Atomic forces from correlation energy functionals based on the adiabatic-connection fluctuation-dissipation theorem

TL;DR

The paper addresses accurate force calculations within correlation-energy functionals based on the adiabatic-connection fluctuation-dissipation theorem (ACFDT) by deriving and implementing analytical atomic forces for RPA and RPAx in plane-wave/pseudopotential DFT. It presents both self-consistent forces via optimized effective potential (OEP) and non-self-consistent forces using density functional perturbation theory (DFPT) starting from PBE, with an eigenvalue formulation that avoids unoccupied-state sums. The results show high numerical accuracy for both scf- and nscf-forces, with self-consistency having little impact on geometries and vibrational frequencies in most cases; RPA generally improves over PBE, while RPAx, including the exact-exchange kernel, achieves near CCSD(T)-level accuracy for molecules and provides excellent phonon predictions for diamond, silicon, and germanium, including anharmonic shifts. This work establishes a robust path for accurate geometry optimization and phonon calculations beyond semi-local functionals and lays groundwork for future extensions to full RPAx forces and related vibrational properties. Key expressions include $E_{\text{c}}^{\text{ACFDT}}$ and its analytic forms for RPA and RPAx, as well as the force decompositions that account for nonlocal pseudopotentials within the OEP framework.

Abstract

We extend the capabilities of correlation energy functionals based on the adiabatic-connection fluctuation-dissipation theorem by implementing the analytical atomic forces within the random phase approximation (RPA), in the context of plane waves and pseudopotentials. Forces are calculated at self-consistency through the optimized effective potential method and the Hellmann-Feynman theorem. In addition, non-self-consistent RPA forces, starting from the PBE generalized gradient approximation, are evaluated using density functional perturbation theory. In both cases, we find forces of excellent numerical quality. Furthermore, for most molecules and solids studied, self-consistency is found to have a negligible impact on the computed geometries and vibrational frequencies. The RPA is shown to systematically improve over PBE and, by including the exact-exchange kernel within RPA + exchange (RPAx), through finite-difference total energy calculations, we obtain an accuracy comparable to advanced wavefunction methods. Finally, we estimate the anharmonic shift and provide accurate theoretical references based on RPA and RPAx for the zone-center optical phonon of diamond, silicon, and germanium.

Figures (11)

• Figure 1: Change in the total force for the fluorane molecule according to a) HF, EXX, and the uncorrected EXX ($-{\bf F}^{\rm scf\text{-}EXX}_{\rm x}$), b) RPA, RPA[PBE], and the uncorrected RPA ($-{\bf F}^{\rm scf\text{-}RPA}_{\rm xc}$).
• Figure 2: Evolution of a) the total energy, b) the total force, and c) the bond distance during the BFGS optimization procedure of the FH molecule.
• Figure 3: Estimated error for the value of the force exerted on a) a hydrogen atom (labeled H$^1$) along the O-H$^1$ bond of the water H$_2$O molecule and b) a boron ion (labeled B$^1$) along the z-direction of the $\omega$-BN crystal. The evolution is monitored with respect to changes in their position, using different approximations.
• Figure 4: Relative error with respect to CCSD(T) for the structural parameters (bond distance, bond angle) listed in Table \ref{['tab:Tab_Molecules_Structure_RPA']}.
• Figure 5: Relative error with respect to CCSD(T) for the various harmonic vibrational frequencies listed in Table \ref{['tab:Tab_Molecules_Frequency_RPA']}.