Finite-size effects in periodic coupled cluster calculations

TL;DR

This paper rigorously analyzes the finite-size errors in periodic coupled cluster doubles (CCD) calculations, establishing convergence rates and highlighting the dominant error sources, with implications for improving accuracy in electronic structure computations.

Contribution

It provides the first rigorous proof of finite-size error scaling in CCD and MP3 theories for periodic systems, linking error convergence to system size and amplitude accuracy.

Findings

Finite-size error in CCD scales as $N_k^{-1/3}$.

Accurate amplitudes can improve energy error scaling to $N_k^{-1}$.

First proof of finite-size error scaling in periodic MP3.

Abstract

We provide the first rigorous study of the finite-size error in the simplest and representative coupled cluster theory, namely the coupled cluster doubles (CCD) theory, for gapped periodic systems. Assuming that the CCD equations are solved using exact Hartree-Fock orbitals and orbital energies, we prove that the convergence rate of finite-size error scales as $\mathscr{O}(N_\mathbf{k}^{-\frac13})$, where $N_{\mathbf{k}}$ is the number of discretization point in the Brillouin zone and characterizes the system size. Our analysis shows that the dominant error lies in the coupled cluster amplitude calculation, and the convergence of the finite-size error in energy calculations can be boosted to $\mathscr{O}(N_\mathbf{k}^{-1})$ with accurate amplitudes. This also provides the first proof of the scaling of the finite-size error in the third order M{\o}ller-Plesset perturbation theory (MP3) for periodic systems.