Dipolar Couplings in Solid-State and Solution NMR

Static Dipolar Couplings

Solid-state spectroscopists calculate the magnetic dipole-dipole coupling constant, also known as the dipolar coupling constant (DCC), with the following equation:

\[\omega_{ij} = - \frac{\mu_0 \hbar \gamma_i \gamma_j}{4 \pi r_{ij}^3}\]
\(\omega_{ij}\)
The dipolar coupling frequency between spins ‘i’ and ‘j’. \(\left( \frac{rad}{T} \right)\)
\(\mu_0\)
Vacuum permeability. \(\left( 4 \pi \cdot 10^{-7} \frac{T \cdot m}{A} \right)\)
\(\hbar\)
Planck’s constant in radial units. \(\left( 1.0545718 \cdot 10^{-34} \frac{J \cdot s}{rad} \right)\)
\(\gamma_i\)
gyromagnetic/magnetogyric ratio of the ‘i’ spin. \(\left( \frac{rad}{s \cdot T} \right)\)
\(\gamma_j\)
gyromagnetic/magnetogyric ratio of the ‘j’ spin. \(\left( \frac{rad}{s \cdot T} \right)\)
\(r_{ij}^3\)
internuclear distance between spins ‘i’ and ‘j’. (meters)

The gyromagnetic ratios of common spin isotopes in NMR are as follows:

Gyromagnetic ratios of common isotopes in NMR

Nucleus \(\gamma\)
1H \(267.513 \cdot 10^6 \frac{rad}{s \cdot T}\)
13C \(67.262 \cdot 10^6 \frac{rad}{s \cdot T}\)
15N \(-27.116 \cdot 10^6 \frac{rad}{s \cdot T}\)

Accordingly, the static-limit dipolar coupling constants (\(\nu_{ij}\)) for common bonds found in proteins can be calculated:

Dipolar coupling constants of common bonds in proteins

Spin Pair \(r_{ij}\) \(\nu_{ij}\)
1H-1H \(1.00 \unicode{x212B}\) -120 kHz
1H-15N \(1.02 \unicode{x212B}\) +11.5 kHz
1H-13C \(1.10 \unicode{x212B}\) -22.7 kHz

A 1H-1H distance of 1.0A is not found in proteins, but it is listed as a reference dipolar coupling.

Since the solid-state Pake (powder) pattern is symmetric, solid-state spectroscopists generally measure the absolute value of the dipolar coupling. This is not the case, however, for aligned solid-state samples.

Sample Calculation

A reference dipolar coupling between two 1H spins separated by 1.00Å is calculated as follows:

\[\begin{align} \omega_{ij} &= - \frac{\mu_0 \hbar \gamma_i \gamma_j}{4 \pi r_{ij}^3} \\ &= - \frac{(4 \pi \cdot 10^{-7} \frac{T \cdot m}{A}) (1.05457 \cdot 10^{-34} \frac{J \cdot s}{rad}) (267.513 \cdot 10^6 \frac{rad}{s \cdot T}) (267.513 \cdot 10^6 \frac{rad}{s \cdot T})} {4 \pi \cdot (1.00 \cdot 10^{-10} m)^3} \\ &= -754.7 \frac{krad}{s} \\ \nu_{ij} &= -120.1 kHz \end{align}\]

I made use of the fact that \(1 T = 1 \frac{kg}{s^2 \cdot A} = 1 \frac{J}{A \cdot m^2}\)

Solution NMR and Residual Dipolar Couplings

RDC sign

The sign of the dipolar coupling can be resolved with residual dipolar couplings (RDCs) since these are measured relative to the J-coupling and the sign of the J-coupling is known. RDCs are measured from partial alignment of the molecule of interest with a liquid crystal, which aligns in the magnetic field.

If we consider a single spin pair aligned along the polar axis (\(\theta=0^{\circ}\)), the RDC (\(D_{ij}\)) is proportional to the degree of alignment (A) and the static dipolar coupling constant.

\[D_{ij} = \nu_{ij} \cdot A\]

The degree of alignment is a positive number. As a result, the RDC for the spin pair aligned along the poles will follow the sign of the static-limit dipolar coupling (\(\nu_{ij}\)).

Since the spin terms for the J-coupling and dipolar coupling are the same, the sum of the two are measured, |\(J_{ij}+D_{ij}\)|, and the sign of the dipolar coupling can be measured if \(J_{ij}\) is known.

For 1H-13C and 15N spin pairs, the \(|J_{ij}+D_{ij}|\) coupling will always be reduced in magnitude for bonds oriented along the poles (\(\theta=0^{\circ}\)).

Example RDCs measured for spin pairs oriented along the poles.

Spin Pair \(J_{ij}\) \(\nu_{ij}\) \(D_{ij}\) 1 \(J_{ij}+D_{ij}\) \(|J_{ij}+D_{ij}|\)
1H-15N -93 Hz +11.5 kHz 12 Hz -81 Hz 81 Hz
1H-13C 145 Hz -22.7 kHz -23 Hz 122 Hz 122 Hz

The distinction in signs is important because you cannot simultaneously ignore the sign of the J-coupling and dipolar coupling and get the right answer.

\[|J_{ij}+D_{ij}| \neq |J_{ij}|+|D_{ij}|\]

NMRPipe Dipolar Couplings Convention

NMRPipe and its RDC fitting program, DC, calculate static dipolar couplings (DI) with the following equation:

\[DI_{ij} = \frac{\mu_0 \hbar \gamma_i \gamma_j} {4 \pi^2 r_{ij}^3}\]

This equation is different from the static dipolar coupling from above (\(\nu_{ij}\)) by a factor of -2.

\[DI_{ij} = -2 \nu_{ij}\]

Producing the following dipolar couplings for H-N and H-C bonds:

Dipolar coupling constants of common bonds in proteins

Spin Pair \(\nu_{ij}\) \(DI_{ij}\)
1H-1H -120 kHz 240 kHz
1H-15N +11.5 kHz -22.0 kHz
1H-13C -22.7 kHz 45.4 kHz

The \(\nu_{ij}\) component is directly related to the \(\delta_{zz}\)-component of the dipolar tensor. The \(DI_{ij}\) coupling can be measured from a Pake pattern as well, but it entails measuring the difference between \(\delta_{xx}/\delta_{yy}\) frequencies of the two doublet components. Stated another way, this is the frequency difference calculated from measuring at the \(0^{\circ}\) edge of the Pake pattern, instead of the \(90^{\circ}\) peaks.

For the \(\nu_{ij}\) coupling measured from the \(90^{\circ}\) peaks:

\[\begin{align} \nu_{ij} &= | (\delta_{zz} 0.5(3cos^2 90^{\circ}-1)) - (-(\delta_{zz} 0.5(3cos^2 90^{\circ}-1))| \\ &= | (-0.5-0.5)\delta_{zz}| \\ &= | \delta_{zz} | \end{align}\]

And for the DI component:

\[\begin{align} DI_{ij} &= | (\delta_{zz} 0.5(3cos^2 0^{\circ}-1)) - (-(\delta_{zz} 0.5(3cos^2 0^{\circ}-1))| \\ &= | (1+1)\delta_{zz}| \\ &= | 2 \delta_{zz} | \end{align}\]

Note that the static dipolar coupling tensor is axially symmetric and that the sign of \(DI_{ij}\) is inferred from the sign of the gyromagnetic ratios.

The factor of 2 is needed for the RDC because it is measured from a splitting (J+D - J).

Example RDCs measured for spin pairs oriented along the poles for \(DI_{ij}\).

Spin Pair \(J_{ij}\) \(DI_{ij}\) \(D_{ij}\) 2 \(J_{ij}+D_{ij}\) \(|-J_{ij}+D_{ij}|\)
1H-15N -93 Hz -22.0 kHz -12 Hz -105 Hz 81 Hz
1H-13C 145 Hz 45.4 kHz 23 Hz 168 Hz 122 Hz

When using the \(DI_{ij}\) definition for the static dipolar coupling, the dipole aligned along the polar axis will consistently have a reduced value of the \(|J_{ij}+D_{ij}|\)-coupling if you use J-couplings that are multiplied by -1 (i.e. 93Hz for NH, -145Hz for CH).

References and Notes

  1. The RDC was calculated with an degree of alignment of 10-3

  2. The RDC was calculated with an degree of alignment of 5-4


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