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.
- Received 18 April 2023
- Revised 17 January 2024
- Accepted 13 February 2024
DOI:https://doi.org/10.1103/PhysRevX.14.011059
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
In a periodic system, replicates of a unit cell span an infinite space. Computations of the theoretical properties of such systems, however, require a finite space. Researchers therefore apply quantum chemistry methods to “computational supercells,” finite boxes that contain some number of unit cells, and extrapolate the properties of the infinite system from there. This results in significant finite-size error, analysis of which is crucial for understanding how the system size affects the calculation accuracy and how to improve it. We present a comprehensive analysis of the finite-size error in a periodic coupled cluster (CC) calculation, the gold-standard ansatz of molecular quantum chemistry.
Specifically, we focus on the correlation energy, the part of the energy of the electrons that is truly many body in its nature. Recent numerical results suggest that the finite-size error of this energy in periodic CC calculations for 3D insulating systems satisfies inverse volume scaling: As the supercell expands to fill the whole space, the error should decay to zero at a rate that is inversely proportional to the volume of the supercell. This is surprising, as the finite-size error of a single component of the CC theory already exhibits much slower decay, scaling inversely with the length of the system.
We provide a rigorous numerical analysis that explains the underlying mechanisms behind this phenomenon. The key idea is that a finite periodic CC calculation of the correlation energy can be decomposed into multiple numerical quadrature calculations—a technique for numerically approximating an integral—that converge to corresponding integrals in the thermodynamic limit. The finite-size error can thus be interpreted as numerical quadrature errors and then be estimated in a rigorous way.
Our findings set the stage for effectively addressing finite-size errors in practical quantum chemistry calculations for periodic systems.
