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Derivation of the second derivatives of PBE correlation

 Background In both my SSB-D ( DOI 10.1063/1.3213193 ) and S12g ( DOI 10.1016/j.cplett.2013.06.045 ) functionals I started from the non-empirical PBE functional , changing mostly the exchange part and adding Grimme dispersion . Recently however I started working on the correlation part as well, with interesting results ( unpublished data ). The problem The code for the correlation part of PBE in ADF/AMS was computer generated, and hence incomprehensible. After writing the code, and checking it, I found differences with the results from the standard (computer generated) code, for spin-polarized systems . It turned out that there was a bug in the code: Bug PBEc The bug for analytical frequencies calculations of spin-polarized molecules using PBEc affects calculations with the exchange-correlation functionals PBE, OPBE, RPBE, revPBE, and S12g. This bug was introduced in ADF2006 and fixed in AMS2020.103. Analytical frequencies using LibXC were not affected. Workaround for older versions i

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