Dipole-quadrupole polarizability term for induced dipole
This question has been around for more than a year without an answer, so probably you have figured it out by now! But anyway, here goes. The standard literature references in this field are Theory of Molecular Fluids vol 1 by CG Gray and KE Gubbins (Clarendon Press 1984) and The Theory of Intermolecular Forces by Anthony Stone (Oxford University Press 2013) and these books explain the various terms in your equation. The dipole-quadrupole polarizability $A_{i,jk}$ may vanish by symmetry (e.g. for centrosymmetric molecules and isolated atoms) but does not vanish for a molecule like water. The corresponding term in your equation is also proportional to the local electric field gradient, so the importance of this term (relative to the other terms) depends on some of the details of the surroundings. In Appendix C of Gray and Gubbins they state 


  For the cases we consider in this book the induction $A$ terms are negligible


but clearly there might be some cases in which they should not be neglected.