arXiv subjects

Alex J. W. Thom

Publications and source records attributed to Alex J. W. Thom.

At least 37 records · Page 2Linked to original sources

arXiv2020

A Stochastic Approach to Unitary Coupled Cluster

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 UCCSD in small systems, we find 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 N$_2$ 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.

physics.chem-ph↗

arXiv2020

Multireference Stochastic Coupled Cluster

We describe a modification of the stochastic coupled cluster algorithm that allows the use of multiple reference determinants. By considering the secondary references as excitations of the primary reference and using them to change the acceptance criteria for selection and spawning, we obtain a simple form of stochastic multireference coupled cluster which preserves the appealing aspects of the single reference approach. The method is able to successfully describe strongly correlated molecular systems using few references and low cluster truncation levels, showing promise as a tool to tackle strong correlation in more general systems. Moreover, it allows simple and comprehensive control of the included references and excitors thereof, and this flexibility can be taken advantage of to gain insight into some of the inner workings of established electronic structure methods.

physics.chem-ph↗

arXiv2020

Theory and Implementation of a Novel Stochastic Approach to Coupled Cluster

We present a detailed discussion of our novel diagrammatic coupled cluster Monte Carlo (diagCCMC) [Scott et al. J. Phys. Chem. Lett. 2019, 10, 925]. The diagCCMC algorithm performs an imaginary-time propagation of the similarity-transformed coupled cluster Schrödinger equation. Imaginary-time updates are computed by stochastic sampling of the coupled cluster vector function: each term is evaluated as a randomly realised 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 realisation 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, independently of truncation level in coupled cluster theory. The algorithm only requires an asymptotic memory costs scaling linearly, as demonstrated previously. These scaling reductions require no ad hoc modifications to the approach.

physics.chem-ph↗

arXiv2020

Holomorphic Density Functional Theory

Self-consistent-field (SCF) approximations formulated using Hartree-Fock (HF) or Kohn-Sham Density Functional Theory (KS-DFT) both have the potential to yield multiple solutions. However, the formal relationship between multiple solutions identified using HF or KS-DFT remains generally unknown. We investigate the connection between multiple SCF solutions for HF or KS-DFT by introducing a parametrised functional that scales between the two representations. Using the hydrogen molecule and a model of electron transfer, we continuously map multiple solutions from the HF potential to a KS-DFT description. We discover that multiple solutions can coalesce and vanish as the functional changes, forming a direct analogy with the disappearance of real HF solutions along a change in molecular structure. To overcome this disappearance of solutions, we develop a complex-analytic extension of DFT - the "holomorphic DFT" approach - that allows every SCF stationary state to be analytically continued across all molecular structures and exchange-correlation functionals.

physics.chem-ph↗

arXiv2020

Reaching Full Correlation through Nonorthogonal Configuration Interaction: A Second-Order Perturbative Approach

Nonorthogonal multireference methods can predict statically correlated adiabatic energies while providing chemical insight through the combination of diabatic reference states. However, reaching quantitative accuracy using nonorthogonal multireference expansions remains a significant challenge. In this work, we present the first rigorous perturbative correction to nonorthogonal configuration interaction, allowing the remaining dynamic correlation to be reliably computed. Our second-order "NOCI-PT2" theory exploits a zeroth-order generalised Fock Hamiltonian and builds the first-order interacting space using single and double excitations from each reference determinant. This approach therefore defines the rigorous nonorthogonal extension to conventional multireference perturbation theory. We find that NOCI-PT2 can quantitatively predict multireference potential energy surfaces and provides state-specific ground and excited states for adiabatic avoided crossings. Furthermore, we introduce an explicit imaginary-shift formalism requiring shift values that are an order of magnitude smaller than those used in conventional multireference perturbation theory.

physics.chem-ph↗

arXiv2019

Symmetry in Multiple Self-Consistent-Field Solutions of Transition-Metal Complexes

We use a method based on metadynamics to locate multiple low-energy Unrestricted Hartree--Fock (UHF) self-consistent-field (SCF) solutions of two model octahedral $d^1$ and $d^2$ transition-metal complexes, $[\mathrm{MF}_6]^{3-} (\mathrm{M} = \mathrm{Ti}, \mathrm{V})$. By giving a group-theoretical definition of symmetry breaking, we classify these solutions in the framework of representation theory and observe that a number of them break spin or spatial symmetry, if not both. These solutions seem unphysical at first, but we show that they can be used as bases for Non-Orthogonal Configuration Interaction (NOCI) to yield multi-determinantal wavefunctions that have the right symmetry to be assigned to electronic terms. Furthermore, by examining the natural orbitals and occupation numbers of these NOCI wavefunctions, we gain insight into the amount of static correlation that they incorporate. We then investigate the behaviors of the most low-lying UHF and NOCI wavefunctions when the octahedral symmetry of the complexes is lowered and deduce that the symmetry-broken UHF solutions must first have their symmetry restored by NOCI before they can describe any vibronic stabilization effects dictated by the Jahn--Teller theorem.

physics.chem-ph↗

arXiv2019

Accelerating Stochastic Quantum Chemistry

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 optimised propagation is O(1), 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.

physics.chem-ph↗

arXiv2019

$\mathcal{PT}$-Symmetry in Hartree-Fock Theory

$\mathcal{PT}$-symmetry --- invariance with respect to combined space reflection $\mathcal{P}$ and time reversal $\mathcal{T}$ --- provides a weaker condition than (Dirac) Hermiticity for ensuring a real energy spectrum of a general non-Hermitian Hamiltonian. $\mathcal{PT}$-symmetric Hamiltonians therefore form an intermediate class between Hermitian and non-Hermitian Hamiltonians. In this work, we derive the conditions for $\mathcal{PT}$-symmetry in the context of electronic structure theory, and specifically, within the Hartree-Fock (HF) approximation. We show that the HF orbitals are symmetric with respect to the $\mathcal{PT}$ operator \textit{if and only if} the effective Fock Hamiltonian is $\mathcal{PT}$-symmetric, and \textit{vice versa}. By extension, if an optimal self-consistent solution is invariant under $\mathcal{PT}$, then its eigenvalues and corresponding HF energy must be real. Moreover, we demonstrate how one can construct explicitly $\mathcal{PT}$-symmetric Slater determinants by forming $\mathcal{PT}$ doublets (i.e. pairing each occupied orbital with its $\mathcal{PT}$-transformed analogue), allowing $\mathcal{PT}$-symmetry to be conserved throughout the self-consistent process. Finally, considering the \ce{H2} molecule as an illustrative example, we observe $\mathcal{PT}$-symmetry in the HF energy landscape and find that the symmetry-broken unrestricted HF wave functions (i.e. diradical configurations) are $\mathcal{PT}$-symmetric, while the symmetry-broken restricted HF wave functions (i.e. ionic configurations) break $\mathcal{PT}$-symmetry.

physics.chem-ph↗

arXiv2019

A General Approach for Multireference Ground and Excited States using Non-Orthogonal Configuration Interaction

A balanced description of ground and excited states is essential for the description of many chemical processes. However, few methods can handle cases where static correlation is present, and often these scale very unfavourably with system size. Recently, multiple Hartree--Fock (HF) solutions have been proposed as a basis for non-orthogonal configuration interaction (NOCI) to provide multireference ground and excited state energies, although applications across multiple geometries have been limited by the coalescence of HF solutions. Holomorphic HF (h-HF) theory allows solutions to be analytically continued beyond the Coulson--Fischer points at which they vanish but, until now, this has only been demonstrated for small model systems. In this work, we propose a general protocol for computing NOCI ground and excited state energies using multiple HF solutions. To do so, we outline an active space variation of SCF metadynamics that allows a chemically relevant set of HF states to be identified, and describe how these states can be routinely traced across all molecular geometries by exploiting the topology of h-HF solutions in the complex plane. Finally, we illustrate our approach using the dissociation of the fluorine dimer and the pseudo-Jahn--Teller distortion of cyclobutadiene, demonstrating its applicability for multireference ground and excited states.

physics.chem-ph↗

arXiv2019

Making the most of data: Quantum Monte Carlo Post-Analysis Revisited

In quantum Monte Carlo (QMC) methods, energy estimators are calculated as the statistical average of the Markov chain sampling of energy estimator along with an associated statistical error. This error estimation is not straightforward and there are several choices of the error estimation methods. We evaluate the performance of three methods, Straatsma, an autoregressive model, and a blocking analysis based on von Neumann's ratio test for randomness, for the energy time-series given by Diffusion Monte Carlo, Full Configuration Interaction Quantum Monte Carlo and Coupled Cluster Monte Carlo methods. From these analyses we describe a hybrid analysis method which provides reliable error estimates for series of all lengths. Equally important is the estimation of the appropriate start point of the equilibrated phase, and two heuristic schemes are tested, establishing that MSER (mean squared error rule) gives reasonable and constant estimations independent of the length of time-series.

physics.comp-ph↗

arXiv2019

Preconditioning and perturbative estimators in full configuration interaction quantum Monte Carlo

We propose the use of preconditioning in FCIQMC which, in combination with perturbative estimators, greatly increases the efficiency of the algorithm. The use of preconditioning allows a time step close to unity to be used (without time-step errors), provided that multiple spawning attempts are made per walker. We show that this approach substantially reduces statistical noise on perturbative corrections to initiator error, which improve the accuracy of FCIQMC but which can suffer from significant noise in the original scheme. Therefore, the use of preconditioning and perturbatively-corrected estimators in combination leads to a significantly more efficient algorithm. In addition, a simpler approach to sampling variational and perturbative estimators in FCIQMC is presented, which also allows the variance of the energy to be calculated. These developments are investigated and applied to benzene (30e,108o), an example where accurate treatment is not possible with the original method.

physics.chem-ph↗

arXiv2019

Diagrammatic Coupled Cluster Monte Carlo

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.

physics.chem-ph↗

arXiv2019

Complex Adiabatic Connection: a Hidden Non-Hermitian Path from Ground to Excited States

Processes related to electronically excited states are central in many areas of science, however accurately determining excited-state energies remains a major challenge in theoretical chemistry. Recently, higher energy stationary states of non-linear methods have themselves been proposed as approximations to excited states, although the general understanding of the nature of these solutions remains surprisingly limited. In this Letter, we present an entirely novel approach for exploring and obtaining excited stationary states by exploiting the properties of non-Hermitian Hamiltonians. Our key idea centres on performing analytic continuations of conventional quantum chemistry methods. Considering Hartree--Fock theory as an example, we analytically continue the electron-electron interaction to expose a hidden connectivity of multiple solutions across the complex plane, revealing a close resemblance between Coulson--Fischer points and non-Hermitian degeneracies. Finally, we demonstrate how a ground-state wave function can be morphed naturally into an excited-state wave function by constructing a well-defined complex adiabatic connection.

physics.chem-ph↗

arXiv2018

Large Scale Parallelization in Stochastic Coupled Cluster

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.

physics.chem-ph↗

arXiv2018

The HANDE-QMC project: open-source stochastic quantum chemistry from the ground state up

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 last 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 semi-stochastic 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.

physics.comp-ph↗

arXiv2018

Exciting determinants in Quantum Monte Carlo: Loading the dice with fast, low memory weights

High-quality excitation generators are crucial to the effectiveness of Coupled cluster Monte Carlo (CCMC) and full configuration interaction Quantum Monte Carlo (FCIQMC) calculations. The heat bath sampling of Holmes et al. [A. A. Holmes, H. J. Changlani, and C. J. Umrigar, J. Chem. Theory Comput. 12, 1561 (2016)] dramatically increases the efficiency of the spawn step of such algorithms but requires memory storage scaling quartically with system size which can be prohibitive for large systems. Alavi et al. [S. D. Smart, G. H. Booth, and A. Alavi, unpublished] then approximated these weights with weights based on Cauchy--Schwarz-like inequalities calculated on-the-fly. While reducing the memory cost, this algorithm scales linearly in system size computationally. We combine both these ideas with the single reference nature of many systems, and introduce a spawn-sampling algorithm that has low memory requirements (quadratic in basis set size) compared to the heat bath algorithm and only scales either independently of system size (CCMC) or linearly in the number of electrons (FCIQMC). On small water chains with localized orbitals, we show that it is equally efficient as the other excitation generators. As the system gets larger, it converges faster than the on-the-fly weight algorithm, while having a much more favourable memory scaling than the heat bath algorithm.

physics.chem-ph↗

arXiv2018

Field-Programmable Gate Arrays and Quantum Monte Carlo: Power Efficient Co-processing for Scalable High-Performance Computing

Massively parallel architectures offer the potential to significantly accelerate an application relative to their serial counterparts. However, not all applications exhibit an adequate level of data and/or task parallelism to exploit such platforms. Furthermore, the power consumption associated with these forms of computation renders "scaling out" for exascale levels of performance incompatible with modern sustainable energy policies. In this work, we investigate the potential for field-programmable gate arrays (FPGAs) to feature in future exascale platforms, and their capacity to improve performance per unit power measurements for the purposes of scientific computing. We have focussed our efforts on Variational Monte Carlo, and report on the benefits of co-processing with an FPGA relative to a purely multicore system.

physics.comp-ph↗

arXiv2018

Modelling Electron Transfers Using Quasidiabatic Hartree-Fock States

Electron transfer processes are ubiquitous in chemistry and of great importance in many systems of biological and commercial interest. The ab-initio description of these processes remains a challenge in theoretical chemistry, partly due to the high scaling of many post-Hartree--Fock computational methods. This poses a problem for systems of interest that are not easily investigated experimentally. We show that readily available Hartree--Fock solutions can be used as a quasidiabatic basis to understand electron transfer reactions in a Marcus framework. Non-orthogonal configuration interaction calculations can be used to quantify interactions between the resulting electronic states, and to investigate the adiabatic electron transfer process. When applied to a titanium-alizarin complex used as a model of a Grätzel-type solar cell, this approach yields a correct description of the electron transfer and provides information about the electronic states involved in the process.

physics.chem-ph↗