J-coupling

J-coupling (also called indirect dipole dipole coupling) is the coupling between two nuclear spins due to the influence of bonding electrons on the magnetic field running between the two nuclei. J-coupling contains information about dihedral angles which can be estimated using Karplus equation.

Discovery
In October 1951, EL Hahn and D.E. Maxwell reported a spin echo experiment which indicates the existence of an interaction between two protons in dichloroacetaldehyde. In the echo experiment, two short, intent pulses of radiofrequency are applied to spin ensemble at the nuclear resonance condition and are separated by time interval of τ. The echo appears with a given maximum amplitude at time 2τ. For each setting of τ, the maximum of the echo signal is measured and plotted as a function of τ. If the spin ensemble consists of magnetic moment, a monotonic decay in the echo envelope is obtained. In Hahn-Maxwell experiment, the decay was modulated by two frequencies: one frequency was in correspondence with the difference in chemical shift between two non equivalent spins and a second frequency, J, that was smaller and independent of magnetic field strength. (J/2π = 0.7 cycle per second)

Such interaction came as a great surprise. The direct interaction between two magnetic dipole is dependent on the relative position of two nuclei in such a way that when averaged on all various orientation of the molecule it equals to zero.

In November 1951, NF Ramsey and EM Purcell, proposed a mechanism that explained the observation and gave rise to an interaction of the form I1.I2. The mechanism is the magnetic interaction between each nucleus and the electron spin of its own atom together with the exchange coupling of the electron spins with each other.

In 1990s, direct evidence has been found for the presence of J-couplings between magnetically active nuclei on both sides of the hydrogen bond. Initially, it was surprising to observe such couplings across hydrogen bonds since we usually associate J-couplings with the presence of purely covalent bonds. However, it is now well established that the H-bond J-couplings follow the same electron-mediated polarization mechanism as their covalent counterparts.

J-coupling Hamiltonian
The hamiltonian of a molecular system may be taken as:

H = D1 +D2 +D3.

D1 = electron orbital-orbital, spin-orbital, spin-spin and electron spin-external field interactions

D2 = magnetic interactions between nuclear spin and electron spin

D3 = direct interaction of nuclei with each other

for a singlet molecular state and frequent molecular collisions, D1 and D3 are almost zero. The full form of J-coupling interaction between spins Ij and Ik on the same molecule is:

''H = 2π Ij. Jjk. Ik''

where Jjk is the j-coupling tensor, a 3x3 real matrix. It depends on molecular orientation. In isotropic liquid it reduces to a number, so called scalar coupling. In 1D NMR, scalar coupling leads to oscillations in FID as well as splitting of lines in the spectrum.

Measurement of J-coupling
The Quantitative J correlation developed by Ad Bax et al. in 1994 is commonly the method of choice for accurate measurements of J couplings.