Mihm TN, Weiler L, Shepherd JJ. How the Exchange Energy Can Affect the Power Laws Used to Extrapolate the Coupled Cluster Correlation Energy to the Thermodynamic Limit.
J Chem Theory Comput 2023;
19:1686-1697. [PMID:
36918372 PMCID:
PMC10061680 DOI:
10.1021/acs.jctc.2c00737]
[Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/16/2022] [Indexed: 03/16/2023]
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) ∼ G2, 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.
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