After this first publication of the equation, Karplus started to get feedback from organic chemists about the use of the equation, and he realized that he should publish a follow-up communication in the Journal of the American Chemical Society. The resulting paper [J. Am. Chem. Soc., 85, 2870 (1963)] is now the 17th most cited article in JACS history.

In the late 1950s, nuclear magnetic resonance was a mushrooming field. Physicists worked on the theory of NMR while chemists began to garner information about molecular structure from NMR shifts and splittings. Karplus realized he could generate theoretical models that linked a molecule's electronic structure to its NMR spectrum.

As he looked deeper, he made an interesting observation: The NMR coupling constant between two atoms not directly bonded to each other is not zero. In other words, the two protons in an HCC9H9 fragment interact--something not predicted by the perfect pairing model of valence bond theory. "It's a small effect," Karplus says, "and it is only because NMR is sensitive to such a small effect that you can actually determine the deviation from perfect pairing.

"Once I was aware of the existence of nonzero coupling constants between atoms that weren't bonded to each other," Karplus says, "I knew what to do--how to attack the problem by using valence bond theory, which I had learned during my graduate research with Linus Pauling. The work that was required was to estimate the molecular integrals from which one could actually calculate the deviation from perfect pairing."

Karplus came up with an equation relating the dihedral angle between H' and H to the coupling constant: J_HH' = A + B cos + C cos 2, where A, B, and C are approximately constant for saturated hydrocarbons. The result calculated for J_HH9 was a smooth hill-and-valley curve.

	RIGID RING Angles between vicinal protons in a cyclohexane can be measured with NMR J-values.

Interestingly, Harold Conroy, then at Yale University, independently obtained a similar relation by approaching the problem through molecular orbital theory [Raphael, Ralph A. et al, editors. "Advances in Organic Chemistry: Methods and Results," Vol. II. New York: Interscience Publishers, 1960].

Karplus' equation quantified something a few chemists had noticed qualitatively: A proton could interact with its vicinal neighbor, and that influence varies depending on the geometry of the two protons. For example, E. J. Corey, an organic chemist then at the University of Illinois and a friend who regularly dined with Karplus, knew from experiment that orthogonal protons (those having a dihedral angle of 90º) had no interaction, whereas protons with dihedral angles between 0º and 90º interacted variably.

"And in 1958, a Canadian chemist, Ray V. Lemieux, came and gave a couple of lectures at the University of Illinois," Corey relates. "He had measured coupling constants between vicinal protons in a large series of sugars. His data were paralleling the rates of vicinal elimination as a function of the geometry of the groups that were leaving during the elimination process."

Karplus also attended Lemieux's lecture. At the time, Karplus was finishing up his 1959 paper and saw that Lemieux's data fit nicely with his theoretical calculations. Even better, Karplus realized that his equation could help assign conformations to organic molecules with some rigidity to them, like Lemieux's sugars.

Once the NMR community got wind of the Karplus equation, citation of Karplus' research became a fixture of any conformational analysis using NMR. Corey may have been the first to use it for the determination of the structure of a biooxidation product of camphor [J. Am. Chem. Soc., 81, 5507 (1959)].

Today, variations of the Karplus equation are heavily used in establishing protein structure from NMR spectra. "In proteins," Karplus says, "the main-chain dihedral angles are the essential element of how the polypeptide chain is folded. You assume when you determine a protein structure that you know the bond lengths and the bond angles and all the connectivity. So determining the structure is not determining the chemical structure, but determining the conformation."

"In many ways, my feeling about the uses and refinements of the 'Karplus equation' is that of a proud father," Karplus wrote in 1996 in an historical chapter for the "Encyclopedia of Nuclear Magnetic Resonance." "I am very pleased to see all the nice things that the equation can do, but it is clear that it has grown up and now is living its own life."

C&EN is celebrating the 125th volume of the Journal of the American Chemical Society by featuring selected papers from among its 125 most cited. This paper was ranked 17th.

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