Towards an Accurate Description of Strongly Correlated Chemical Systems with Phaseless Auxiliary-Field Quantum Monte Carlo - Methodological Advances and Applications

The exact and phaseless variants of auxiliary-field quantum Monte Carlo (AFQMC) have been shown to be capable of producing accurate ground-state energies for a wide variety of systems including those which exhibit substantial electron correlation effects. The first chapter of this thesis will provide an overview of the relevant electronic structure problem, and the phaseless AFQMC (ph-AFQMC) methodology. The computational cost of performing these calculations has to date been relatively high, impeding many important applications of these approaches. In Chapter 2 we present a correlated sampling methodology for AFQMC which relies on error cancellation to dramatically accelerate the calculation of energy differences of relevance to chemical transformations. In particular, we show that our correlated sampling-based ph-AFQMC approach is capable of calculating redox properties, deprotonation free energies, and hydrogen abstraction energies in an efficient manner without sacrificing accuracy. We validate the computational protocol by calculating the ionization potentials and electron affinities of the atoms contained in the G2 test set and then proceed to utilize a composite method, which treats fixed-geometry processes with correlated sampling-based AFQMC and relaxation energies via MP2, to compute the ionization potential, deprotonation free energy, and the O-H bond dissociation energy of methanol, all to within chemical accuracy. We show that the efficiency of correlated sampling relative to uncorrelated calculations increases with system and basis set size and that correlated sampling greatly reduces the required number of random walkers to achieve a target statistical error. This translates to reductions in wall-times by factors of 55, 25, and 24 for the ionization potential of the K atom, the deprotonation of methanol, and hydrogen abstraction from the O-H bond of methanol, respectively. In Chapter 3 we present an implementation of ph-AFQMC utilizing graphical processing units (GPUs). The AFQMC method is recast in terms of matrix operations which are spread across thousands of processing cores and are executed in batches using custom Compute Unified Device Architecture kernels and the hardware-optimized cuBLAS matrix library. Algorithmic advances include a batched Sherman-Morrison-Woodbury algorithm to quickly update matrix determinants and inverses, density-fitting of the two-electron integrals, an energy algorithm involving a high-dimensional precomputed tensor, and the use of single-precision floating point arithmetic. These strategies result in dramatic reductions in wall-times for both single- and multi-determinant trial wavefunctions. For typical calculations we find speed-ups of roughly two orders of magnitude using just a single GPU card. Furthermore, we achieve near-unity parallel efficiency using 8 GPU cards on a single node, and can reach moderate system sizes via a local memory-slicing approach. We illustrate the robustness of our implementation on hydrogen chains of increasing length, and through the calculation of all-electron ionization potentials of the first-row transition metal atoms. We compare long imaginary-time calculations utilizing a population control algorithm with our previously published correlated sampling approach, and show that the latter improves not only the efficiency but also the accuracy of the computed ionization potentials. Taken together, the GPU implementation combined with correlated sampling provides a compelling computational method that will broaden the application of ph-AFQMC to the description of realistic correlated electronic systems. In Chapter 4 the bond dissociation energies of a set of 44 3d transition metal-containing diatomics are computed with ph-AFQMC utilizing the correlated sampling technique. We investigate molecules with H, N, O, F, Cl, and S ligands, including those in the 3dMLBE20 database first compiled by Truhlar and co-workers with calculated and experimental values that have since been revised by various groups. In order to make a direct comparison of the accuracy of our ph-AFQMC calculations with previously published results from 10 DFT functionals, CCSD(T), and icMR-CCSD(T), we establish an objective selection protocol which utilizes the most recent experimental results except for a few cases with well-specified discrepancies. With the remaining set of 41 molecules, we find that ph-AFQMC gives robust agreement with experiment superior to that of all other methods, with a mean absolute error (MAE) of 1.4(4) kcal/mol and maximum error of 3(3) kcal/mol (parenthesis account for reported experimental uncertainties and the statistical errors of our ph-AFQMC calculations). In comparison, CCSD(T) and B97, the best performing DFT functional considered here, have MAEs of 2.8 and 3.7 kcal/mol, respectively, and maximum errors in excess of 17 kcal/mol (for the CoS diatomic). While a larger and more diverse data set would be required to demonstrate that ph-AFQMC is truly a benchmark method for transition metal systems, our results indicate that the method has tremendous potential, exhibiting unprecedented consistency and accuracy compared to other approximate quantum chemical approaches. The energy gap between the lowest-lying singlet and triplet states is an important quantity in chemical photocatalysis, with relevant applications ranging from triplet fusion in optical upconversion to the design of organic light-emitting devices. The ab initio prediction of singlet-triplet (ST) gaps is challenging due to the potentially biradical nature of the involved states, combined with the potentially large size of relevant molecules. In Chapter 5, we show that ph-AFQMC can accurately predict ST gaps for chemical systems with singlet states of highly biradical nature, including a set of 13 small molecules and the ortho-, meta-, and para- isomers of benzyne. With respect to gas-phase experiments, ph-AFQMC using CASSCF trial wavefunctions achieves a mean averaged error of ~1 kcal/mol. Furthermore, we find that in the context of a spin-projection technique, ph-AFQMC using unrestricted single-determinant trial wavefunctions, which can be readily obtained for even very large systems, produces equivalently high accuracy. We proceed to show that this scalable methodology is capable of yielding accurate ST gaps for all linear polyacenes for which experimental measurements exist, i.e. naphthalene, anthracene, tetracene, and pentacene. Our results suggest a protocol for selecting either unrestricted Hartree-Fock or Kohn-Sham orbitals for the single-determinant trial wavefunction, based on the extent of spin-contamination. These findings provide a reliable computational tool with which to investigate specific photochemical processes involving large molecules that may have substantial biradical character. We compute the ST gaps for a set of anthracene derivatives which are potential triplet-triplet annihilators for optical upconversion, and compare our ph-AFQMC predictions with those from DFT and CCSD(T) methods. We conclude with a discussion of ongoing projects, further methodological improvements on the horizon, and future applications of ph-AFQMC to chemical systems of interest in the fields of biology, drug-discovery, catalysis, and condensed matter physics

Efficient ab initio auxiliary-field quantum Monte Carlo calculations in Gaussian bases via low-rank tensor decomposition

We describe an algorithm to reduce the cost of auxiliary-field quantum Monte Carlo (AFQMC) calculations for the electronic structure problem. The technique uses a nested low-rank factorization of the electron repulsion integral (ERI). While the cost of conventional AFQMC calculations in Gaussian bases scales as $\mathcal{O}(N^4)$ where $N$ is the size of the basis, we show that ground-state energies can be computed through tensor decomposition with reduced memory requirements and sub-quartic scaling. The algorithm is applied to hydrogen chains and square grids, water clusters, and hexagonal BN. In all cases we observe significant memory savings and, for larger systems, reduced, sub-quartic simulation time.Comment: 14 pages, 13 figures, expanded dataset and tex

Optimizing the Regularization in Size-Consistent Second-Order Brillouin-Wigner Perturbation Theory

Despite its simplicity and relatively low computational cost, second-order M{\o}ller-Plesset perturbation theory (MP2) is well-known to overbind noncovalent interactions between polarizable monomers and some organometallic bonds. In such situations, the pairwise-additive correlation energy expression in MP2 is inadequate. Although energy-gap dependent amplitude regularization can substantially improve the accuracy of conventional MP2 in these regimes, the same regularization parameter worsens the accuracy for small molecule thermochemistry and density-dependent properties. Recently, we proposed a repartitioning of Brillouin-Wigner perturbation theory that is size-consistent to second order (BW-s2), and a free parameter ({\alpha}) was set to recover the exact dissociation limit of H$_2$ in a minimal basis set. Alternatively {\alpha} can be viewed as a regularization parameter, where each value of {\alpha} represents a valid variant of BW-s2, which we denote as BW-s2({\alpha}). In this work, we semi-empirically optimize {\alpha} for noncovalent interactions, thermochemistry, alkane conformational energies, electronic response properties, and transition metal datasets, leading to improvements in accuracy relative to the ab initio parameterization of BW-s2 and MP2. We demonstrate that the optimal {\alpha} parameter ({\alpha} = 4) is more transferable across chemical problems than energy-gap-dependent regularization parameters. This is attributable to the fact that the BW-s2({\alpha}) regularization strength depends on all of the information encoded in the t amplitudes rather than just orbital energy differences. While the computational scaling of BW-s2({\alpha}) is iterative $\mathcal{O}(N^5)$, this effective and transferable approach to amplitude regularization is a promising route to incorporate higher-order correlation effects at second-order cost.Comment: 7 pages main text, 7 pages supporting information, 10 figure