I'm studing DFT and I would like to know if there is an analytic expression for the first and second order derivatives of the PES obtained through the functional B3LYP. If so, I would need some articles about it.


1 Answer 1


Yes, the expressions for the first and second order derivatives of most common functionals are known. However they include a term (the XC term) that requires numerical integration to calculate. So, although the expressions are analytic, they cannot be evaluated analytically, but rather have to be evaluated numerically.

The formulas of DFT energy first order derivatives, including explicit expressions for B3LYP, were reported here, while those of the second order derivatives were reported here.

  • $\begingroup$ If you look at the paper cited above for first derivatives, it actually cites the well-known 1992 paper on implementing KS-DFT in Gaussian basis sets, doi.org/10.1016/0009-2614(92)85009-Y. However, this was also done before e.g. by Baerends et al in the 1970s as far as I remember. $\endgroup$ Commented May 3, 2022 at 20:18
  • $\begingroup$ Also, density functional approximations like B3LYP have closed-form expressions and their derivatives are evaluated analytically. The analytical calculation does include numerical quadrature, which is already necessary for evaluating the exchange-correlation energy as the integrals aren't expressable in a closed form. The quadrature can, however, be made very accurate by using sufficiently large grids. $\endgroup$ Commented May 3, 2022 at 20:20

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