How the Exchange Energy Can Affect the Power Laws Used to Extrapolate the Coupled Cluster Correlation Energy to the Thermodynamic Limit

Abstract

Finite size error is commonly removed from coupled cluster theory calculations by N −1 extrapolations over correlation energy calculations of different system sizes (N), where the N −1 scaling comes from the total energy rather than the correlation energy. However, previous studies in the quantum Monte Carlo community suggest an exchange-energy-like power law of N −2/3 should also be present in the correlation energy when using the conventional Coulomb interaction. The rationale for this is that the total energy goes as N −1 and the exchange energy goes as N −2/3 ; thus, the correlation energy should be a combination of these two power laws. Further, in coupled cluster theory, these power laws are related to the low G scaling of the transition structure factor, S(G), which is a property of the coupled cluster wave function calculated from the amplitudes. We show here that data from coupled cluster doubles calculations on the uniform electron gas fit a function with a low G behavior of S(G) ∼ G. The prefactor for this linear term is derived from the exchange energy to be consistent with an N −2/3 power law at large N. Incorporating the exchange structure factor into the transition structure factor results in a combined structure factor of S(G) ∼ G 2 , consistent with an N −1 scaling of the exchange-correlation energy. We then look for the presence of an N −2/3 power law in the energy. To do so, we first develop a plane-wave cutoff scheme with less noise than the traditional basis set used for the uniform electron gas. Then, we collect data from a wide range of electron numbers and densities to systematically test five methods using N −1 scaling, N −2/3 scaling, or combinations of both scaling behaviors. We find that power laws that incorporate both N −1 and N −2/3 scaling perform better than either alone, especially when the prefactor for N −2/3 scaling can be found from exchange energy calculations.

Highly accurate electronic structure of metallic solids from coupled-cluster theory with nonperturbative triple excitations

“…Specifically, we perform CCSD and DCSD calculations on systems containing up to N = 1404 electrons and estimate the complete basis set limit using calculations on smaller system sizes. These results are then used to extrapolate to the thermodynamic limit assuming that finite-size errors in the correlation energy decay asymptotically as N −2/3 -a functional form that has also been proposed in recent work [60]. Our final CCSD correlation energies agree within about 1 mE h with previous studies that targeted the thermodynamic limit [6,7], despite different technical details, providing a validation of our methods.…”

Investigating the Basis Set Convergence of Diagrammatically Decomposed Coupled-Cluster Correlation Energy Contributions for the Uniform Electron Gas

“…The static structure factor is a pivotal quantity in the context of periodic electronic structure theory . In CC approaches, a related quantity, the so-called transition structure factor S , was employed in a number of recent works. − The transition structure factor is directly related to the correlation energy contribution at a given momentum transfer q , according to the following equation E c = prefix∑ bold-italicq bold-italicυ ( q ) S ( q ) For energy expressions in the form of eq , the transition structure factor can be written as S ( q ) = ∑ i j a b false( 2 t i j a b − t j i a b false) δ bold-italicq , k a − k i δ bold-italicq , k j − k b where the amplitudes t ij ab have been obtained from calculations with a given finite virtual basis set.…”

“…51 In CC approaches, a related quantity, the so-called transition structure factor S , was employed in a number of recent works. 52 − 54 The transition structure factor is directly related to the correlation energy contribution at a given momentum transfer q , according to the following equation For energy expressions in the form of eq 9 , the transition structure factor can be written as where the amplitudes t ij ab have been obtained from calculations with a given finite virtual basis set.…”