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We present a robust strategy to numerically sample the Coulomb potential in reciprocal space for periodic Born-von Karman cells of general shape. Our approach tackles two common issues of plane-wave based implementations of Coulomb integrals under periodic boundary conditions, the treatment of the singularity at the Brillouin-zone center, as well as quadrature errors, which can cause severe convergence problems in anisotropic cells, necessary for the calculation of low-dimensional systems. We apply our strategy to the Hartree-Fock (HF) and coupled cluster (CC) theory and discuss the consequences of different sampling strategies on the different theories. We show that sampling the Coulomb potential via the widely used probe-charge Ewald method is unsuitable for CC calculations in anisotropic cells. To demonstrate the applicability of our developed approach, we study two representative, low-dimensional use cases: the infinite carbon chain, for which we report the first periodic CCSD(T) potential energy surface, as well as a surface slab of lithium hydride, for which we demonstrate the impact of different sampling strategies for calculating surface energies. We find that our Coulomb sampling strategy serves as a vital solution, addressing the critical need for improved accuracy in plane-wave based CC calculations for low-dimensional systems., Comment: 7 pages, 3 figures

We present a machine learning approach to calculating electronic specific heat capacities for a variety of benchmark molecular systems. Our models are based on data from density matrix quantum Monte Carlo, which is a stochastic method that can calculate the electronic energy at finite temperature. As these energies typically have noise, numerical derivatives of the energy can be challenging to find reliably. In order to circumvent this problem, we use Gaussian process regression to model the energy and use analytical derivatives to produce the specific heat capacity. From there, we also calculate the entropy by numerical integration. We compare our results to cubic splines and finite differences in a variety of molecules whose Hamiltonians can be diagonalized exactly with full configuration interaction. We finally apply this method to look at larger molecules where exact diagonalization is not possible and make comparisons with more approximate ways to calculate the specific heat capacity and entropy.

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 \emph{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 and has a size of space which is the square of the number of Slater determinants. In principle this means DMQMC needs to sample a space which scales in the system size, $N$, as $\mathcal{O}[(\exp(N))^2]$. In practice, there is a system-dependent critical walker population ($N_c$) which must be exceeded in order to remove the sign problem, and this imposes limitations by way of storage and computer 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) only is directly proportionate to the $N_c$ for FCIQMC. We find that this comes from the asymmetric propagation of IP-DMQMC compared to the symmetric propagation of canonical DMQMC. An asymmetric mode of propagation is prohibitively expensive for DMQMC because it has a much greater stochastic error. 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 basis sets and other systems. This suggests IP-DMQMC offers a way to ameliorate the cost of moving between a Slater determinant space and an outer product basis., Comment: 18 pages, 23 figures

We recently developed a scheme to use low-cost calculations to find a single twist angle where the couple cluster doubles energy of a single calculation matches the twist-averaged coupled cluster doubles energy in a finite unit cell. We used initiator full configuration interaction quantum Monte Carlo ($i$-FCIQMC) as an example of an exact method beyond coupled cluster doubles theory to show that this selected twist angle approach had comparable accuracy in methods beyond coupled cluster. Further, at least for small system sizes, we show that the same twist angle can also be found by comparing the energy directly (at the level of second-order Moller-Plesset theory) suggesting a route toward twist angle selection which requires minimal modification to existing codes which can perform twist averaging., Comment: 9 figures, 9 pages