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

Shubhang Krishnakant Trivedi, Phanish Suryanarayana

TL;DR

This work develops a spectral finite-element framework for the optimized effective potential (OEP) method in atomic structure calculations using random phase approximation (RPA) exchange–correlation. It employs a Chebyshev–Gauss–Lobatto mesh with $\,\mathcal{C}^0$-continuous high-order bases and Gauss–Legendre quadrature, allowing distinct interpolation degrees for orbitals, Hartree potential, and xc potential, and it discretizes the governing equations into standard FE matrices for efficient self-consistent solutions. The study validates accuracy against reference cubic-spline results, investigates one-parameter double-hybrid functionals built with RPA correlation, and introduces a kernel-based, GGAs-level machine-learned $V_{xc}$ trained on a small atomic dataset, showing competitive spectral predictions and reasonable energetics. The results demonstrate a scalable, accurate approach for RPA–OEP atomic calculations and point to ML-driven acceleration as a practical route for broader applications in electronic-structure theory.

Abstract

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.

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

TL;DR

This work develops a spectral finite-element framework for the optimized effective potential (OEP) method in atomic structure calculations using random phase approximation (RPA) exchange–correlation. It employs a Chebyshev–Gauss–Lobatto mesh with -continuous high-order bases and Gauss–Legendre quadrature, allowing distinct interpolation degrees for orbitals, Hartree potential, and xc potential, and it discretizes the governing equations into standard FE matrices for efficient self-consistent solutions. The study validates accuracy against reference cubic-spline results, investigates one-parameter double-hybrid functionals built with RPA correlation, and introduces a kernel-based, GGAs-level machine-learned trained on a small atomic dataset, showing competitive spectral predictions and reasonable energetics. The results demonstrate a scalable, accurate approach for RPA–OEP atomic calculations and point to ML-driven acceleration as a practical route for broader applications in electronic-structure theory.

Abstract

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 -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.

Paper Structure

This paper contains 8 sections, 34 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. RPA--OEP atomic structure formalism
  3. Spectral Finite-element Framework
  4. Results and Discussion
  5. Accuracy
  6. Double-hybrid functional
  7. Machine-learned model
  8. Concluding Remarks

Figures (4)

  • Figure 1: Illustration of the spectral finite-element framework developed for RPA--OEP atomic-structure calculations. The example shown uses a domain size of $R_{max} = 10$ bohr, $N_e=5$ elements, polynomial degree $p_1=p_2=p_3=4$ for interpolation, and a quadrature order of $20$.
  • Figure 2: RPA--OEP correlation potentials obtained using the spectral finite-element (FE) framework, along with the cubic-spline (CS) results reported in Ref. gwasphericalatomshellgren, for (a) He, (b) Be, (c) Ne, (d) Mg, and (e) Ar.
  • Figure 3: Variation of the error in ionization potential (IP), HOMO-LUMO gap (Gap), total energy ($E$), and its difference from the Hartree-Fock total energy ($E - E^{HF}$), with parameter $\alpha$ in the double-hybrid functional for (a) He, (b) Be, (c) Ne, (d) Mg, and (e) Ar. The reference values for IP, $E - E_{\mathrm{HF}}$, and $E$ are taken from full Configuration Interaction (CI) calculations chakravarthyberylliumandotherschakravarthyhelium, and the reference values for the Gap are obtained from inversion of quantum Monte Carlo (QMC) densities umrigargonzegwasphericalatomshellgren.
  • Figure 4: Comparison of the RPA--OEP exchange--correlation potential with that predicted from the machine-learned model (RPA-ML) for (a) He, (b) Be, (c) Ne, (d) Mg, and (e) Ar.