Department of Chemistry, University of Iowa, Iowa City, Iowa 52242-1294, United States.
J Chem Theory Comput. 2023 Mar 28;19(6):1686-1697. doi: 10.1021/acs.jctc.2c00737. Epub 2023 Mar 14.
Finite size error is commonly removed from coupled cluster theory calculations by extrapolations over correlation energy calculations of different system sizes (), where the 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 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 and the exchange energy goes as ; 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 scaling of the transition structure factor, (), 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 behavior of () ∼ . The prefactor for this linear term is derived from the exchange energy to be consistent with an power law at large . Incorporating the exchange structure factor into the transition structure factor results in a combined structure factor of () ∼ , consistent with an scaling of the exchange-correlation energy. We then look for the presence of an 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 scaling, scaling, or combinations of both scaling behaviors. We find that power laws that incorporate both and scaling perform better than either alone, especially when the prefactor for scaling can be found from exchange energy calculations.
有限尺寸误差通常通过对不同系统大小的相关能计算进行外推来从耦合簇理论计算中消除（），其中标度来自总能量而不是相关能。然而，量子蒙特卡罗社区的先前研究表明，当使用常规库仑相互作用时，相关能中也应该存在类似于交换能的幂律。这样做的理由是总能量与交换能量呈 关系，因此相关能应该是这两种幂律的组合。此外，在耦合簇理论中，这些幂律与跃迁结构因子的低 标度（）有关，（）是从振幅计算得出的耦合簇波函数的一个性质。我们在这里表明，均匀电子气上的耦合簇双计算的数据符合具有低 行为的函数（） ∼ 。此线性项的前因子是从交换能推导出的，以便在大 时与 幂律一致。将交换结构因子纳入跃迁结构因子会导致组合结构因子（） ∼ ，这与交换相关能的 标度一致。然后，我们寻找能量中是否存在 幂律。为此，我们首先开发了一种平面波截止方案，与传统用于均匀电子气的基组相比，该方案噪声更小。然后，我们从广泛的电子数和密度收集数据，以系统地测试五种使用 标度、 标度或两者的组合标度行为的方法。我们发现，包含 和 标度的幂律比单独使用任何一种都表现更好，尤其是当 标度的前因子可以从交换能计算中找到时。
