Spectral finite-element formulation of the optimized effective potential method for atomic structure in the random phase approximation

We present a spectral finite-element formulation of the optimized effective potential (OEP) method for atomic structure calculations in the random phase approximation (RPA). In particular, we develop a finite-element framework that employs a polynomial mesh with element nodes placed according to the Chebyshev-Gauss-Lobatto scheme, high-order $\mathcal{C}^0$-continuous Lagrange polynomial basis functions, and Gauss-Legendre quadrature for spatial integration. We employ distinct polynomial degrees for the orbitals, Hartree potential, and RPA-OEP exchange-correlation potential. Through representative examples, we verify the accuracy of the developed framework, assess the fidelity of one-parameter double-hybrid functionals constructed with RPA correlation, and develop a machine-learned model for the RPA-OEP exchange-correlation potential at the level of the generalized gradient approximation, based on the kernel method and linear regression.