# What is the polarization catastrophe?

I heard about the polarization catastrophe in a talk about polarizable force fields.

The speaker talked about how the dipole induction is damped at short range to avoid the “polarization catastrophe". For example, as in this paper.

Apparently if two induced dipoles get two close together this leads to a singularity in the $$T$$ matrix, which is the polarization matrix, because two rows of the matrix are degenerate.

But what is the polarization catastrophe?

Apparently this is also important in other contexts as well so any details would be appreciated.

In short, the polarization catastrophe is the fact that the classical description of polarization applied to point dipoles contains a singularity which occurs at a distance which should regularly be sampled in the course of ordinary classical dynamics.

More specifically, the polarization catastrophe is the failure that happens when one tries to describe the interactions of molecules with polarizable atoms without any kind of short-range damping. There is a very popular solution to this problem called Thole-damping[1]. Thole explains the basics of the problem in Ref. [1], so I will just paraphrase some portions of what is explained there, but the whole paper is quite a good read.

The polarizability is the dipole response of a molecule to an applied field, $$\mu_{mol}=\alpha_{mol}E \tag{1}$$

Getting $$\alpha_{mol}$$ is actually somewhat non-trivial as it requires contracting a tensor of all the atomic polarizabilities into a single tensor describing the molecular response. Thole provides the analytic solution to this for the case of a diatomic both parallel and perpendicular to the bond axis:

$$\alpha_{\mathbin{\|}}=\frac{\alpha_A+\alpha_B+4\alpha_A\alpha_B/r^3}{1-4\alpha_A\alpha_B/r^6}\tag{2}$$ $$\alpha_{\perp}=\frac{\alpha_A+\alpha_B-2\alpha_A\alpha_B/r^3}{1-\alpha_A\alpha_B/r^6}\tag{3}$$

This has a singularity in $$\alpha_{\mathbin{\|}}$$ when $$r=(4\alpha_A\alpha_B)^{1/6}$$. So, in other words, the polarizability blows up at certain atom-atom distances, which tends to be catastrophic for the results of your simulation. Hence, polarization catastrophe.

This is somewhat interesting because clearly this doesn't happen in nature, so some kind of damping is required. Thole damping is, in my opinion, a very elegant way of solving the problem. The paper explains it better than I will. Essentially, the method changes the length scale associated with the dipole interactions that nonetheless follow the appropriate limits and makes infinite polarizabilities much less likely. In the last paragraph of section 3, some approximate bounds on the stability of the polarization are given.

Note that there are other methods of calculating polarization that also avoid the polarization catastrophe. Drude oscillators tend to be good at avoiding this problem I think. Don't take my word for that though.

References:

[1] Thole, B. T. (1981). Molecular polarizabilities calculated with a modified dipole interaction. Chemical Physics, 59(3), 341-350.