Communication: An efficient and accurate perturbative correction to initiator full configuration interaction quantum Monte Carlo

Distributed Implementation of Full Configuration Interaction for One Trillion Determinants

Full configuration interaction (FCI) can provide an exact molecular ground-state energy within a given basis set and serve as a benchmark for approximate methods in quantum chemical calculations, including the emerging variational quantum eigensolver. However, its exponential computational and memory requirements easily exceed the capability of a single server and limit its applicability to large molecules. In this paper, we present a distributed FCI implementation employing a hybrid parallelization scheme with multithreading and multiprocessing to expand FCI's applicability. We optimize this scheme to minimize the bottlenecks arising from interprocess communications and interthread data management. Our implementation achieves higher scalability than the naive combination of prior works and successfully calculates the exact energy of C3H8/STO-3G with 1.31 trillion determinants, which is the largest FCI calculation to the best of our knowledge. Furthermore, we provide a comprehensive list of FCI results with 136 combinations of molecules and basis sets for future evaluation and development of approximate methods.

Full Configuration Interaction Excited-State Energies in Large Active Spaces from Subspace Iteration with Repeated Random Sparsification

We present a stable and systematically improvable quantum Monte Carlo (QMC) approach to calculating excited-state energies, which we implement using our fast randomized iteration method for the full configuration interaction problem (FCI-FRI). Unlike previous excited-state quantum Monte Carlo methods, our approach, which is based on an asymmetric variant of subspace iteration, avoids the use of dot products of random vectors and instead relies upon trial vectors to maintain orthogonality and estimate eigenvalues. By leveraging recent advances, we apply our method to calculate ground- and excited-state energies of challenging molecular systems in large active spaces, including the carbon dimer with 8 electrons in 108 orbitals (8e,108o), an oxo-Mn(salen) transition metal complex (28e,28o), ozone (18e,87o), and butadiene (22e,82o). In the majority of these test cases, our approach yields total excited-state energies that agree with those from state-of-the-art methods─including heat-bath CI, the density matrix renormalization group approach, and FCIQMC─to within sub-milliHartree accuracy. In all cases, estimated excitation energies agree to within about 0.1 eV.

Fixed- and Partial-Node Approximations in Slater Determinant Space for Molecules

We present a study of fixed- and partial-node approximations in Slater determinant basis sets, using full configuration interaction quantum Monte Carlo (FCIQMC) to perform sampling. Walker annihilation in the FCIQMC method allows partial-node simulations to be performed, relaxing the nodal constraint to converge to the FCI solution. This is applied to ab initio molecular systems, using symmetry-projected Jastrow mean-field wave functions for complete active space (CAS) problems. Convergence and the sign problem within the partial-node approximation are studied, which is shown to eventually be limited in its use due to the large walker populations required. However, the fixed-node approximation results in an accurate and practical method. We apply these approaches to various molecular systems and active spaces, including ferrocene and acenes. This also provides a test of symmetry-projected Jastrow mean-field wave functions in variational Monte Carlo for a new set of problems. For trans-polyacetylene molecules and acenes, we find that the time to perform a constant number of fixed-node FCIQMC iterations scales as O(N1.44) and O(N1.75), respectively, resulting in an efficient method for CAS-based problems that can be applied accurately to large active spaces.

The Sign Problem in Density Matrix Quantum Monte Carlo

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.

The Shape of Full Configuration Interaction to Come

We present a Perspective on what the future holds for full configuration interaction (FCI) theory, with an emphasis on conceptual rather than technical details. Upon revisiting the early history of FCI, a number of its key contemporary approximations are compared on as equal a footing as possible, using a recent blind challenge on the benzene molecule as a testbed [Eriksen et al., J. Phys. Chem. Lett., 2020 11, 8922]. In the process, we review the scope of applications for which FCI continues to prove indispensable, and the required traits in terms of robustness, efficacy, and reliability its modern approximations must satisfy are discussed. We close by conveying a number of general observations on the merits offered by the state-of-the-art alongside some of the challenges still faced to this day. While the field has altogether seen immense progress over the years-the past decade, in particular-it remains clear that our community as a whole has a substantial way to go in enhancing the overall applicability of near-exact electronic structure theory for systems of general composition and increasing size.

The Ground State Electronic Energy of Benzene

We report on the findings of a blind challenge devoted to determining the frozen-core, full configuration interaction (FCI) ground-state energy of the benzene molecule in a standard correlation-consistent basis set of double-ζ quality. As a broad international endeavor, our suite of wave function-based correlation methods collectively represents a diverse view of the high-accuracy repertoire offered by modern electronic structure theory. In our assessment, the evaluated high-level methods are all found to qualitatively agree on a final correlation energy, with most methods yielding an estimate of the FCI value around -863 mE_H. However, we find the root-mean-square deviation of the energies from the studied methods to be considerable (1.3 mE_H), which in light of the acclaimed performance of each of the methods for smaller molecular systems clearly displays the challenges faced in extending reliable, near-exact correlation methods to larger systems. While the discrepancies exposed by our study thus emphasize the fact that the current state-of-the-art approaches leave room for improvement, we still expect the present assessment to provide a valuable community resource for benchmark and calibration purposes going forward.

Stochastic perturbation theory in a limited configuration space

A general-order stochastic perturbation algorithm is obtained from the order-by-order expansion of the imaginary-time evolution of a configuration interaction wave function. A truncation of configuration space that is required for the practical treatment of the perturbative corrections, however, does not preserve size-consistency as is the case for a truncated configuration interaction. To circumvent this problem, we formulate a linked variant of stochastic perturbation theory based on the coupled-cluster ansatz. The implementation based on the linearized coupled-cluster is compared with several full configuration interaction results. We also compare the results with those obtained from deterministic coupled-cluster and many-body perturbation theories.