H\) or \(^{13}\text C\)), microwave spectroscopy, infrared (IR) spectroscopy, UV-visible spectroscopy, x-ray/electron/neutron diffraction, and mass spectrometry (with or without electrospray ionization). Notice that the spectroscopic methods are ordered in increasing energy \(E\) along the electromagnetic spectrum. Mass spectrometry is sort of an outlier in that it is not a spectroscopic method, but rather purely spectrometric. However, the spectroscopic methods are arguably the most important ones these days because they convey the richest amount of information, and several Nobel prizes in chemistry have basically gone to people who used these or similar techniques to determine the structures of very complicated (often biologically significant) molecules.
- In NMR spectroscopy, the idea is to place a molecule in an external magnetic field \(\textbf B_{\text{ext}}\) (this is often actually the superposition of three magnetic fields, one uniform and constant, one uniform but oscillating, and one non-uniform but constant, details can be found here). Then, an NMR-active nucleus will have to first-order a Zeeman Hamiltonian of the form \(H_{\text{NMR-active nucleus}}=-\boldsymbol{\mu}\cdot\textbf B_{\text{ext}}\) where the nuclear magnetic dipole moment operator is \(\boldsymbol{\mu}=\gamma\textbf S\). Furthermore, define the dimensionless nuclear spin angular momentum operator \(\textbf I:=\textbf S/\hbar\) so that its spectrum is of the form \(\sqrt{I(I+1)}\) for \(I=0,1/2,1,3/2,…\). Thus, for both \(^1\text H\) and \(^{13}\text C\) nuclei which are NMR-active with \(I=1/2\), one has \(2I+1=2\) energy eigenstates (\(m_I=\pm 1/2\) or “spin-up” and “spin-down”) arising from Zeeman splitting, separated by \(\Delta E=\gamma\hbar|\textbf B_{\text{ext}}|\). However, this wouldn’t make NMR particularly useful as all identical nuclei would just experience the same \(\textbf B_{\text{ext}}\) and hence have identical energies and be indistinguishable. This is where two important perturbations are then added to the Hamiltonian \(H\) of the NMR-active nucleus. Specifically, one has \(H=H_{\text{Zeeman}}+H_{\text{screening}}+H_{\text{J-coupling}}\), where \(H_{\text{screening}}=-\boldsymbol{\mu}\cdot\delta\textbf B_{\text{ext}}\) with \(\delta\) the chemical shift tensor field and \(\delta\textbf B_{\text{ext}}\) thought of as the induced screening magnetic field at the NMR-active nucleus due to functional groups nearby (e.g. electronegative elements, diamagnetic shielding from \(e^-\) in an aromatic ring, forced hybridizations of valence atomic orbitals on the atom associated to that NMR-active nucleus, etc.). Meanwhile, \(H_{\text{J-coupling}}=\sum_{\text{all non-identical NMR-active nuclei } i}2\pi\hbar\textbf I\cdot(J\textbf I_i)\) where the sum runs over all NMR-active nuclei (or at least the ones close to the NMR-active nucleus of interest) including different elements (and excluding identical nuclei in identical magnetic environments). Here, \(J\) is the \(J\)-coupling tensor field, and this \(H_{\text{J-coupling}}\) interaction is mediated by the Fermi contact interaction, Pauli exclusion principle, and Hund’s rule. For viscinal \(^3J_{^1\text H-^1\text H}\) coupling of \(^1\text H\) nuclei, it is always possible to draw a Newman projection with an associated dihedral angle \(\theta\) between the two \(^1\text H\) nuclei, and the Karplus equation asserts that \(^3J_{^1\text H-^1\text H}(\theta)=f_2\cos(2\theta)+f_1\cos(\theta)+f_0\) for some empirical fitting parameters \(f_0,f_1,f_2\in\textbf R\). The intuition is thus that trans/anti protons \(\theta=180^{\circ}\) have the greatest viscinal coupling constant, followed by cis/eclipsed protons \(\theta=0\), and worst coupling is for orthogonal protons \(\theta=90^{\circ}\). There is pretty much an infinite rabbit hole of interactions that can arise with NMR spectroscopy (e.g. not just diamagnetic but also paramagnetic screening interactions in \(H_{\text{screening}}\), or interactions of the nuclear spin angular momentum with rotation of the molecule, or for NMR-active nuclei with \(I\geq 1\), a quadrupolar interaction? It’s a deep subject. Final comment: under isotropic motional averaging and within the secular approximation, one has \(\delta\mapsto\frac{1}{3}\text{Tr}(\delta)\) and likewise \(J\mapsto\frac{1}{3}\text{Tr}(J)\)and this is what is actually measurable and plotted on an NMR spectrum. Larger nuclear masses (e.g. \(^{13}\text{C}\)) will have larger chemical shift ranges.
- For \(^{13}\text C\) NMR spectroscopy:
Also, spin angular momentum coupling between \(^{13}\text C\) nuclei and \(^1\text H\) nuclei are typically suppressed via broadband proton decoupling. Also, due to low isotopic abundance, spin angular momentum coupling among \(^{13}\text C\) nuclei may only be visible as satellite peaks (unless the compound is \(^{13}\text C\)-enriched) and vice versa (i.e. in an \(^1\text H\)-NMR spectrum, the spin angular momentum coupling of protons with \(^{13}\text C\) nuclei would also only maybe appear as satellites). There is also a variant known as \(^{13}\text C\) attached proton test (APT) NMR spectroscopy which is essentially just the regular \(^{13}\text C\) NMR spectrum but it is also sensitive to the parity of the number of attached \(\text H\) atoms (look at which way the deuterated chloroform \(\text{CDCl}_3\) solvent points to determine this).
- For \(^1\text{H}\)-NMR spectra \(n_{^1\text H}(\delta)\), the integral trace \(\int_{\delta_1}^{\delta_2}n_{^1\text H}(\delta)d\delta\) gives the number of protons with chemical shifts \(\delta\in[\delta_1,\delta_2]\). This is not the case for \(^{13}\text C\)-NMR spectra (why though?). Another quirk peculiar to \(^1\text{H}\)-NMR spectra is roofing (why does it happen?).
- Exchangeable protons \(\text O-\text H\) and \(\text N-\textbf H\) can be made to disappear from a \(^1\text H\)-NMR spectrum using a \(\text D_2\text O\) shake (as \(\text H\to\text D\) which resonates at very different wavenumbers due to twice the nuclear mass).
- In IR spectroscopy, no external field is needed. Just irradiate the sample with a bunch of infrared light and see if certain frequencies are absorbed more than others by the sample. These would correspond to normal mode/vibrational eigenstates of the molecule, which themselves tend to be dominated by particular functional groups. Although it’s really a quantum harmonic oscillator, one can think of a normal mode as a classical harmonic oscillator with \(\omega_0=\sqrt{\frac{k}{\mu}}\) where \(\mu=\frac{m_1m_2}{m_1+m_2}\) is the reduced mass (and as its name suggests, its generally closer to the smaller mass). An IR spectrum is then a plot of \(|\Delta\textbf p|(\nu)\), where \(\Delta\textbf p\) is the change in electric dipole moment associated with a given normal mode. Thus, purely covalent/non-polar bonds do not show up on IR spectra.