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Stochastic Approaches to Coupled Cluster Theory

Charles J. C. Scott

Published Jun 26, 2019• 0 citations
Publisher

Abstract

This thesis is concerned with the development and extension of various new approaches to solving the Coupled Cluster (CC) equations through Monte Carlo (MC) integration. Firstly, exact importance sampling of the CC wavefunction to arbitrary order is defined and applied as a modification to the extant unlinked Coupled Cluster Monte Carlo (CCMC) method. This is shown to be vital for the stability of the method while reducing both computational and memory costs on test calculations. Stochastic approaches utilising the diagrammatic formalism of deterministic CC theory in its unfactorised and totally factorised forms are then developed. These are found to show reduced computational costs compared to prior approaches. It is demonstrated that sampling only the connected components of the similarity-transformed Hamiltonian obtains a solution with memory cost proportional to system size for perfectly local systems and a fixed granularity of representation. For a fixed errorbar per replica calculation costs scale as quartic and linear in the number of replicas for the unfactorised and factorised approaches, respectively. This behaviour is independent of truncation level, and shown to translate into approximately local physical systems. The presence of various systematic biases resulting from wavefunction nonlinearity within these approaches is then investigated. All metrics introduced are found to be negligible compared to the accuracy required for chemical applications in an example system, and possible reasons for this suggested. By considering developments of Full Configuration Interaction Quantum Monte Carlo (FCIQMC) extensions of the totally factorised approach are formulated based upon the semistochastic and initiator approximations, and are shown to provide reduced computational and memory costs. Finally, these new approaches are combined with probabilistic considerations to allow exact solutions to the CC equations to be obtained while only considering approximate update steps.

DOI
10.17863/cam.47142

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This thesis is concerned with the development and extension of various new approaches to solving the Coupled Cluster (CC) equations through Monte Carlo (MC) integration. Firstly, exact importance sampling of the CC wavefunction to arbitrary order is defined and applied as a modification to the extant unlinked Coupled Cluster Monte Carlo (CCMC) method. This is shown to be vital for the stability of the method while reducing both computational and memory costs on test calculations. Stochastic approaches utilising the diagrammatic formalism of deterministic CC theory in its unfactorised and totally factorised forms are then developed. These are found to show reduced computational costs compared to prior approaches. It is demonstrated that sampling only the connected components of the similarity-transformed Hamiltonian obtains a solution with memory cost proportional to system size for perfectly local systems and a fixed granularity of representation. For a fixed errorbar per replica calculation costs scale as quartic and linear in the number of replicas for the unfactorised and factorised approaches, respectively. This behaviour is independent of truncation level, and shown to translate into approximately local physical systems. The presence of various systematic biases resulting from wavefunction nonlinearity within these approaches is then investigated. All metrics introduced are found to be negligible compared to the accuracy required for chemical applications in an example system, and possible reasons for this suggested. By considering developments of Full Configuration Interaction Quantum Monte Carlo (FCIQMC) extensions of the totally factorised approach are formulated based upon the semistochastic and initiator approximations, and are shown to provide reduced computational and memory costs. Finally, these new approaches are combined with probabilistic considerations to allow exact solutions to the CC equations to be obtained while only considering approximate update steps.
We propose a modified coupled cluster Monte Carlo algorithm that stochastically samples connected terms within the truncated Baker-Campbell-Hausdorff expansion of the similarity-transformed Hamiltonian by construction of coupled cluster diagrams on the fly. Our new approach-diagCCMC-allows propagation to be performed using only the connected components of the similarity-transformed Hamiltonian, greatly reducing the memory cost associated with the stochastic solution of the coupled cluster equations. We show that for perfectly local, noninteracting systems diagCCMC is able to represent the coupled cluster wavefunction with a memory cost that scales linearly with system size. The favorable memory cost is observed with the only assumption of fixed stochastic granularity and is valid for arbitrary levels of coupled cluster theory. Significant reduction in memory cost is also shown to smoothly appear with dissociation of a finite chain of helium atoms. This approach is also shown not to break down in the presence of strong correlation through the example of a stretched nitrogen molecule. Our novel methodology moves the theoretical basis of coupled cluster Monte Carlo closer to deterministic approaches.
We present a detailed discussion of our novel diagrammatic coupled cluster Monte Carlo (diagCCMC) [Scott et al. J. Phys. Chem. Lett. 10, 925 (2019)]. The diagCCMC algorithm performs an imaginary-time propagation of the similarity-transformed coupled cluster Schrödinger equation. Imaginary-time updates are computed by the stochastic sampling of the coupled cluster vector function: each term is evaluated as a randomly realized diagram in the connected expansion of the similarity-transformed Hamiltonian. We highlight similarities and differences between deterministic and stochastic linked coupled cluster theory when the latter is re-expressed as a sampling of the diagrammatic expansion and discuss details of our implementation that allow for a walker-less realization of the stochastic sampling. Finally, we demonstrate that in the presence of locality, our algorithm can obtain a fixed errorbar per electron while only requiring an asymptotic computational effort that scales quartically with system size, independent of the truncation level in coupled cluster theory. The algorithm only requires an asymptotic memory cost scaling linearly, as demonstrated previously. These scaling reductions require no ad hoc modifications to the approach.
In the last decade, stochastic versions of quantum chemistry methods such as coupled cluster Monte Carlo (CCMC) or full configuration interaction quantum Monte Carlo (FCIQMC) have made highly accurate energy calculations possible that are not accessible to the corresponding deterministic methods (full configuration interaction and coupled cluster) at the same accuracy. CCMC and FCIQMC parallelize well and exploit the sparsity in the wavefunction which decreases memory costs and makes calculations in larger systems tractable. With CCMC it is straightforward to set up high order coupled cluster calculations, such as CCSDTQ5, which includes quintuple excitations explicitly. In this thesis, the convergence of the energy accuracy with the coupled cluster levels up to CCSDTQ5 was tested on the uniform electron gas, a model solid system, for various degrees of electron correlation. This gave information on what coupled cluster level is needed to reach sufficient accuracy when modelling a solid system. Before large solid systems can be modelled, the CCMC and FCIQMC algorithms need to be optimised. The efficiency in one of the crucial steps in these algorithms, the spawnspawnspawn step, was improved, keeping computational and memory costs as low as possible. Furthermore, the convergence of CCMC and FCIQMC was accelerated by employing a quasi-Newton propagation. Using the model system information of what coupled cluster level is needed and having made great progress towards accelerating these methods, the computation of highly accurate energies in solid or large molecular systems should be more feasible in the future.
We consider a new formulation of the stochastic coupled cluster method in terms of the similarity transformed Hamiltonian. We show that improvement in the granularity with which the wavefunction is represented results in a reduction in the critical population required to correctly sample the wavefunction for a range of systems and excitation levels and hence leads to a substantial reduction in the computational cost. This development has the potential to substantially extend the range of the method, enabling it to be used to treat larger systems with excitation levels not easily accessible with conventional deterministic methods.
We consider the sampling of the coupled cluster expansion within stochastic coupled cluster theory. Observing the limitations of previous approaches due to the inherently non-linear behavior of a coupled cluster wavefunction representation, we propose new approaches based on an intuitive, well-defined condition for sampling weights and on sampling the expansion in cluster operators of different excitation levels. We term these modifications even and truncated selections, respectively. Utilising both approaches demonstrates dramatically improved calculation stability as well as reduced computational and memory costs. These modifications are particularly effective at higher truncation levels owing to the large number of terms within the cluster expansion that can be neglected, as demonstrated by the reduction of the number of terms to be sampled when truncating at triple excitations by 77% and hextuple excitations by 98%.
Coupled cluster theory is a vital cornerstone of electronic structure theory and is being applied to ever-larger systems. Stochastic approaches to quantum chemistry have grown in importance and offer compelling advantages over traditional deterministic algorithms in terms of computational demands, theoretical flexibility or lower scaling with system size. We present a highly parallelizable algorithm of the coupled cluster Monte Carlo method involving sampling of clusters of excitors over multiple time steps. The behaviour of the algorithm is investigated on the uniform electron gas and the water dimer at CCSD, CCSDT and CCSDTQ levels. We also describe two improvements to the original sampling algorithm, full non-composite and multi-spawn sampling. A stochastic approach to coupled cluster results in an efficient and scalable implementation at arbitrary truncation levels in the coupled cluster expansion.