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.

💡 Research Summary

The paper introduces a spectral finite‑element (SFE) framework for solving the optimized effective potential (OEP) equations in atomic structure calculations that employ the random phase approximation (RPA) for exchange‑correlation. The authors first review the need for fifth‑rung density‑functional approximations, emphasizing that RPA provides exact exchange and a non‑local correlation functional capable of describing van‑der‑Waals forces, self‑interaction correction, and metallic as well as small‑gap systems. However, the non‑local nature of the RPA correlation makes self‑consistent calculations prohibitively expensive unless the OEP method is used to replace the non‑local operator with a local multiplicative potential Vₓc(r).

To discretize the radial Kohn‑Sham equations, the Hartree potential, and the OEP potential, the authors partition the semi‑infinite radial domain