This script calculates electron-phonon coupling contribution to exchange interaction $J_{ij}$ for one band model given by nn hopping parameter $t$ and on-site splitting $\Delta$. Exchange coupling is calculated using Green's function technique (see 1, 2):
$$J_{ij}(T) = -\frac{1}{2 \pi S^2} \int \limits_{-\infty}^{E_F} d \omega \,{\rm Im} \left[ \widetilde{\Delta}_i (\omega, T) \widetilde{G}_{ij}^{\downarrow} (\omega, T) \widetilde{\Delta}_j (\omega, T) \widetilde{G}_{ji} ^{\uparrow} (\omega, T) \right],$$
Here $\widetilde{\Delta} (\omega, T)$ and $\widetilde{G}(\omega, T)$ stands for Fourier transformed component of correlated intra-orbital spin-splitting energy and Green's function due to electron-phonon coupling:
$$\widetilde{\Delta}_i (\omega, \mathbf{k}, T) = H_{i}^{\uparrow}(\mathbf{k}) - H_{i}^{\downarrow}(\mathbf{k}) + \Sigma^\uparrow(\omega, \mathbf{k}, T) - \Sigma_{\textrm{elph}}^\downarrow (\omega, \mathbf{k}, T),$$
$$\widetilde{G}_{ij}^{-1} (\omega, \mathbf{k}, T) = G_{ij}^{-1} (\omega, \mathbf{k}) - \Sigma_{\textrm{elph}}(\omega, \mathbf{k}, T).$$
The electron self-energy of electron-phonon interaction in Migdal approximation has the form
$$\Sigma^\sigma_{\textrm{elph}}(\omega, \mathbf{k} T) = g^2 \sum_{\mathbf{q}} \left[ \frac{b_{\mathbf{q}} + f^\sigma_{\mathbf{k + q}}}{\omega - \varepsilon^\sigma_{\mathbf{k + q}} + \hbar \omega_{\mathbf{q}} - i\eta} + \frac{b_{\mathbf{q}} +1 - f^\sigma_{\mathbf{k + q}}}{\omega - \varepsilon^\sigma_{\mathbf{k + q}} - \hbar \omega_{\mathbf{q}} - i\eta} \right] ,$$
where $g$ is electron-phonon coupling approximated with constant value. $b_{\mathbf{q}\nu} = (\exp[\hbar \omega_{\mathbf{q}}/k_BT] - 1)^{-1}$ corresponds to the Bose occupation function for phonons with wave vector $\mathbf{q}$ and frequency $\omega_{\mathbf{q \nu}}$. Phonon spectra is introduced as a single linear branch $\omega_{\mathbf{q}} = v q$ with sound velocity $v$. In turn, $f^\sigma_{\mathbf{k}} = (\exp[(\varepsilon^\sigma_{\mathbf{k}})/k_BT] + 1)^{-1}$ is the Fermi occupation function for the electron states with energy $\varepsilon^\sigma_{\mathbf{k}}$ given respect to the Fermi level.
