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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.

Density matrix quantum Monte Carlo (DMQMC) is a recently developed method for stochastically sampling the N -particle thermal density matrix to obtain exact-on-average energies for model and ab initio systems. We report a systematic numerical study of the sign problem in DMQMC based on simulations of atomic and molecular systems. In DMQMC, the density matrix is written in an outer product basis of Slater determinants. In principle, this means that DMQMC needs to sample a space that scales in the system size, N , as O[(exp(N)) 2 ]. In practice, removing the sign problem requires a total walker population that exceeds a system-dependent critical walker population ( N c ), imposing limitations on both storage and compute time. We establish that N c for DMQMC is the square of N c for FCIQMC. By contrast, the minimum N c in the interaction picture modification of DMQMC (IP-DMQMC) is only linearly related to the N c for FCIQMC. We find that this difference originates from the difference in propagation of IP-DMQMC versus canonical DMQMC: the former is asymmetric, whereas the latter is symmetric. When an asymmetric mode of propagation is used in DMQMC, there is a much greater stochastic error and is thus prohibitively expensive for DMQMC without the interaction picture adaptation. Finally, we find that the equivalence between IP-DMQMC and FCIQMC seems to extend to the initiator approximation, which is often required to study larger systems with large basis sets. This suggests that IP-DMQMC offers a way to ameliorate the cost of moving between a Slater determinant space and an outer product basis.

Recent calculations using coupled cluster on solids have raised the 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 the basis set incompleteness error and use a selected twist angle approach to mitigate the 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 .

We here apply the recently developed initiator density matrix quantum Monte Carlo (i-DMQMC) to a variety of atoms and molecules in vacuum. i-DMQMC samples the exact density matrix of a Hamiltonian at finite temperature and combines the accuracy of full configuration interaction quantum Monte Carlo (FCIQMC)-full configuration interaction (FCI) or exact energies in a finite basis set-with finite temperature. In order to explore the applicability of i-DMQMC for molecular systems, we choose to study a recently developed test set by Rubenstein and co-workers: Be, H 2 O, and H 10 at near-equilibrium and stretched geometries. We find that, for Be and H 2 O, i-DMQMC delivers energies with submillihartree accuracy when compared with finite temperature FCI. For H 2 O and both geometries of H 10 , we examine the difference between FT-AFQMC and i-DMQMC, which, in turn, estimates the difference in canonical versus grand canonical energies. We close with two discussions: one of simulation settings (initiator error, the interaction picture, and different basis sets), and another of energy difference calculations in the form of specific heat capacity and ionization potential calculations.

We here develop a fully quantum embedded version of initiator full configuration interaction quantum Monte Carlo ( i -FCIQMC) and apply it to study an ionic bond (lithium hydride, LiH) and a covalent bond (hydrogen flouride, HF) physisorbed to a benzene molecule. The embedding is performed using a recently developed Huzinaga projection operator approach, which affords good synergy with i -FCIQMC by minimizing the number of orbitals in the calculation. When considering the dissociation energy of these bonds into closed-shell ionic fragments, we find that i -FCIQMC embedded in density functional theory ( i -FCIQMC-in-DFT) delivers comparable accuracy with coupled cluster singles and doubles with perturbative triples embedded in density functional theory (CCSD(T)-in-DFT). In treating the bond dissociation energy curve of HF, i -FCIQMC-in-DFT has improved accuracy over CCSD(T)-in-DFT due to the presence of strong correlation. We discuss the implications of the new i -FCIQMC-in-DFT method as applied to bond breaking in catalysis.

Building on the success of Quantum Monte Carlo techniques such as diffusion Monte Carlo, alternative stochastic approaches to solve electronic structure problems have emerged over the past decade. The full configuration interaction quantum Monte Carlo (FCIQMC) method allows one to systematically approach the exact solution of such problems, for cases where very high accuracy is desired. The introduction of FCIQMC has subsequently led to the development of coupled cluster Monte Carlo (CCMC) and density matrix quantum Monte Carlo (DMQMC), allowing stochastic sampling of the coupled cluster wave function and the exact thermal density matrix, respectively. In this Article, we describe the HANDE-QMC code, an open-source implementation of FCIQMC, CCMC and DMQMC, including initiator and semistochastic adaptations. We describe our code and demonstrate its use on three example systems; a molecule (nitric oxide), a model solid (the uniform electron gas), and a real solid (diamond). An illustrative tutorial is also included.