Automatic differentiation approach for obtaining exchange-correlation functional derivatives*

Obtaining explicit analytical expressions for exchange-correlation (XC) functional derivatives in density functional theory (DFT) can be tedious. Automatic differentiation methods like those implemented in the JAX framework allow us to take derivatives to the n-th order of a defined function. Here, we start from the analytical form of a known exchange-correlation energy density (ε_xc) and use automatic differentiation to get function descriptions of the first (v_xc) and second (f_xc) derivatives, known as the potential and kernel, respectively. The kernel is required for excited-state linear-response calculations in the Casida formulation of time-dependent density functional theory (TD-DFT). Simulated electronic excitation spectra using our functionals obtained from automatic differentiation show excellent agreement with the same using analytical descriptions stored within the widely-used libxc library. We scale this framework for use with meta-GGAs, orbital (OEP) and hybrid functionals, many-body dispersion functionals, as well as functionals that include electron-photon interactions (QEDFT). Newly derived functionals that only have an ε_xc expression could also make use of this framework for easier implementation in higher-order applications.