How the exchange energy can affect the power laws used to extrapolate the coupled cluster correlation energy to the thermodynamic limit (2302.05051v1)

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. However, previous studies in the quantum Monte Carlo community suggest an exchange-energy-like power law of $N^{-2/3}$ is 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 as $N^{-2/3}$; so, the correlation energy should be a combination of the 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 wavefunction calculated from the amplitudes. We show that data from coupled cluster doubles calculations on the uniform electron gas(UEG) fit a function with a low $G$ behavior of $S(G)$~$G$. The pre-factor 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. First, we develop a plane-wave cutoff scheme with less noise than the traditional basis set used for the UEG. Then, we collect data from a wide range of electron numbers and densities to systematically test 5 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 pre-factor for $N^{-2/3}$ scaling can be found from exchange energy calculations.