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.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.

Nov 28, 201813 citations

Accelerating and Converging Stochastic Quantum Chemistry

Verena A. Neufeld

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 $spawn$ 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.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 $spawn$ 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.

Dec 20, 20190 citations

Theory and implementation of a novel stochastic approach to coupled cluster

Charles J. C. Scott, Roberto Di Remigio, T. Daniel Crawford +1 more

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.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.

arXiv:2008.10514The Journal of Chemical PhysicsOct 14, 20209 citations

Stochastic coupled cluster theory: Efficient sampling of the coupled cluster expansion

Charles J. C. Scott, Alex J. W. Thom

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%.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%.

arXiv:1706.07017The Journal of Chemical PhysicsSep 26, 201735 citations

A stochastic approach to unitary coupled cluster.

Maria-Andreea Filip, Alex J. W. Thom

Unitary coupled cluster (UCC), originally developed as a variational alternative to the popular traditional coupled cluster method, has seen a resurgence as a functional form for use on quantum computers. However, the number of excitors present in the Ansatz often presents a barrier to implementation on quantum computers. Given the natural sparsity of wavefunctions obtained from quantum Monte Carlo methods, we consider here a stochastic solution to the UCC problem. Using the coupled cluster Monte Carlo framework, we develop cluster selection schemes that capture the structure of the UCC wavefunction, as well as its Trotterized approximation, and use these to solve the corresponding projected equations. Due to the fast convergence of the equations with order in the cluster expansion, this approach scales polynomially with the size of the system. Unlike traditional UCC implementations, our approach naturally produces a non-variational estimator for the energy in the form of the projected energy. For unitary coupled cluster singles and doubles (UCCSD) in small systems, we find that this agrees well with the expectation value of the energy and, in the case of two electrons, with full configuration interaction results. For the larger N2 system, the two estimators diverge, with the projected energy approaching the coupled cluster result, while the expectation value is close to results from traditional UCCSD.Unitary coupled cluster (UCC), originally developed as a variational alternative to the popular traditional coupled cluster method, has seen a resurgence as a functional form for use on quantum computers. However, the number of excitors present in the Ansatz often presents a barrier to implementation on quantum computers. Given the natural sparsity of wavefunctions obtained from quantum Monte Carlo methods, we consider here a stochastic solution to the UCC problem. Using the coupled cluster Monte Carlo framework, we develop cluster selection schemes that capture the structure of the UCC wavefunction, as well as its Trotterized approximation, and use these to solve the corresponding projected equations. Due to the fast convergence of the equations with order in the cluster expansion, this approach scales polynomially with the size of the system. Unlike traditional UCC implementations, our approach naturally produces a non-variational estimator for the energy in the form of the projected energy. For unitary coupled cluster singles and doubles (UCCSD) in small systems, we find that this agrees well with the expectation value of the energy and, in the case of two electrons, with full configuration interaction results. For the larger N2 system, the two estimators diverge, with the projected energy approaching the coupled cluster result, while the expectation value is close to results from traditional UCCSD.

Nov 19, 202029 citations

Diagrammatic Coupled Cluster Monte Carlo

Charles J. C. Scott, Roberto Di Remigio, T. Daniel Crawford +1 more

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 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.

arXiv:1901.02753The Journal of Physical Chemistry LettersFeb 6, 201919 citations

spawn

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.