Karplus equation

The Karplus equation, named after Martin Karplus, describes the correlation between 3J-coupling constants and dihedral torsion angles in nuclear magnetic resonance spectroscopy: :J(\phi) = A \cos,2\phi + B \cos,\phi + C

where J is the 3J coupling constant, \phi is the dihedral angle, and A, B, and C are empirically derived parameters whose values depend on the atoms and substituents involved. The relationship may be expressed in a variety of equivalent ways e.g. involving cos 2φ rather than cos2 φ —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 H atoms bonded to neighbouring carbon atoms are termed vicinal). 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.