Karplus equation

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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, $φ$ is the dihedral angle, and A, B, and C are empirically-derived parameters whose values depend on the atoms and substituents involved.^[1] The relationship may be expressed in an a variety of equivalent ways e.g. involving cos 2 φ rather than cos²φ —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 i.e. J-couplings between two ¹H nuclei through three bonds (such hydrogens bonded to neighbouring 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 back bone torsion angles in protein NMR studies.