# 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