Connection between DFPT and cDFPT [1] $${\tilde{\omega }}_{q\mu \nu }^2={\omega }_{q\mu \nu }^2{\delta }_{\mu \nu }+2\sqrt{{\omega }_{q\mu }{\omega }_{q\nu }}{\Pi }_{q\mu \nu }$$ Phonon self-energy $${\Pi }_{q\mu \nu }=\frac{2}{N}\sum _k{g}_{qk\mu }^{*}\frac{f\left({\epsilon }_{k+q}\right)-f\left({\epsilon }_k\right)}{{\epsilon }_{k+q}-{\epsilon }_k}{\tilde{g}}_{qk\nu }$$

Add phonon self-energy

$\mu =$	eV
$kT=$	eV
$f\left(\epsilon \right)=$	Fermi–Dirac Gauß Marzari–Vanderbilt Methfessel–Paxton Lorentz

[1] Nomura and Arita, PRB 92, 245108 (2015)