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Inverse Volume Scaling of Finite-Size Error in Periodic Coupled Cluster Theory

Xin Xing, Lin Lin

TL;DR

This paper addresses finite-size errors in periodic coupled-cluster calculations for insulating solids, revealing an unexpected inverse-volume scaling of the CCD correlation-energy error even without corrections. It develops a two-step framework that links Madelung-constant corrections to singularity-subtraction in a rigorous quadrature-error analysis, showing that applying corrections to both orbital energies and ERI contractions reduces the finite-size error to $O(N_{ extbf{k}}^{-1})$ for CCD$(n)$, while convergence of the amplitude equations leads to cancellations and maintains inverse-volume scaling without corrections. The authors validate the theory with CCD/CCD$(n)$ calculations on a 3D periodic hydrogen-dimer system, demonstrating that dual corrections are essential for achieving the fastest convergence and confirming the sharpness of the predicted rates. The work provides practical guidance for reducing finite-size errors in solids, clarifies prior conflicting findings, and offers a solid foundation for extending finite-size analyses to more advanced CC theories and other periodic systems.

Abstract

Coupled cluster theory is one of the most popular post-Hartree-Fock methods for ab initio molecular quantum chemistry. The finite-size error of the correlation energy in periodic coupled cluster calculations for three-dimensional insulating systems has been observed to satisfy the inverse volume scaling, even in the absence of any correction schemes. This is surprising, as simpler theories that utilize only a subset of the coupled cluster diagrams exhibit much slower decay of the finite-size error, which scales inversely with the length of the system. In this study, we review the current understanding of finite-size error in quantum chemistry methods for periodic systems. We introduce new tools that elucidate the mechanisms behind this phenomenon in the context of coupled cluster doubles calculations. This reconciles some seemingly paradoxical statements related to finite-size scaling. Our findings also show that singularity subtraction can be a powerful method to effectively reduce finite-size errors in practical quantum chemistry calculations for periodic systems.

Inverse Volume Scaling of Finite-Size Error in Periodic Coupled Cluster Theory

TL;DR

This paper addresses finite-size errors in periodic coupled-cluster calculations for insulating solids, revealing an unexpected inverse-volume scaling of the CCD correlation-energy error even without corrections. It develops a two-step framework that links Madelung-constant corrections to singularity-subtraction in a rigorous quadrature-error analysis, showing that applying corrections to both orbital energies and ERI contractions reduces the finite-size error to for CCD, while convergence of the amplitude equations leads to cancellations and maintains inverse-volume scaling without corrections. The authors validate the theory with CCD/CCD calculations on a 3D periodic hydrogen-dimer system, demonstrating that dual corrections are essential for achieving the fastest convergence and confirming the sharpness of the predicted rates. The work provides practical guidance for reducing finite-size errors in solids, clarifies prior conflicting findings, and offers a solid foundation for extending finite-size analyses to more advanced CC theories and other periodic systems.

Abstract

Coupled cluster theory is one of the most popular post-Hartree-Fock methods for ab initio molecular quantum chemistry. The finite-size error of the correlation energy in periodic coupled cluster calculations for three-dimensional insulating systems has been observed to satisfy the inverse volume scaling, even in the absence of any correction schemes. This is surprising, as simpler theories that utilize only a subset of the coupled cluster diagrams exhibit much slower decay of the finite-size error, which scales inversely with the length of the system. In this study, we review the current understanding of finite-size error in quantum chemistry methods for periodic systems. We introduce new tools that elucidate the mechanisms behind this phenomenon in the context of coupled cluster doubles calculations. This reconciles some seemingly paradoxical statements related to finite-size scaling. Our findings also show that singularity subtraction can be a powerful method to effectively reduce finite-size errors in practical quantum chemistry calculations for periodic systems.
Paper Structure (20 sections, 14 theorems, 150 equations, 1 figure, 1 table)

This paper contains 20 sections, 14 theorems, 150 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. CCD theory
  4. Madelung constant correction
  5. Main statements
  6. Key ideas
  7. Singularity structure and quadrature error estimate
  8. Madelung constant correction as a quadrature error reduction method
  9. Numerical examples
  10. Discussion
  11. CCD amplitude equation
  12. Proof of Theorem \ref{['thm:error_ccdn']}
  13. Proof Outline
  14. Proof of Lemma \ref{['lem:error_amplitude_contraction_app']}: error in ERI contractions
  15. Proof of \ref{['lem:error_orbital_energy']}: error in orbital energies
  16. ...and 5 more sections

Key Result

Theorem 1

In CCD($n$) calculation, the finite-size error in the correlation energy scales as $\mathcal{O}(N_\mathbf{k}^{-\frac{1}{3}})$ in each of the following scenarios (1) there is no finite-size correction, (2) the Madelung constant correction is only applied to the ERI contraction $\mathcal{A}_{N_\mathbf

Figures (1)

  • Figure 5.1: Convergence of the CCD(1), CCD(2), CCD(3), and CCD correlation energies for a 3D periodic system of hydrogen dimers with increasing $N_\mathbf{k}$. The settings are distinguished by the presence or absence of the Madelung constant correction to the orbital energies ("$\varepsilon_{n\mathbf{k}}^{N_\mathbf{k},\xi}$") and the ERI contraction ("$\mathcal{A}_{N_\mathbf{k},\xi}$"). In (a)-(d), the dashed curves show the power-law fitting using $C_0 + C_1N_\mathbf{k}^{-\frac{1}{3}}$ and data points $N_\mathbf{k}^{\frac{1}{3}} = 3,4,5$ for the two cases with partial Madelung constant correction. Subfigures (e)-(h) plot the curve fittings using $N_\mathbf{k}^{-\frac{1}{3}}$ and $N_\mathbf{k}^{-1}$ over the calculations with correction to both components, numerically corroborating the inverse volume scaling.

Theorems & Definitions (27)

  • Theorem 1
  • proof
  • Theorem 2
  • Corollary 3
  • Definition 4
  • Lemma 5
  • Lemma 6: Singularity structure of the amplitude, Lemma 4 in XingLin2023
  • Lemma 7: Error in ERI contractions
  • Definition 8: Algebraic singularity for univariate functions
  • Definition 9: Algebraic singularity for multivariate functions
  • ...and 17 more