Role of Fermi Contact Interaction in J-Coupling

[Steven Hanna]

I'm having a hard time understanding the mechanism of J coupling in NMR. Why is coupling information only transmitted through bonding electrons with nonzero s-character? For example, why can't coupling information be transmitted through a bond with no s-character, e.g. a retrodative bond between a metal and a ligand? It's been explained to me that J coupling occurs through the Fermi contact interaction, in which the magnetic moment of the nucleus interacts with those of bonding electrons, and that this interaction can only occur with s-electrons since they are the only electrons with nonzero probability density at the nucleus. However, if the nucleus makes a magnetic moment in space, why shouldn't that magnetic moment interact with p or d electrons? I found the following formula for the magnitude of the Fermi contact interaction on Wikipedia.

Do these brackets <> represent an inner product? And if so, is the Fermi contact interaction zero for p electrons because this inner product is somehow zero?

 

[DrClaude]

However, if the nucleus makes a magnetic moment in space, why shouldn't that magnetic moment interact with p or d electrons?

The magnetic field created by the magnetic moment nucleus is extremely weak, so to first order only the electron density at the nucleus interacts significantly with that magnetic field.

Do these brackets <> represent an inner product? And if so, is the Fermi contact interaction zero for p electrons because this inner product is somehow zero?

The brackets represent the expectation value. But what is zero is $\Psi(0)$ for orbitals other than s.

 

[Steven Hanna]

Thanks for your quick reply! Two followup questions:
1) Is this formula just an approximation, i.e. could a nuclear magnetic moment weakly interact with that of a p-electron through space?
2) Is the expectation value of this dot product necessarily zero for p electrons?