# GGA functionals requiring second derivatives?

In "Lecture Notes in Quantum Chemistry II" from the European Summer School in Quantum Chemistry, the authors note that when discussing the LYP functional and other GGA functionals, that

Thus one can immediately see that an evaluation of $$v_{xc}$$ demands an evaluation fo the second derivatives of basis functions. (pp102).

Indeed, when (for a functional like B88) I start with the functional itself and compute the functional derivative (either by hand or with a CAS), I arrive at a (messy) expression that involves second derivatives with respect to space. However, libraries like libxc only need scalar functions of the gradient to be passed, in particular $$\nabla \rho \cdot \nabla \rho$$ , in order to compute $$v_{xc}$$. Now libxc contains the expressions for its functionals in Maple and then performs symbolic differentiation on them and then uses the maple-to-C translator; thus the actual C code in libxc is uninterpretable by a human.

Is there a known expression for $$v_{xc}$$ for B88 or another GGA-like functional that might give me a hint as to how to simplify the functional derivative to only depend on these parameters?

• The NWChem code for PW91 and B88 looks handwritten and may be understandable. I do not have the papers at hand, so I cannot judge. – TAR86 Aug 31 '20 at 5:41

It turns out that my original reference is now somewhat out of date, and it is quite possible to compute the relevant properties without resorting to second spatial derivatives. [Johnson1993] has a nice overview of the relevant math, including clean derivations of both YLP and B88 functional terms, and states:

Note that Eq. (7) does not require evaluation of the spin density second de- rivatives as is necessary in many other formulations, a major computational advantage [48]. The fact that these can be avoided has been previously noted by Kobayashi et al.

Johnson, B. G., Gill, P. M. W., & Pople, J. A. (1993). The performance of a family of density functional methods. The Journal of Chemical Physics, 98(7), 5612–5626. https://doi.org/10.1063/1.464906

and

Kobayashi, K., Kurita, N., Kumahora, H., & Tago, K. (1991). Bond-Energy calculations of Cu2, Ag2, and CuAg with the generalized gradient approximation. Physical Review A, 43(11), 5810–5813. https://doi.org/10.1103/PhysRevA.43.5810