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Accelerating and Converging Stochastic Quantum Chemistry

Verena A. Neufeld

Published Dec 20, 2019• 0 citations
Publisher

Abstract

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.

DOI
10.17863/cam.47261

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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.
The convergence of full configuration interaction quantum Monte Carlo (FCIQMC) is accelerated using a quasi-Newton propagation (QN) which can also be applied to coupled cluster Monte Carlo (CCMC). The computational scaling of this optimized propagation is, keeping the additional computational cost to a bare minimum. Its effects are investigated deterministically and stochastically on a model system, the uniform electron gas, with Hilbert space size up to 10 40, and shown to accelerate convergence of the instantaneous projected energy by over an order of magnitude in the FCIQMC test case. Its capabilities are then demonstrated with FCIQMC on an archetypical quantum chemistry problem, the chromium dimer, in an all-electron basis set with Hilbert space size of about 10 22 yielding highly accurate FCI energies.
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.
Building on the success of Quantum Monte Carlo techniques such as diffusion Monte Carlo, alternative stochastic approaches to solve electronic structure problems have emerged over the past decade. The full configuration interaction quantum Monte Carlo (FCIQMC) method allows one to systematically approach the exact solution of such problems, for cases where very high accuracy is desired. The introduction of FCIQMC has subsequently led to the development of coupled cluster Monte Carlo (CCMC) and density matrix quantum Monte Carlo (DMQMC), allowing stochastic sampling of the coupled cluster wave function and the exact thermal density matrix, respectively. In this Article, we describe the HANDE-QMC code, an open-source implementation of FCIQMC, CCMC and DMQMC, including initiator and semistochastic adaptations. We describe our code and demonstrate its use on three example systems; a molecule (nitric oxide), a model solid (the uniform electron gas), and a real solid (diamond). An illustrative tutorial is also included.
In this thesis we investigate the recently developed Full Configuration Interaction Quantum Monte Carlo (FCIQMC) method. This method, unlike the traditional methods in Quantum Monte Carlo (QMC), doesn’t require the use of the uncontrolled fixed node approximation and so potentially it can yield far more accurate results. This means that it can be thought of as a hybrid between the methods used by quantum chemists and QMC, and thus has spawned a new field of stochastic quantum chemistry. The work described in this thesis can be split into three distinct but interrelated parts. We begin with an investigation of the underlying FCIQMC stochastic process. We show that FCIQMC is an example of Markov Chain Monte Carlo. This means we can compute a stochastic matrix from which all details about the Monte-Carlo simulation can be obtained. Unfortunately the size of the space scales unfavourably as a function of system size meaning that we have only managed to compute the matrix for the smallest interesting two determinant system. We then use these results to quantify population control bias in FCIQMC for a two determinant system supplementing these analytical results with empirical results to investigate more realistic systems. We then attempt to quantify the efficiency of the FCIQMC algorithms, defining a measure of efficiency. After this we investigate the dependence of our measure on the system size. The error bar of the most efficient FCIQMC algorithm will decay fastest as a function of computer time. We then draw conclusions about the applicability of the FCIQMC method. Finally we describe an implementation of FCIQMC on a novel data flow computer architecture. In our implementation we made a modification to the FCIQMC algorithm to fit the data flow paradigm. We investigate the efficiency of the modified FCIQMC algorithm.
We propose to accelerate convergence toward full configuration interaction (FCI) energetics by using the coupled-cluster approach, in which singly and doubly excited clusters, needed to determine the energy, are iterated in the presence of their three- and four-body counterparts extracted from FCI quantum Monte Carlo (FCIQMC) propagations. Preliminary calculations for the water molecule at the equilibrium and stretched geometries show that we can accurately extrapolate the FCI energetics based on the early stages of FCIQMC propagations.
For many decades, quantum chemical method development has been dominated by algorithms which involve increasingly complex series of tensor contractions over one-electron orbital spaces. Procedures for their derivation and implementation have evolved to require the minimum amount of logic and rely heavily on computationally efficient library-based matrix algebra and optimised paging schemes. In this regard, the recent development of exact stochastic quantum chemical algorithms to reduce computational scaling and memory overhead requires a contrasting algorithmic philosophy, but one which when implemented efficiently can achieve higher accuracy/cost ratios with small random errors. Additionally, they can exploit the continuing trend for massive parallelisation which hinders the progress of deterministic high-level quantum chemical algorithms. In the Quantum Monte Carlo community, stochastic algorithms are ubiquitous but the discrete Fock space of quantum chemical methods is often unfamiliar, and the methods introduce new concepts required for algorithmic efficiency. In this paper, we explore these concepts and detail an algorithm used for Full Configuration Interaction Quantum Monte Carlo (FCIQMC), which is implemented and available in MOLPRO and as a standalone code, and is designed for high-level parallelism and linear-scaling with walker number. Many of the algorithms are also in use in, or can be transferred to, other stochastic quantum chemical methods and implementations. We apply these algorithms to the strongly correlated chromium dimer to demonstrate their efficiency and parallelism.