Potential of a Quadrupole is given as,
$$V(r) = \frac{1}{4\pi e_0}\left(\frac{1}{R}q+\frac{1}{R^2}\sum_{i=x,y,z}\hat{R_i}\vec{p_i} + \frac{1}{R^3}\sum_{i,j=x,y,z}\hat{R_i}\hat{R_j}Q_{ij}\right)$$
where $$Q_{ij} = \frac{1}{2}\sum^N_{n = 1}q_n(3r_{n,i}r_{n,j}-(r_n)^2\delta_{ij})$$
I am kind of confused with the last term
$$\frac{1}{R^3}\sum_{i,j=x,y,z}\hat{R_i}\hat{R_j}Q_{ij}$$
So if we have 4 particles, How can I expand this term ?
$$\frac{1}{R^5}[x^2Q_{xx}+y^2Q_{yy}+z^2Q_{zz}+2xyQ_{xy}+2xzQ_{xz}+2yzQ_{yz}]$$
Then $$Q_{xx} = \frac{1}{2}[q_1(3r_{1,x}r_{1,x}-r_1^2)+q_2(3r_{2,x}r_{2,x}-r_2^2)+q_3(3r_{3,x}r_{3,x}-r_3^2)+q_4(3r_{4,x}r_{4,x}-r_4^2)]$$ $$Q_{yy} = \frac{1}{2}[q_1(3r_{1,y}r_{1,y}-r_1^2)+q_2(3r_{2,y}r_{2,y}-r_2^2)+q_3(3r_{3,y}r_{3,y}-r_3^2)+q_4(3r_{4,y}r_{4,y}-r_4^2)]$$ and so on.
Is these expansions true ?
I thought that $\hat{R_i}\hat{R_j}=\delta_{ij}$ and what is exactly $\hat{R_i}$ ? I am kind of lost.