NMR

Nuclear magnetic resonance (NMR) spectroscopy is one of the most important analytical techniques in chemistry and related fields. It is widely used to identify molecules, but also to obtain information about their structure, dynamics, and chemical environment. Some fundamental aspects of NMR are summarized below.

NMR spectrometers place the sample into a strong, but constant magnetic field and use a weak, oscillating magnetic field to perturb the nuclei. At or near resonance, when the oscillation frequency matches the intrinsic frequency of a nucleus, the system responds by producing an electromagnetic signal with a frequency characteristic of the magnetic field at the respective nucleus.

Zeeman interaction

For the simulation of NMR spectra for molecules, the most important part is the Zeeman effect describing the interaction of the nuclear spin with the external magnetic field $\mathbf{B}$. The corresponding Hamiltonian can be written as

$${\hat{H}}_{\text{Z}}=-\gamma \mathbf{B}\cdot \hat{\mathbf{S}},$$

where $\gamma$ is the gyromagnetic ratio, the ratio of a system's magnetic moment to its angular momentum, and $\hat{\mathbf{S}}=({\hat{S}}^x,{\hat{S}}^y,{\hat{S}}^z{)}^{\text{T}}$ is the total spin operator. Assuming that the strong (and constant) magnetic field is in $z$-direction only, meaning $\mathbf{B}=(0,0,B^z)$, the Hamiltonian simplifies to

$${\hat{H}}_{\text{Z}}=-\gamma B^z{\hat{S}}^z={\omega }_0{\hat{S}}^z,$$

where ${\omega }_0=-\gamma B^z$ is the so-called Larmor frequency, the angular frequency corresponding to the precession of the spin magnetization around the magnetic field at the position of the nucleus.

As an example, consider the simplest nucleus, ${}^1$H, consisting of only one proton, for which the gyromagnetic ratio is $\gamma /2\pi =42.6$ MHz T${}^{-1}$, meaning that a 500 MHz NMR spectrometer has a static magnetic field of about 11.7 Tesla. The energy of radiation of the Larmor frequency $\nu =500$ MHz ($h\nu \approx 3.3\cdot 10^{-25}$ J) is several orders of magnitude smaller than the average thermal energy of a molecule at a temperature of $T=298$ K ($k_{\text{B}}T\approx 4.1\cdot 10^{-21}$ J). Therefore, the occupations of the spin states are almost equal at room temperature, only a small surplus is responsible for the sample magnetization.

Chemical shift

Perhaps the most important aspect for NMR spectroscopy in chemistry is that the nuclei in molecules are shielded against the external magnetic field by the electrons surrounding them. This can be expressed by adding a correction term to the Hamiltonian as

$${\hat{H}}_{\text{Z}}=-\gamma (1-\mathbf{\sigma })\mathbf{B}\cdot \hat{\mathbf{S}},$$

where $\mathbf{\sigma }$ is referred to as the shielding tensor quantifying the change in the local magnetic field experienced by the nucleus in the molecule relative to a bare nucleus in vacuum. However, if the molecules of interest are in solution, or in liquid phase in general, they can rotate freely and only the isotropic chemical shift $\sigma =\frac{1}{3}\text{Tr}(\mathbf{\sigma })$ is of interest,

$${\hat{H}}_{\text{Z}}=-\gamma (1-\sigma )B^z{\hat{S}}^z.$$

In practice, chemical shifts are normally used instead of chemical shieldings: instead of invoking the Larmor frequency of a nucleus in vacuum, shifts are defined with respect to the resonance frequency ${\nu }_{\text{ref}}$ of a reference compound:

$$\delta =\frac{\nu -{\nu }_{\text{ref}}}{{\nu }_{\text{ref}}}\approx {\sigma }_{\text{ref}}-\sigma ,$$

The standard reference for ${}^1$H-NMR is the Larmor frequency of the protons in TMS [tetramethylsilane, Si(CH${}_3$)${}_4$]. Chemical shifts are normally reported on a scale of ppm (parts per million): most ${}^1$H chemical shifts are observed in the range between 0 and 10 ppm, and most ${}^{13}$C chemical shifts between 0 and 200 ppm. Since the scale of chemical shieldings is so small in absolute terms $\left(|\sigma |\ll 1\right)$, for practical intents and purposes the chemical shift can be substituted directly into the Hamiltonian:

$${\hat{H}}_{\text{Z}}=-\gamma (1+\delta )B^z{\hat{S}}^z.$$

Spin-spin coupling

Up to this point, the nuclear spins have been regarded to be isolated from each other. However, their magnetic moments have an effect on neighboring spins. The interaction of the nuclear spins can happen through two different mechanisms. The first one is the direct (or through space) spin-spin coupling, where the interaction strength depends on the distance of the two nuclei and the angle of their distance vector relative to the external field. As it comes from the direct interaction of two magnetic dipoles, it is also referred to as dipolar coupling. However, the effect is generally not observable in liquid phase since the free rotation of the molecules averages over all orientations and thus results in a vanishing average coupling.

An effect observable in the NMR spectrum is indirect spin-spin coupling, which is mediated by the electrons of a chemical bond. Due to the Pauli principle, the electrons of a covalent bond always have an anti-parallel spin orientation, and one electron will be closer to one nucleus than to the other, preferring an anti-parallel orientation with the nearby nucleus. Depending on the number of electrons involved in the transmission of the interaction, either a parallel or an anti-parallel orientation of two nuclei may result in a lower energy. Importantly, this interaction does not average out in solution since it mainly depends on the electron density at the position of the nucleus and not on the orientation of the distance vector relative to the field, which is why it is also referred to as scalar coupling. Since only s-orbitals have a finite electron density at the nucleus, the coupling depends on the electron density in those orbitals alone.

The interaction Hamiltonian in the case of homonuclear coupling is given by

$${\hat{H}}_J=2\pi J{\hat{\mathbf{I}}}_1\cdot {\hat{\mathbf{S}}}_2,$$

where $\hat{\mathbf{I}}={\hslash }^{-1}\hat{\mathbf{S}}$. In heteronuclear coupling, where the difference in Larmor frequencies is much larger in magnitude than the corresponding coupling constant, $|{\omega }_1-{\omega }_2|\gg |J|$, the Hamiltonian can be written in terms of the $z$-components only,

$${\hat{H}}_J=2\pi J{\hat{I}}_1^z{\hat{S}}_2^z.$$

It should be noted that the $\mathbf{J}$-coupling tensor is a real $3\times 3$ matrix that depends on the molecular orientation, but in liquid phase only its isotropic part $J$ is observed due to motional averaging. Typical $J$-coupling strengths between protons in ${}^1$H-NMR amount to a few Hz.

NMR spin Hamiltonian for molecules in liquid phase

The spin Hamiltonian in a static magnetic field in frequency units (rad s${}^{-1}$) is given by

$${\hslash }^{-1}\hat{H}=-\sum _k{\gamma }_k(1+{\delta }_k)B_0{\hat{I}}_k^z+2\pi \sum _{k<l}{J}_{kl}{\hat{\mathbf{I}}}_k\cdot {\hat{\mathbf{I}}}_l,$$

where the sum runs over all nuclear spins of interest.

There are several interactions that have not been taken into account here. As already mentioned, the direct dipolar spin-spin interaction vanishes in liquids due to motional averaging. Beyond dipolar coupling, such as quadrupolar interactions, for instance, are relevant only for nuclei with spin quantum number $I\ge 1$. Furthermore, interactions with unpaired electrons need a special treatment as well. While most organic compounds are diamagnetic (closed-shell), paramagnetic NMR also exists.