Efficient implementation of analytical Raman intensities

In this study, we report an analytical derivative implementation of static polarizability gradients, needed for the prediction of Raman spectra, in the ORCA program. Our implementation is capable of handling density functionals up to the meta-GGA level and makes use of RIJ and COSX integral approximations to improve the scaling of calculations using both pure and hybrid functionals. Solvent effects can be included with CPCM. We examine the errors in RIJ and COSX Raman activities and find both approximations safe to use in routine calculations. In an effort to further systematically reduce errors resulting from RIJ, we introduce an automatic procedure to minimally augment auxiliary basis sets with diffuse functions if these are detected in the orbital basis set. Compared with numerical derivatives previously available in ORCA, we achieve speedups of up to 1-2 orders of magnitude. Finally, we demonstrate that these developments allow us to calculate fully analytical harmonic Raman spectra of systems containing more than 100 atoms, with the polarizability gradient only introducing a moderate overhead (lower than 10%) to the Hessian calculation.