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 Recently however I started working on the correlation part as well, with interesting results (unpublished data).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 is to use numerical frequencies. For AMS2020 one should use AMS2020.103 or later. For ADF2019.3 one should use ADF2019.307.

The solution

The bug was solved, and the new comprehensible code is now the default. I have written all equations into the following PDF document, for those who are interested in the derivation of the formulas, and those who need to implement it in some other code. 

Additional information

The "Implementing the PBEc Hessian into ADF" document can be found here.