Power laws used to extrapolate the coupled cluster correlation energy to the thermodynamic limit

TL;DR

This study investigates the appropriate power law for extrapolating coupled cluster correlation energies to the thermodynamic limit in solids, finding that $N^{-1}$ applies at large sizes despite initial appearances of $N^{-1/3}$ fit.

Contribution

The paper reconciles differing views on the power law used for extrapolation by analyzing large-scale coupled cluster calculations and transition structure factors.

Findings

$N^{-1/3}$ fits small systems better initially.

$N^{-1}$ applies to large system sizes.

Transition structure factor analysis supports $N^{-1}$ applicability.

Abstract

Recent calculations using coupled cluster on solids have raised discussion of using a $N^{-1/3}$ power law to fit the correlation energy when extrapolating to the thermodynamic limit, an approach which differs from the more commonly used $N^{-1}$ power law which is (for example) often used by quantum Monte Carlo methods. In this paper, we present one way to reconcile these viewpoints. Coupled cluster doubles calculations were performed on uniform electron gases reaching system sizes of $922$ electrons for an extremely wide range of densities ($0.1<r_s<100.0$) to study how the correlation energy approaches the thermodynamic limit. The data were corrected for basis set incompleteness error and use a selected twist angle approach to mitigate finite size error from shell filling effects. Analyzing these data, we initially find that a power law of $N^{-1/3}$ appears to fit the data better than a $N^{-1}$ power law in the large system size limit. However, we provide an analysis of the transition structure factor showing that $N^{-1}$ still applies to large system sizes and that the apparent $N^{-1/3}$ power law occurs only at low $N$.