Dipole-quadrupole polarizability term for induced dipole For describing an induced dipole, I have usually seen the following equation,
$$
P_{i} = \alpha_{ij}E_{j} + \frac{1}{2}\beta_{ijk}E_{j}E_{k}
$$
where $P_{i}$ is the $i^{\text{th}}$ component of the induced dipole moment, $\alpha$ is the (dipole) polarizability and $\beta$ is the first (dipole) hyperpolarizability.

However, I recently read a paper with quadrupole-coupled terms coming in, like the dipole-quadrupole polarizability as shown below:
$$
P_{i} = \alpha_{ij}E_{j} + \frac{1}{3}A_{ijk} \frac{dE_{j}}{dr_{k}}+ \frac{1}{2}\beta_{ijk}E_{j}E_{k}
$$
where in the second term, $A_{ijk}$ is the dipole-quadrupole polarizability term. This equation is introduced for the case of polar molecules in the condensed phase, and in the related paper it is for water.

It appears that the contribution of dipole-quadrupole polarizability is rather small, but maybe larger than the hyperpolarizability.

Could someone explain or refer some literature so as to describe


the importance of the cross-coupled terms like the dipole-quadrupole polarizability, and
if there are some cases where the cross-coupled terms (the second term in the second equation) become negligible?




Reference: Batista, E. R.; Xantheas, S. S.; Jónsson, H. Molecular multipole moments of water molecules in ice Ih. J. Chem. Phys. 1998, 109 (11), 4546-4551. DOI: 10.1063/1.477058.
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.