NMR Partial Alignment 1  Theory of RDCs (Introduction)
 Part 1: Theory of RDCs (Introduction)
 Part 2: Preparing Liquid Crystal Samples
 Part 3: Collecting ²H Spectra and Shimming
Overview
Residual dipolar couplings (RDCs) and residual anisotropic chemical shifts (RACS) offer highresolution orientation information of bonds and nuclei in a molecular alignment frame. In contrast to nuclear Overhauser effect (NOE) distance constrains, which generally offer lowresolution distances between ^{1}H atoms in a molecule and are used to correctly pack a molecule, RDCs are useful in orienting secondary structure units in a refinement of a molecular structure. Stated another way, NOEs provide structures with a resolution of a molten globule, and RDCs and RACSs are used to refine NMR structures to high resolution.
The objective of this series of posts is to introduce researchers unfamiliar with RDCs and RACSs with their collection, analysis and refinement in structures. Excellent, comprehensive reviews can be found at the following references.^{1}^{,}^{2}
Definitions and Abbreviations
 NOE
 Nuclear Overhauser Effect. The crossrelaxation rate measured between nuclei, usually between ¹H nuclei. Ignoring the effects of spin diffusion, this rate is proportional to the r^{6} distance between nuclei. NOEs can be used to solve a molecular structure.
 RACS
 Residual Anisotropic Chemical Shift. The small contribution of the chemical shift anisotropy tensor measured from partially aligned samples. These are commonly described as residual chemical shift anisotropies (RCSAs). However, I believe this is a misnomer since the anisotropy of the chemical shift tensor is not directly measured in partially aligned samples.
 RDC
 Residual Dipolar Coupling. The small contribution of the magnetic dipoledipole coupling measured from partially aligned samples.
Theory
Introduction
RDCs change the magnitude of measured Jcouplings whereas RACSs change the frequency of peaks in a spectrum.
Jcouplings are readily measured in NMR, and they are the basis of coherent magnetization transfers in solutionstate NMR. Jcouplings arise from the polarization induced in the electron cloud between NMR active nuclei, and they tend to cluster in a small range of values, depending on bond lengths, torsion angles, hydrogen bonding and other factors. In proteins, the backbone ^{1}H^{15}N Jcoupling for backbone amides tend to adopt values between 90 and 95 Hz. Backbone ^{1}H^{α}^{13}C^{α} Jcouplings are typically around 145 Hz. Jcouplings are accurately measured from the spacing of split peaks in a spectrum or by using a quantitative Jcoupling experiment.
For a heteronuclear spin pair, the Jcoupling Hamiltonian is represented as follows.
\[\begin{equation} \label{eq:Hamiltonian_J} H_J = 2 I_z S_z \cdot J_{IS} \end{equation}\]The \(I_z\) and \(S_z\) operators are the nuclear spin operators for spins ‘I’ and ‘S’, respectively, and \(J_{IS}\) is the Jcoupling between these spins. Likewise, the heteronuclear dipolar coupling Hamiltonian can be written:
\[\begin{equation} \label{eq:Hamiltonian_D} H_D = 2 I_z S_z \cdot D_{IS} \end{equation}\]The Hamiltonian is identical to the heteronuclear Jcoupling Hamiltonian, with the exception of the dipolar coupling, \(D_{IS}\). The Jcoupling above is orientation independent^{3}, and it remains despite the rapid and isotropic tumbling^{4} of the molecule in solution. The dipolar coupling does not have an orientationindependent contribution, and it only has an orientationdependent component. Consequently, it cannot be directly observed from the spectrum of a molecular that is tumbling (rotating in solution) isotropically.
For a simple dipolar coupling in a powder sample^{5}, the orientation dependence is evaluated as follows.
\[\begin{equation} \label{eq:dipolar_PAS_to_lab} D_{IS} = D_{IS}^0 \frac{1}{2} \left(3 cos^2\theta  1 \right) \end{equation}\]The \(D_{IS}^0\) coupling is the static dipolar coupling constant, and it is the largest value the dipolar coupling can adopt. The static dipolar coupling depends on the distance between nuclei.
\[\begin{equation} \label{eq:static_dipole} D_{IS}^0 = \frac{\mu_0 \hbar \gamma_i \gamma_j}{4 \pi r_{IS}^3} \end{equation}\]See the following post for a more detailed discussion on the calculation of dipolar couplings.
In the absence of motion, the dipolar coupling is maximum (and equal to its static dipolar coupling constant) when the dipole is colinear with the applied magnetic field, \(B_0\), such that \(\theta = 0^\circ\). Likewise, it is equal to zero at the magic angle (\(\theta = 54.7^\circ\)) and it is equal to 0.5 of the static dipolar coupling at \(\theta = 90^\circ\).
\[\begin{align} \label{eq:dipolar_at_multiple_angles} D_{IS}(\theta = 0^\circ) &= D_{IS}^0 \nonumber \\ D_{IS}(\theta \approx 54.7^\circ) &= 0 \nonumber \\ D_{IS}(\theta = 90^\circ) &= \frac{1}{2} D_{IS}^0 \nonumber \end{align}\]Integrating all of the dipolar couplings over a sphere produces a Pake doublet spectrum from solidstate NMR.
If the molecule, and indeed the dipole, rotate rapidly, only an average value of the dipolar coupling can be measured. If all angles \(\theta\) are sampled equally and isotropically, then the average is equal to zero.
\[\begin{equation} \label{eq:2nd_legendre_integral} \int_{0^\circ}^{180^\circ} \left< \frac{1}{2} (3 cos^2\theta 1) sin\theta d \theta \right> = 0 \end{equation}\]The \(sin \theta\) factor comes from the Jacobian of integration. It’s role is to account for the fact that there are more vectors at the equator of a sphere than at the poles.
It follows that the dipolar coupling isn’t measured directly in the solution state spectrum because the time average is zero.^{6}
Of course, if the time average of the dipolar coupling is not zero, it may be observed directly in the spectrum. This is easily achieved by making the molecule tumble slightly nonisotropically (anisotropically) with partial alignment. In this case, the above integral is not equal to zero, and the dipolar coupling adopts a single, average value.
Since the Jcoupling and dipolar Hamiltonians share the same spin operator terms, they combine in the measured Hamiltonian.
\[\begin{equation} \label{eq:Hamiltonian_J+D} H_J + H_D = 2 I_z S_z \left ( J_{IS} + D_{IS} \right) \end{equation}\]Consequently, if partial alignment can be achieved in a sample, the \(\ J_{IS} + D_{IS}\\) coupling can be measured, analogously to the Jcoupling, and the Jcoupling can be subtracted from this new coupling to collect the RDC, \(D_{IS}\).
In practice, the magnitude depends on how the molecule is aligned, and therefore its value depends on the orientation of the dipole vector with respect to this molecular (alignment) frame. This transformation requires at least two rotations, and the final result represents the RDC equation.^{2}
\[\begin{equation} \label{eq:RDC} D_{IS} = D_{a} \left [ (3 cos^2 \theta 1 ) + \frac{3}{2} R sin^2\theta cos2\phi \right ] \end{equation}\]This equation assumes the absence of motion and the same \(D_a\), for a given interaction type, throughout the molecule. Two rotational transformations are needed in describing the RDC, one from the dipole^{7} to the frame of the molecule or alignment frame and one from the molecular frame to the laboratory frame, due to uniaxial diffusion of the liquid crystal.
The \(D_{a}\) represents the static dipolar coupling constant, \(D_{IS}^0\), scaled by the degree of alignment. To produce highresolution spectra with partial alignment, samples must only be weakly aligned–on the order of 10^{3}. Altogether, the static dipolar coupling (11.5 kHz for a ^{1}H^{15}N) and degree of alignment should produce a \(D_{a}\) of 515 Hz for an ^{1}H^{15}N coupling. If the \(D_{a}\) is larger than this value, other couplings may become larger and further split peaks, and the average tumbling time of molecules may become too large to produce highresolution spectra. An example of the former effect is the splitting of amide peaks in the HSQC of a protein due to appreciable ^{1}H^{N}^{1}H^{α} dipolar couplings.
The \(R\) represents the rhombicity of alignment. The rhombicity, \(D_{a}\) and the θ and φ angles are dictated by the nature of the interaction between the molecule and the alignment medium.
In the following post, we discuss general considerations in preparing liquid crystal samples for NMR.
References and Notes

Torchia, D. A. Prog. Nucl. Magn. Reson. Spectrosc. 2015, 84–85, 14. ↩

The Jcoupling actually has an orientation independent 0^{th} rank component as well as a 2^{nd} rank orientationdependent component. However, the orientation dependence of the Jcoupling is suppressed by isotropic tumbling of the molecule, and it is difficult to resolve from the much larger dipolar coupling in solidstate samples. ↩

Isotropic tumbling (rotation) means that the molecule rotates in solution freely and about all three axes with the same rate. ↩

A powder sample is simply a solid state sample in which the molecules are randomly oriented. A powder sample is distinct from a crystalline sample in which molecules have fixed orientations with respect to the sample (crystal). A powder sample is also distinct from a solution in which molecules tumble rapidly on the NMR timescale, typically nanoseconds: molecules in powder samples are fixed in place. ↩

Note that the dipolar coupling can be measured from isotropic solution state sample from the 2^{nd} order Hamiltonian, relaxation and crossrelaxation (NOEs). ↩

Specifically, from the Principal Axis System of the dipolar interaction, where the tensor is diagonal. ↩