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:

The dipolar coupling frequency between spins ‘i’ and ‘j’.
Vacuum permeability.
Planck’s constant in radial units.
gyromagnetic/magnetogyric ratio of the ‘i’ spin.
gyromagnetic/magnetogyric ratio of the ‘j’ spin.
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
1H
13C
15N

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

Dipolar coupling constants of common bonds in proteins

Spin Pair
1H-1H -120 kHz
1H-15N +11.5 kHz
1H-13C -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:

I made use of the fact that

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 (), the RDC () is proportional to the degree of alignment (A) and the static dipolar coupling constant.

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 ().

Since the spin terms for the J-coupling and dipolar coupling are the same, the sum of the two are measured, ||, and the sign of the dipolar coupling can be measured if is known.

For 1H-13C and 15N spin pairs, the coupling will always be reduced in magnitude for bonds oriented along the poles ().

Example RDCs measured for spin pairs oriented along the poles.

Spin Pair 1
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.

NMRPipe Dipolar Couplings Convention

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

This equation is different from the static dipolar coupling from above () by a factor of -2.

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

Dipolar coupling constants of common bonds in proteins

Spin Pair
1H-1H -120 kHz 240 kHz
1H-15N +11.5 kHz -22.0 kHz
1H-13C -22.7 kHz 45.4 kHz

The component is directly related to the -component of the dipolar tensor. The coupling can be measured from a Pake pattern as well, but it entails measuring the difference between frequencies of the two doublet components. Stated another way, this is the frequency difference calculated from measuring at the edge of the Pake pattern, instead of the peaks.

For the coupling measured from the peaks:

And for the DI component:

Note that the static dipolar coupling tensor is axially symmetric and that the sign of 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 .

Spin Pair 2
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 definition for the static dipolar coupling, the dipole aligned along the polar axis will consistently have a reduced value of the -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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