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Using density functional perturbation theory (DFPT) we computed the phononfrequencies, Raman and IR activities of hafnia polymorphs (P4$_{2}$nmc,Pca2$_{1}$, Pmn2$_{1}$, Pbca OI, brookite, and baddeleyite) for phaseidentification. We investigated the evolution of Raman and IR activities withrespect to epitaxial strain and provide plots of frequency differences as afunction of strain for experimental calibration and identification of thestrain state of the sample. We found Raman signatures of different hafniapolymorphs: $\omega(A_{1g})=300$ cm$^{-1}$ for P4$_{2}$nmc, $\omega(A_{1})=343$cm$^{-1}$ for Pca2$_{1}$, $\omega(B_{2})=693$ cm$^{-1}$ for Pmn2$_{1}$,$\omega(A_{g})=513$ cm$^{-1}$ for Pbca (OI), $\omega(A_{g})=384$ cm$^{-1}$ forbrookite, and $\omega (A_{g}) = 496$ cm$^{-1}$ for baddeleyite. We alsoidentified the Raman $B_{1g}$ mode, an anti-phase vibration of dipole moments,( $\omega(B_{1g}) = 758$ cm$^{-1}$ for OI, $\omega (B_{1g})= 784$ cm$^{-1}$ forbrookite) as the Raman signature of antipolar Pbca structures. We calculated alarge splitting between longitudinal optical (LO) and transverse optical (TO)modes ($\Delta{\omega_{\text{LO-TO}}(A^{z}_{1})}=255$ cm$^{-1}$ in Pca2$_{1}$,and $\Delta{\omega_{\text{LO-TO}}(A_{1})}=263$ cm$^{-1}$ in Pmn2$_{1}$) to thesame order as those observed in perovskite ferroelectrics, and related them tothe anomalously large Born effective charges of Hf atoms ($Z^{*}(\text{Hf}) =5.54$).

First-principles Calculations of Raman and Infrared Spectroscopy For Phase Identification and Strain Calibration of Hafnia