I'm looking for an explanation of how the calculation of nuclear gradients works in DFT in the context of geometric optimization. Mostly, I want to know what equations are used to calculate gradients (for example, using one of the functionals LDA, BLYP ...) in analytical form, and especially how changing the exchange-correlation functional can affect the result
The equations used in Hartree-Fock and Kohn-Sham-DFT are the same, they are just applied to the different results:
Article An efficient implementation of second analytical derivatives...
With the question how changing the xc functional changes the outcome you run into a typical issue of KS-DFT: it is not systematically improvable. Therefore, swapping from VWN-LDA to B3LYP will usually provide an increase of result quality, but there's no guarantee for that.
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