= Quadrupole interaction
An electrostatic interaction involving an <electric quadrupole moment>, the second-order term in a <multipole expansion>. About a specified origin, use the convention
$$
Q_{ij}=\sum_a q_a(3r_{a,i}r_{a,j}-|\mathbf r_a|^2\delta_{ij}).
$$
Here $q_a$ is an <electric charge>, $\mathbf r_a$ its position, and $\delta_{ij}$ the <Kronecker delta>. Far from the <electric charges>, the quadrupole contribution to <electrostatic potential> is
$$
\phi_Q(\mathbf r)=\frac{\sum_{ij}Q_{ij}n_i n_j}{8\pi\varepsilon_0 r^3},\qquad \mathbf n=\mathbf r/r,
$$
where $\varepsilon_0$ is <vacuum permittivity>. Thus a point <electric charge> has quadrupole interaction <potential energy> proportional to $r^{-3}$ at fixed orientation. A second <electric quadrupole moment> couples to second derivatives of this <electrostatic potential>, giving quadrupole–quadrupole <potential energy> proportional to $r^{-5}$ at fixed orientations. These are far-field terms, rather than the whole <intermolecular force> at close contact.
<Benzene> has no permanent <electric dipole moment> by <symmetry (physics)>, but its <electric quadrupole moment> need not vanish. Its electrostatic interaction therefore depends on the relative molecular orientations even without permanent <electric dipole moments>. Published <electric quadrupole moment> conventions can differ by a factor of two; the tensor convention above must be matched before inserting tabulated values.<Electric quadrupole moments in the CCCBDB by National Institute of Standards and Technology>[Tutorial: Primitive? Traceless?; experimental quadrupoles: benzene entry; far-field powers derived here from the multipole expansion]{ref}
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