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:[2]

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

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

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 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 ³J_H,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.