are small.

Estimating $J_{1}$ for SrTiO$_{3}$ {#sec:J1}
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As was shown in , $J_{1}$ sets the scale of the gradient term $C$ in the LGD expansion, and it can therefore be obtained from quantities related to spatial gradients of the polarization. In perovskites, the polarization is closely connected to an optical phonon mode [@cowley64; @atkinson17], pictured in . One can therefore obtain $J_1$ from the phonon dispersion.

Key to this analysis is that the optical phonon has a large dipole moment that is represented by the TIM pseudospins. The phonon spectrum can therefore be obtained from the dynamical pseudospin correlation function. In the paraelectric phase, the term proportional to $\Omega$ in equation  ensures that the pseudospins lie primarily along the (1)-axis. Perturbations of this state can be viewed as the magnons of a fictitious ferromagnetic material in which the magnetic moments align along the (1)-axis. The phonons can then be described as spin-wave excitations.

The spin operators are difficult to work with, however, and it is useful to bosonize them. This is achieved with the Holstein-Primakoff transformation [@holstein; @holstein40]. This transformation maps the pseudospin operators on to the boson creation and annihilation operators, $\hat{a}^{\dag}_{i}$ and $\hat{a}_{i}$. Pseudospin projections on the 