# 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 | \(\gamma\) |
---|---|

^{1}H |
\(267.513 \cdot 10^6 \frac{rad}{s \cdot T}\) |

^{13}C |
\(67.262 \cdot 10^6 \frac{rad}{s \cdot T}\) |

^{15}N |
\(-27.116 \cdot 10^6 \frac{rad}{s \cdot T}\) |

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 | \(r_{ij}\) | \(\nu_{ij}\) |
---|---|---|

^{1}H-^{1}H |
\(1.00 \unicode{x212B}\) | -120 kHz |

^{1}H-^{15}N |
\(1.02 \unicode{x212B}\) | +11.5 kHz |

^{1}H-^{13}C |
\(1.10 \unicode{x212B}\) | -22.7 kHz |

A ^{1}H-^{1}H 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 ^{1}H spins
separated by 1.00Å is calculated as follows:

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 ^{1}H-^{13}C and ^{15}N 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} |
\(J_{ij}+D_{ij}\) | \(|J_{ij}+D_{ij}|\) |
---|---|---|---|---|---|

^{1}H-^{15}N |
-93 Hz | +11.5 kHz | 12 Hz | -81 Hz | 81 Hz |

^{1}H-^{13}C |
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 | \(\nu_{ij}\) | \(DI_{ij}\) |
---|---|---|

^{1}H-^{1}H |
-120 kHz | 240 kHz |

^{1}H-^{15}N |
+11.5 kHz | -22.0 kHz |

^{1}H-^{13}C |
-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} |
\(J_{ij}+D_{ij}\) | \(|-J_{ij}+D_{ij}|\) |
---|---|---|---|---|---|

^{1}H-^{15}N |
-93 Hz | -22.0 kHz | -12 Hz | -105 Hz | 81 Hz |

^{1}H-^{13}C |
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).