Karplus equation

{{Short description|Correlation used in NMR spectroscopy}}

File:Karplus.svg | volume = 6 |pages=41–56 | date =1994 | first=M. J.|last=Minch| title = Orientational Dependence of Vicinal Proton-Proton NMR Coupling Constants: The Karplus Relationship}}]]

The Karplus equation, named after Martin Karplus, describes the correlation between ³J-coupling constants and dihedral torsion angles in nuclear magnetic resonance spectroscopy:{{cite journal | journal = Chemical & Engineering News | volume = 81 | issue = 51 |page=37 | date =2003-12-22 | first=Louisa |last=Dalton | title = Karplus Equation | url = http://pubs.acs.org/cen/science/8151/8151karplus.html | doi = 10.1021/cen-v081n036.p037| doi-access = free }}

:$J(\backslash phi)\; =\; A\; \backslash cos\backslash ,2\backslash phi\; +\; B\; \backslash cos\backslash ,\backslash phi\; +\; C$

where J is the ³J coupling constant, $\backslash phi$ is the dihedral angle, and A, B, and C are empirically derived parameters whose values depend on the atoms and substituents involved.{{cite journal |last=Karplus |first=Martin | title = Contact Electron-Spin Coupling of Nuclear Magnetic Moments | date = 1959 | journal = J. Chem. Phys. | volume = 30 | issue = 1 | pages = 11–15 | doi = 10.1063/1.1729860|bibcode = 1959JChPh..30...11K }} 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 H atoms bonded to neighbouring carbon atoms are termed vicinal).{{cite journal |last=Karplus |first=Martin |title = Vicinal Proton Coupling in Nuclear Magnetic Resonance | date = 1963 | journal = J. Am. Chem. Soc. | volume = 85 | issue = 18 | pages = 2870–2871 | doi = 10.1021/ja00901a059}} 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.