Molecular Electric Fields and Field Gradients

Exercise 4.6 Derive the expression (4.87) for the molecular electric-field gradient. 

The molecular electric field gives rise to a force F acting on the charges in the charge distribution, where the contribution to the electric field from the charge in question has to be excluded, of course. For a charge distribution in equilibrium this force should obviously be zero. For example, the force acting on a nucleus K in a. molecule would then be 

Although the fields and field gradients are well defined for any point in space, it is not possible to measure them at an arbitrary point within the charge distribution. Fields can be probed by dipole moments and field gradients by quadrupole moments (see e.g. Eq. (4.18)). In order to measure the field at an arbitrary point one would have to bring a dipole moment there, which is of course not possible within a molecule. Only via the interaction with the nuclei in a molecule is it therefore possible to get information about some of these field quantities and only at the positions of the nuclei. 

But nuclei do not have electric dipole moments and the molecular electric field can thus not be investigated in this way. 

However, nuclei with a spin quantum number I possess an electric quadrupole moment 0 and one can study the molecular electric field gradient at the positions of the nuclei, via the interaction with the nuclear electric quadrupole moment 

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