# Karplus Equations

Coupling is mediated by the interaction of orbitals within the bonding framework. It is therefore dependent upon overlap, and hence upon dihedral angle. The relationship between the dihedral angle and the vicinal coupling constant ^{3}J (as observed from ^{1}H NMR spectra) is given *theoretically* by the generalised **Karplus equations**:

^{3}J_{ab} = J^{0}cos^{2}φ-0.28 (0^{o} < φ < 90^{o})

^{3}J_{ab} = J^{180}cos^{2}φ-0.28 (90^{o} < φ < 180^{o})

where J^{0} = 8.5 and J^{180} = 9.5 are constants which depend upon the substituents on the carbon atoms and φ (*'phi'*) is the dihedral angle. The dihedral angle is defined by:

An approximate calculated relationship (ignoring the small constant of 0.28 in this graph) between the dihedral angle and the coupling constant may be illustrated below:

In some cases the axial-axial coupling constant for an antiperiplanar 180^{o} H-C-C-H configuration may be more than 9.5 Hz. Indeed for rigid cyclohexanes it is around 9-13 Hz, because the dihedral angle is close to 180^{o}, where the orbitals overlap most efficiently.

Enter the coupling constant in box as a numeral (e.g. 2.38) to calculate the theoretical dihedral angle in the molecule to assist with molecular modelling.*

Although the generalised Karplus equations generally work well for rigid bicyclics (e.g. camphor and its derivatives), for other molecules the situation is sometimes better served by alternative equations, e.g. the Bothner-By equation. As an illustrated example, the authors^{1} describe the near-axial and equatorial (ca. 180^{o} and 60^{o}) relationship between the ^{3}J dihedral couplings between C-2 and C-3 respectively for the trans-1,4-benzoxazepine below (its lowest energy *conformer A* shows the axial and equatorial relationships). The ^{3}J 2H-3Hax = 11.5 Hz and ^{3}J 2H-3Heq = 3.2 Hz corresponding to a dihedral angle of *ca.* 158 and 66^{o} using the Bothner-By equation; using the Karplus equations it is slightly out-of-range for the axial and the equatorial coupling is 50^{o}. From XYZ co-ordinates from the geometry-optimised structures using computational chemistry calculations (DFT B3LYP 6-31G(d), Gaussian 09) the dihedral angles are 179.7 and 64.6^{o} respectively, revealing the Bothner-By equation to be the better choice.

What's New: A book chapter has recently been published (Recent Advances in Asymmetric Diels-Alder Reactions; author J.P. Miller) that may be of interest in organic chemistry:

publication; ISBN: 9781622579112 or here.

1. L.Tóth, Y. Fu, H. Y. Zhang, A. Mándi, K. E. Kövér, T.-Z. Illyés, A. Kiss-Szikszai, B. Balogh, T. Kurtán, S. Antus, P. Mátyus, *Beilstein J. Org. Chem.* 2014, **10**, 2594-2602; doi:10.3762/bjoc.10.272.

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** Author J.P. Miller.**

This article originally available from: “(H-C-C-H) Coupling Constant to Dihedral Angle Converter” Main Page, www.jonathanpmiller.com/Karplus.html [Accessed 08/05/2017].