We present efficient algorithms for using selected configuration interaction (sCI) trial wave functions in phaseless auxiliary field quantum Monte Carlo (ph-AFQMC). These advances, geared toward optimizing computational performance for longer configuration interaction expansions, allow us to use up to a million configurations in the trial state for ph-AFQMC. In one example, we found the cost of ph-AFQMC per sample to increase only by a factor of about 3 for a calculation with 104 configurations compared to that with a single one, demonstrating the tiny computational overhead due to a longer expansion. This favorable scaling allows us to study the systematic convergence of the phaseless bias in auxiliary field quantum Monte Carlo calculations with an increasing number of configurations and provides a means to gauge the accuracy of ph-AFQMC with other trial states. We also show how the scalability issues of sCI trial states for large system sizes could be mitigated by restricting them to a moderately sized orbital active space and leveraging the near-cancellation of out of active space phaseless errors.
We present efficient algorithms for using selected configuration interaction (sCI) trial wave functions in phaseless auxiliary field quantum Monte Carlo (ph-AFQMC). These advances, geared toward optimizing computational performance for longer configuration interaction expansions, allow us to use up to a million configurations in the trial state for ph-AFQMC. In one example, we found the cost of ph-AFQMC per sample to increase only by a factor of about 3 for a calculation with 104 configurations compared to that with a single one, demonstrating the tiny computational overhead due to a longer expansion. This favorable scaling allows us to study the systematic convergence of the phaseless bias in auxiliary field quantum Monte Carlo calculations with an increasing number of configurations and provides a means to gauge the accuracy of ph-AFQMC with other trial states. We also show how the scalability issues of sCI trial states for large system sizes could be mitigated by restricting them to a moderately sized orbital active space and leveraging the near-cancellation of out of active space phaseless errors.
We present an approach that uses the doubly occupied configuration interaction (DOCI) wave function as the trial wave function in phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC). DOCI is a seniority-zero method focused on electron pairs. Although DOCI considers much fewer electron configurations than the complete active space (CAS) configuration interaction method, it efficiently captures the static correlation, while the consequent ph-AFQMC recovers the dynamical correlation across all orbitals. We also explore an orbital-optimized version (OO-DOCI) to further improve accuracy. We test this approach on several chemical systems, including single O-H bond breaking in water and polymer additives. In these cases, OO-DOCI-AFQMC closely matches CAS-based ph-AFQMC and even outperforms coupled-cluster singles, doubles, and perturbative triples. However, for strongly correlated systems, such as the carbon dimer and multibond dissociation in hydrogen systems and water, the method's accuracy drops. This suggests that seniority-zero space models may be insufficient as trial wave functions in ph-AFQMC for strongly correlated systems, suggesting the need for trial wave functions in an extended space. Despite such a limitation, our study demonstrates that DOCI- and OO-DOCI-based ph-AFQMC can reduce the steep cost of CAS approaches, offering a path to accurate multireference calculations for larger, more complex systems.
We present an approach that uses the doubly occupied configuration interaction (DOCI) wave function as the trial wave function in phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC). DOCI is a seniority-zero method focused on electron pairs. Although DOCI considers much fewer electron configurations than the complete active space (CAS) configuration interaction method, it efficiently captures the static correlation, while the consequent ph-AFQMC recovers the dynamical correlation across all orbitals. We also explore an orbital-optimized version (OO-DOCI) to further improve accuracy. We test this approach on several chemical systems, including single O-H bond breaking in water and polymer additives. In these cases, OO-DOCI-AFQMC closely matches CAS-based ph-AFQMC and even outperforms coupled-cluster singles, doubles, and perturbative triples. However, for strongly correlated systems, such as the carbon dimer and multi-bond dissociation in hydrogen systems and water, the method's accuracy drops. This suggests that seniority-zero space models may be insufficient as trial wave functions in ph-AFQMC for strongly correlated systems, suggesting the need for trial wave functions in an extended space. Despite such a limitation, our study demonstrates that DOCI- and OO-DOCI-based ph-AFQMC can reduce the steep cost of CAS approaches, offering a path to accurate multi-reference calculations for larger, more complex systems.
For optimal accuracy, auxiliary-field quantum Monte Carlo (AFQMC) requires trial states consisting of multiple Slater determinants. We develop an efficient algorithm to select the determinants from an AFQMC random walk eliminating the need for other methods. When determinants contribute significantly to the nonorthogonal configuration interaction energy, we include them in the trial state. These refined trial wave functions significantly reduce the phaseless bias and sampling variance of the local energy estimator. With 100 to 200 determinants, we lower the error of AFQMC by up to a factor of 10 for second-row elements that are not accurately described with a Hartree-Fock trial wave function. For the HEAT set, we improve the average error to within chemical accuracy. For benzene, the largest studied system, we reduce AFQMC error by 80% with 214 Slater determinants and find a 10-fold increase of the time to solution. We show that phaseless errors prevail in systems with static correlation or strong spin contamination. For such systems, improved trial states enable stable free-projection AFQMC calculations, achieving chemical accuracy even in the strongly correlated regime.
To make useful connections with experimental measurements, correlated electronic structure theories must accurately predict chemical properties in addition to energies. We present a finite-difference based algorithm to compute first-order response properties with phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) that relies on a branching correlated sampling approach. Focusing on electric dipole moments, we show that mean-field trial wave functions are sufficient to obtain high accuracy relative to CCSD(T) and experimental measurements for a set of 21 molecules, ranging in size from 2 to 18 atoms in their equilibrium geometries. As with energies, the quality of predicted dipole moments can be systematically improved with the use of correlated trial wave functions, even (or especially) in strongly correlated regimes. We show that the challenges faced by low-order perturbation theories in predicting the dipole moment of hydrogen fluoride across its dissociation coordinate are overcome with ph-AFQMC when using relatively simple trials. The key advantage of our approach over those previously reported for ph-AFQMC is its scalability to large system sizes with a phaseless bias no worse than that of a typical ground-state energy calculation; we routinely converge dipole moments for systems with more than 1000 basis functions.
Phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) has emerged as a promising electronic structure method for correlated electronic systems. However, the quality of its predictions depends critically on the choice of trial wave function, and it is not obvious how to make an optimal choice, especially for strongly correlated states of large systems. Mean-field wave functions are compelling trial wave function candidates, as they map directly to chemical concepts and can be obtained withO(N4) cost. Yet in the strongly correlated regime, one faces a symmetry dilemma and the existence of multiple nearly degenerate solutions. In this work, we investigate active space models of [2Fe-2S]2+, mixed-valent [4Fe-4S]2+, and [4Fe-4S]4+ and explore the sensitivity of ph-AFQMC to the choice of unrestricted Hartree-Fock trial wave function. We find that surprisingly accurate ground-state energies for these systems can be obtained when trial selection is guided by chemical properties and physical symmetries, rather than the variational energy. However, in all cases, we find a rapidly decaying overlap between the stochastic wave function and the UHF trial, indicating that the trials are acting as suboptimal importance functions. By analogy to a similar situation in the stretched helium dimer cation, we show how this sampling bias pushes ph-AFQMC with UHF trials toward artificially negative energies, which evidently can be compensated for by the phaseless bias in certain cases.
Quantum Monte Carlo (QMC) methods are some of the most accurate methods for simulating correlated electronic systems. We investigate the compatibility, strengths, and weaknesses of two such methods, namely, diffusion Monte Carlo (DMC) and auxiliary-field quantum Monte Carlo (AFQMC). The multideterminant trial wave functions employed in both approaches are generated using the configuration interaction using a perturbative selection made iteratively (CIPSI) technique. Complete basis-set full configuration interaction energies estimated with CIPSI are used as a reference in this comparative study between DMC and AFQMC. By focusing on a set of canonical finite--size solid-state systems, we show that both QMC methods can be made to systematically converge towards the same energy once basis-set effects and systematic biases have been removed. AFQMC shows a much smaller dependence on the trial wave function than DMC while simultaneously exhibiting a much larger basis-set dependence. We outline some of the remaining challenges and opportunities for improving these approaches.