# Karplus equation

Graph of the Karplus relation JHH(φ) = 12 cos^2φ - cosφ+2 obtained for ethane derivatives [1]

The Karplus equation, named after Martin Karplus, describes the correlation between 3J-coupling constants and dihedral torsion angles in nuclear magnetic resonance spectroscopy:[2]

${\displaystyle J(\phi )=C\cos 2\phi +B\cos \,\phi +A}$

where J is the 3J coupling constant, ${\displaystyle \phi }$ is the dihedral angle, and A, B, and C are empirically derived parameters whose values depend on the atoms and substituents involved.[3] The relationship may be expressed in a variety of equivalent ways e.g. involving cos2 φ rather than cos 2φ —these lead to different numerical values of A, B, and C but do not change the nature of the relationship.

The relationship is used for 3JH,H coupling constants. The superscript "3" indicates that a 1H atom is coupled to another 1H atom three bonds away, via H-C-C-H bonds. (Such hydrogens bonded to neighbouring carbon atoms are termed vicinal).[4] The magnitude of these couplings are generally smallest when the torsion angle is close to 90° and largest at angles of 0 and 180°.

This relationship between local geometry and coupling constant is of great value throughout nuclear magnetic resonance spectroscopy and is particularly valuable for determining backbone torsion angles in protein NMR studies.

## References

1. ^ Minch, M. J. (1994). "Orientational Dependence of Vicinal Proton-Proton NMR Coupling Constants: The Karplus Relationship". Concepts in Magnetic Resonance. 6: 41–56. doi:10.1002/cmr.1820060104.
2. ^ Dalton, Louisa (2003-12-22). "Karplus Equation". Chemical & Engineering News. 81 (51): 37. doi:10.1021/cen-v081n036.p037.
3. ^ Karplus, Martin (1959). "Contact Electron-Spin Coupling of Nuclear Magnetic Moments". J. Chem. Phys. 30 (1): 11–15. Bibcode:1959JChPh..30...11K. doi:10.1063/1.1729860.
4. ^ Karplus, Martin (1963). "Vicinal Proton Coupling in Nuclear Magnetic Resonance". J. Am. Chem. Soc. 85 (18): 2870–2871. doi:10.1021/ja00901a059.