Scalar or J-couplings (also called indirect dipole dipole coupling) are mediated through chemical bonds connecting two spins. It is an indirect interaction between two nuclear spins which arises from hyperfine interactions between the nuclei and local electrons. J-coupling contains information about bond distance and angles. Most importantly, J-coupling provides information on the connectivity of molecules. In NMR spectroscopy, it is responsible for the appearance of many signals in the NMR spectra of fairly simple molecules.
Vector model and manifestations for chemical structure assignments 
The origin of J-coupling can be visualized by a vector model for a simple molecule such as hydrogen fluoride (HF). In HF, the two nuclei have spin 1/2. Four states are possible, depending on the relative alignment of the H and F nuclear spins with the external magnetic field. The selection rules of NMR spectroscopy dictate that ΔI = 1, which means that a given photon (in the radio frequency range) can affect ("flip") only one of the two nuclear spins.
J-coupling provides three parameters: the multiplicity (the "number of lines"), the magnitude of the coupling (strong, medium, weak), and the sign of the coupling.
The multiplicity provides information on the number of centers coupled to the signal of interest, and their nuclear spin. For simple systems, as in 1H-1H coupling in NMR spectroscopy, the multiplicity reflects the number of adjacent, magnetically nonequivalent protons. Nuclei with spins >1/2, which are called quadrupolar, can give rise to greater splitting, although in many cases coupling to quadrupolar nuclei is not observed. Many elements consist of nuclei with nuclear spin and without. In these cases the observed spectrum is the sum of spectra for each isotopomer. One of the great conveniences of NMR spectroscopy for organic molecules is that the many lighter elements are nearly monoisotopic: 1H, 19F, and 31P each have spin 1/2. 12C and 16O have no nuclear spin.
Magnitude of J-coupling 
For a given set of isotopes, say 1H-1H coupling, the magnitude of J provides information on the proximity of the coupling partners. Generally speaking 2-bond coupling (i.e. 1H-C-1H) is stronger than three-bond coupling (1H-C-C-1H). The magnitude of the coupling also provides information on the dihedral angles relating the coupling partners, as described by the Karplus relationship.
For heteronuclear coupling, the magnitude of J is related to the relative nuclear magnetic moments of the coupling partners. 19F, with a high nuclear magnetic moment, gives rise to large coupling to protons. 103Rh, with a very small nuclear magnetic moment, gives only small couplings to 1H.
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.
In October 1951, E. L. 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, intense pulses of radiofrequency magnetic field 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 the Hahn-Maxwell experiment, the decay was modulated by two frequencies: one frequency corresponded with the difference in chemical shift between the 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 depends 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, N. F. Ramsey and E. M. 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 the 1990s, direct evidence was 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 J-couplings are usually associated 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.
- E. L. Hahn and D. E. Maxwell (1952). "Spin Echo Measurements of Nuclear Spin Coupling in Molecules". Phys. Rev. 88 (5): 1070–1084. doi:10.1103/PhysRev.88.1070.
- P. Blake, B. Lee, M. Summers, M. Adams, J.-B. Park, Z. Zhou and A. Bax (1992). "Quantitative measurement of small through-hydrogen-bond and 'through-space' 1H-113Cd and 1H-199Hg J couplings in metal-substituted rubredoxin from Pyrococcus furiosus". Journal of Biomolecular NMR 2 (5): 527–533. doi:10.1007/BF02192814.
- P. R. Blake, J. B. Park, M. W. W. Adams and M. F. Summers (1992). "Novel observation of NH--S(Cys) hydrogen-bond-mediated scalar coupling in cadmium-113 substituted rubredoxin from Pyrococcus furiosus". J. Am. Chem. Soc. 114 (12): 4931–4933. doi:10.1021/ja00038a084.
- Andrew J. Dingley, Florence Cordier and Stephan Grzesiek (2001). "An introduction to hydrogen bond scalar couplings". Concepts in Magnetic Resonance 13 (2): 103–127. doi:10.1002/1099-0534(2001)13:2<103::AID-CMR1001>3.0.CO;2-M.
Further reading 
- H. S. Gutowsky, D. W. McCall, C. P. Slichter (1951). "Coupling among Nuclear Magnetic Dipoles in Molecules". Physical Review 84 (3): 589–90. Bibcode:1951PhRv...84..589G. doi:10.1103/PhysRev.84.589.2.
- E. L. Hahn and D. E. Maxwell (1951). "Chemical Shift and Field Independent Frequency Modulation of the Spin Echo Envelope". Physical Review 84 (6): 1246. Bibcode:1951PhRv...84.1246H. doi:10.1103/PhysRev.84.1246.
- N. F. Ramsey and E. M. Purcell (1952). "Interactions between Nuclear Spins in Molecules". Physical Review 85 (1): 143. Bibcode:1952PhRv...85..143R. doi:10.1103/PhysRev.85.143.
See also 
- Earth's field NMR
- Exclusive correlation spectroscopy (ECOSY)
- Magnetic dipole-dipole interaction (dipolar coupling)
- Nuclear Magnetic Resonance
- Nuclear magnetic resonance spectroscopy of carbohydrates
- Nuclear magnetic resonance spectroscopy of nucleic acids
- Nuclear magnetic resonance spectroscopy of proteins
- Proton NMR
- Relaxation (NMR)
- Residual dipolar coupling