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$\textit{Ab initio}$ correlated calculations without finite basis-set error: Numerically precise all-electron RPA correlation energies for diatomic molecules

Hao Peng, Xinguo Ren

TL;DR

The authors present a numerically exact framework to obtain basis-error-free all-electron RPA correlation energies for diatomic molecules by solving the Sternheimer equations on a real-space prolate spheroidal grid and iteratively diagonalizing the density-response operator $\chi^0(i\omega)v$. This approach eliminates single-particle basis-set errors, enabling meV-level convergence and providing unbiased reference data to benchmark BSSE corrections, frozen-core approximations, and CBS extrapolations, while guiding the development of improved basis sets. Key contributions include a detailed convergence analysis with respect to grid density, $M_{\max}$, $r_{\infty}$, and occupied-state treatment; a rigorous assessment of BSSE and FC errors, and the demonstration that the method yields highly accurate FC-RPA binding energies in a broad diatomic set, with extension to SOS-MP2 and, via finite-element methods, to general polyatomic systems. The work offers a robust, transferable standard for evaluating correlated methods and guiding the design of numerically precise, basis-set-free quantum-chemical procedures applicable to molecules and beyond.

Abstract

In wavefunction-based $\textit{ab-initio}$ quantum mechanical calculations, achieving absolute convergence with respect to the one-electron basis set is a long-standing challenge. In this work, using the random phase approximation (RPA) electron correlation energy as an example, we show how to compute the basis-error-free RPA correlation energy for diatomic molecules by iteratively solving the Sternheimer equations for first-order wave functions in the prolate spheroidal coordinate system. Our approach provides RPA correlation energies across the periodic table to any desired precision; in practice, the convergence of the absolute RPA energies to the meV-level accuracy can be readily attained. Our method thus provides unprecedented reference numbers that can be used to assess the reliability of the commonly used computational procedures in quantum chemistry, such as the counterpoise correction to the basis set superposition errors, the frozen-core approximation, and the complete-basis-set extrapolation scheme. Such reference results can also be used to guide the development of more efficient correlation-consistent basis sets. The numerical techniques developed in the present work also have direct implications for the development of basis error-free schemes for the GW method or the \textit{ab initio} quantum chemistry methods such as MP2.

$\textit{Ab initio}$ correlated calculations without finite basis-set error: Numerically precise all-electron RPA correlation energies for diatomic molecules

TL;DR

The authors present a numerically exact framework to obtain basis-error-free all-electron RPA correlation energies for diatomic molecules by solving the Sternheimer equations on a real-space prolate spheroidal grid and iteratively diagonalizing the density-response operator . This approach eliminates single-particle basis-set errors, enabling meV-level convergence and providing unbiased reference data to benchmark BSSE corrections, frozen-core approximations, and CBS extrapolations, while guiding the development of improved basis sets. Key contributions include a detailed convergence analysis with respect to grid density, , , and occupied-state treatment; a rigorous assessment of BSSE and FC errors, and the demonstration that the method yields highly accurate FC-RPA binding energies in a broad diatomic set, with extension to SOS-MP2 and, via finite-element methods, to general polyatomic systems. The work offers a robust, transferable standard for evaluating correlated methods and guiding the design of numerically precise, basis-set-free quantum-chemical procedures applicable to molecules and beyond.

Abstract

In wavefunction-based quantum mechanical calculations, achieving absolute convergence with respect to the one-electron basis set is a long-standing challenge. In this work, using the random phase approximation (RPA) electron correlation energy as an example, we show how to compute the basis-error-free RPA correlation energy for diatomic molecules by iteratively solving the Sternheimer equations for first-order wave functions in the prolate spheroidal coordinate system. Our approach provides RPA correlation energies across the periodic table to any desired precision; in practice, the convergence of the absolute RPA energies to the meV-level accuracy can be readily attained. Our method thus provides unprecedented reference numbers that can be used to assess the reliability of the commonly used computational procedures in quantum chemistry, such as the counterpoise correction to the basis set superposition errors, the frozen-core approximation, and the complete-basis-set extrapolation scheme. Such reference results can also be used to guide the development of more efficient correlation-consistent basis sets. The numerical techniques developed in the present work also have direct implications for the development of basis error-free schemes for the GW method or the \textit{ab initio} quantum chemistry methods such as MP2.

Paper Structure

This paper contains 18 sections, 60 equations, 9 figures, 16 tables.

Table of Contents

  1. Numerical methods
  2. Prolate spheroidal coordinates system
  3. Sternheimer equation in prolate spheroidal coordinates system
  4. Iterative diagonalization of density response function
  5. Implementation details
  6. Obtaining Hamiltonian on real space grids
  7. Grids finite difference and sparse matrix linear equation system
  8. Scalar-relativistic Sternheimer equation within atomic ZORA approxiation.
  9. Convergence test
  10. Convergence with respect to the real-space grid density.
  11. Derivation of the $M_{\text{max}}$-dependence behavior of the RPA correlation energy
  12. Convergence with respect to the $r_{\infty}$.
  13. Convergence with respect to the occupied states given by numerical atomic orbital.
  14. Assessing the basis set superposition error
  15. Error due to the frozen-core approximation
  16. ...and 3 more sections

Figures (9)

  • Figure 1: RPA@PBE binding energy curves of Kr$_2$ obtained using different basis sets, in comparison with the reference curves obtained using the approach developed in this work. BSSEs are corrected for cc-pV5Z and aug-cc-pV5Z basis sets; for NAO-VCC-5Z, both CP-corrected and uncorrected results are presented. The FC approximation is used for all calculations.
  • Figure S1: Schematic diagram of prolate spheroidal coordinates system
  • Figure S2: The convergence behavior of correlation energy with respect to $M_{\text{max}}$ for the N atom
  • Figure S3: The convergence behavior of correlation energy with respect to $M_{\text{max}}$ for N$_2$.
  • Figure S4: BSSEs for the N$_2$ dimer for three series of AO basis sets. The value of BSSE is obtained by subtracting the energy of an atom with a ghost atom around from that of an isolated atom. The black line represents the BSSE of the RPA correlation energy obtained using the traditional SOS method, while the red line represents the BSSE of the RPA correlation energy obtained based on the present method. The blue line represents the BSSE error of the (non-self-consistent) Hartree-Fock energy component.
  • ...and 4 more figures