Interaction between an Ion and a Water Quadrupole

Instead of presenting a sophisticated general treatment for ion-quadrupole interactions, a particular case of these interactions will be worked out. The special case to be worked out is that corresponding to the water molecule being oriented with respect to a positive ion so that the interaction energy is a minimum. 

In this orientation (see Fig. A2.3.1), the oxygen atom and a positive ion are on the y axis, which bisects the H-O-H angle. Further, the positive ion, the oxygen atom, and the two hydrogen atoms are all considered in the xy plane. The origin of the xy coordinate system is located at the point Q, which is the center of the water molecule. The ion is at a distance r from the origin. 

The ion-quadrupole interaction energy is simply given by the charge on the ion times the potential v r at the site of the ion due to the charges of the quadrupole. 

Each of these potentials is given by the usual Coulombic expression for the potential 

The first term on the right-hand side ofEq. (A2.3.18) can be rearranged as follows  

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