## Publications

87 results found

Azadi S, Drummond ND, Foulkes WMC, 2021, Quasiparticle effective mass of the three-dimensional fermi liquid by quantum Monte Carlo, *Physical Review Letters*, Vol: 127, Pages: 1-6, ISSN: 0031-9007

According to Landau's Fermi liquid theory, the main properties of thequasiparticle excitations of an electron gas are embodied in the effective mass$m^*$, which determines the energy of a single quasiparticle, and the Landauinteraction function, which indicates how the energy of a quasiparticle ismodified by the presence of other quasiparticles. This simple paradigmunderlies most of our current understanding of the physical and chemicalbehavior of metallic systems. The quasiparticle effective mass of thethree-dimensional homogeneous electron gas has been the subject of theoreticalcontroversy and there is a lack of experimental data. In this work, we deploydiffusion Monte Carlo (DMC) methods to calculate $m^*$ as a function of densityfor paramagnetic and ferromagnetic three-dimensional homogeneous electrongases. The DMC results indicate that $m^*$ decreases when the density isreduced, especially in the ferromagnetic case. The DMC quasiparticle energybands exclude the possibility of a reduction in the occupied bandwidth relativeto that of the free-electron model at density parameter $r_s=4$, whichcorresponds to Na metal.

Fagerholm ED, Foulkes W, Friston KJ,
et al., 2021, Rendering neuronal state equations compatiblewith the principle of stationary action, *Journal of Mathematical Neuroscience*, Vol: 11, Pages: 1-15, ISSN: 2190-8567

The principle of stationary action is a cornerstone of modern physics, providing a powerful framework for investigating dynamical systems found in classical mechanics through to quantum field theory. However, computational neuroscience, despite its heavy reliance on concepts in physics, is anomalous in this regard as its main equations of motion are not compatible with a Lagrangian formulation and hence with the principle of stationary action. Taking the Dynamic Causal Modelling (DCM) neuronal state equation as an instructive archetype of the first-order linear differential equations commonly found in computational neuroscience, we show that it is possible to make certain modifications to this equation to render it compatible with the principle of stationary action. Specifically, we show that a Lagrangian formulation of the DCM neuronal state equation is facilitated using a complex dependent variable, an oscillatory solution, anc a Hermitian intrinsic connectivity matrix. We first demonstrate proof of principle by using Bayesian model inversion to show that both the original and modified models can be correctly identified via in silico data generated directly from their respective equations of motion. We then provide motivation for adopting the modified models in neuroscience by using three different types of publicly available in vivo neuroimaging datasets, together with open source MATLAB code, to show that the modified (oscillatory) model provides a more parsimonious explanation for some of these empirical timeseries. It is our hope that this work will, in combination with existing techniques, allow people to explore the symmetries and associated conservation laws within neural systems – and to exploit the computational expediency facilitated by direct variational techniques.

Fagerholm ED, Foulkes W, Gallero-Salas Y,
et al., 2021, Neural systems under change of scale, *Frontiers in Computational Neuroscience*, Vol: 15, ISSN: 1662-5188

We derive a theoretical construct that allows for the characterisation of both scalable and scale free systems within the Dynamic Causal Modelling framework. We define a dynamical system to be ‘scalable’ if the same equation of motion continues to apply as the system changes in size. As an example of such a system, we simulate planetary orbits varying in size and show that our proposed methodology can be used to recover Kepler’s third law from the timeseries. In contrast, a ‘scale free’ system is one in which there is no characteristic length scale, meaning that images of such a system are statistically unchanged at different levels of magnification. As an example of such a system, we use calcium imaging collected in murine cortex and show that the dynamical critical exponent, as defined in renormalization group theory, can be estimated in an empirical biological setting. We find that a task-relevant region of the cortex is associated with higher dynamical critical exponents in task vs. spontaneous states and vice versa for a task-irrelevant region.

Pfau D, Spencer JS, Matthews AGDG,
et al., 2020, Ab-initio solution of the many-electron Schrödinger equation with deepneural networks, *Physical Review Research*, Vol: 2, ISSN: 2643-1564

Given access to accurate solutions of the many-electron Schr\"odingerequation, nearly all chemistry could be derived from first principles. Exactwavefunctions of interesting chemical systems are out of reach because they areNP-hard to compute in general, but approximations can be found usingpolynomially-scaling algorithms. The key challenge for many of these algorithmsis the choice of wavefunction approximation, or Ansatz, which must trade offbetween efficiency and accuracy. Neural networks have shown impressive power asaccurate practical function approximators and promise as a compact wavefunctionAnsatz for spin systems, but problems in electronic structure requirewavefunctions that obey Fermi-Dirac statistics. Here we introduce a novel deeplearning architecture, the Fermionic Neural Network, as a powerful wavefunctionAnsatz for many-electron systems. The Fermionic Neural Network is able toachieve accuracy beyond other variational quantum Monte Carlo Ans\"atze on avariety of atoms and small molecules. Using no data other than atomic positionsand charges, we predict the dissociation curves of the nitrogen molecule andhydrogen chain, two challenging strongly-correlated systems, to significantlyhigher accuracy than the coupled cluster method, widely considered the mostaccurate scalable method for quantum chemistry at equilibrium geometry. Thisdemonstrates that deep neural networks can improve the accuracy of variationalquantum Monte Carlo to the point where it outperforms other ab-initio quantumchemistry methods, opening the possibility of accurate direct optimisation ofwavefunctions for previously intractable molecules and solids.

Foulkes WMC, 2020, Variational Wave Functions for Molecules and Solids, Topology, Entanglement, and Strong Correlations, Editors: Pavarini, Koch, Publisher: Forschungszentrum Juelich GmbH, Institute for Advanced Simulations, Pages: 2.1-2.1, ISBN: 978-3-95806-466-9

Fagerholm ED, Foulkes W, Gallero-Salas Y,
et al., 2020, Conservation laws by virtue of scale symmetries in neural systems, *PLoS Computational Biology*, Vol: 16, ISSN: 1553-734X

In contrast to the symmetries of translation in space, rotation in space, and translation in time, the known laws of physics are not universally invariant under transformation of scale. However, a special case exists in which the action is scale invariant if it satisfies the following two constraints: 1) it must depend upon a scale-free Lagrangian, and 2) the Lagrangian must change under scale in the same way as the inverse time, . Our contribution lies in the derivation of a generalised Lagrangian, in the form of a power series expansion, that satisfies these constraints. This generalised Lagrangian furnishes a normal form for dynamic causal models–state space models based upon differential equations–that can be used to distinguish scale symmetry from scale freeness in empirical data. We establish face validity with an analysis of simulated data, in which we show how scale symmetry can be identified and how the associated conserved quantities can be estimated in neuronal time series.

Azadi S, Foulkes WMC, 2019, Efficient method for grand-canonical twist averaging in quantum Monte Carlo calculations, *Physical Review B: Condensed Matter and Materials Physics*, Vol: 100, ISSN: 1098-0121

We introduce a simple but efficient method for grand-canonical twistaveraging in quantum Monte Carlo calculations. By evaluating the thermodynamic grand potential instead of the ground state total energy, we greatly reduce the sampling errors caused by twist-dependent fluctuations in the particle number. We apply this method to the electron gas and to metallic lithium, aluminum, and solid atomic hydrogen. We show that, even when using a small number of twists, grand-canonical twist averaging of the grand potential produces better estimates of ground state energies than the widely used canonical twist-averaging approach.

Wells T, Horsfield A, Foulkes WMC,
et al., 2019, The microscopic Einstein-de Haas effect, *Journal of Chemical Physics*, Vol: 150, ISSN: 0021-9606

The Einstein-de Haas (EdH) effect, where the spin angular momentum of electrons is transferred to the mechanical angular momentum of atoms, was established experimentally in 1915. While a semiclassical explanation of the effect exists, modern electronic structure methods have not yet been applied to model the phenomenon. In this paper, we investigate its microscopic origins by means of a noncollinear tight-binding model of an O2 dimer, which includes the effects of spin-orbit coupling, coupling to an external magnetic field, and vector Stoner exchange. By varying an external magnetic field in the presence of spin-orbit coupling, a torque can be generated on the dimer, validating the presence of the EdH effect. The avoided energy level crossings and the rate of change of magnetic field determine the evolution of the spin. We also find that the torque exerted on the nuclei by the electrons in a time-varying B field is not only due to the EdH effect. The other contributions arise from field-induced changes in the electronic orbital angular momentum and from the direct action of the Faraday electric field associated with the time-varying magnetic field.

Spencer JS, Blunt NS, Choi S,
et al., 2019, The HANDE-QMC project: open-source stochastic quantum chemistry from the ground state up, *Journal of Chemical Theory and Computation*, Vol: 15, Pages: 1728-1742, ISSN: 1549-9618

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.

Coury MEA, Dudarev SL, Foulkes WMC,
et al., 2018, Erratum: Hubbard-like Hamiltonians for interacting electrons in s, p, and d orbitals (vol 93, 075101, 2016), *Physical Review B*, Vol: 98, ISSN: 2469-9950

Davies PAG, Foulkes WMC, 2018, A two-phase Hessian approach improves the DFT relaxation of slabs, *Journal of Physics: Condensed Matter*, Vol: 30, Pages: 315901-315901, ISSN: 0953-8984

A two-phase Hessian approach to DFT slab relaxation of slabs has been implemented and tested. It addresses weaknesses in the modified Broyden and Pfrommer BFGS algorithms specific to relaxing slabs. Complete Hessian and then inverse Hessian matrices with no strain/stress components are first constructed at high force signal-to-noise ratios with no accompanying relaxation. In a second phase the static inverse Hessian is used to relax the slab down to a low force tolerance.

Groth S, Dornheim T, Sjostrom T,
et al., 2017, Ab initio exchange-correlation free energy of the uniform electron gas at warm dense matter conditions, *Physical Review Letters*, Vol: 119, ISSN: 0031-9007

In a recent Letter [T.~Dornheim \textit{et al.}, Phys. Rev. Lett.\textbf{117}, 156403 (2016)], we presented the first \textit{ab initio} quantumMonte-Carlo (QMC) results of the warm dense electron gas in the thermodynamiclimit. However, a complete parametrization of the exchange-correlation freeenergy with respect to density, temperature, and spin polarization remained outof reach due to the absence of (i) accurate QMC results below$\theta=k_\text{B}T/E_\text{F}=0.5$ and (ii) of QMC results for spinpolarizations different from the paramagnetic case. Here we overcome bothremaining limitations. By closing the gap to the ground state and by performingextensive QMC simulations for different spin polarizations, we are able toobtain the first complete \textit{ab initio} exchange-correlation free energyfunctional; the accuracy achieved is an unprecedented $\sim 0.3\%$. This alsoallows us to quantify the accuracy and systematic errors of various previousapproximate functionals.

Dornheim T, Groth S, Malone FD,
et al., 2017, Ab initio quantum Monte Carlo simulation of the warm dense electron gas, *Physics of Plasmas*, Vol: 24, Pages: 056303-1-056303-10, ISSN: 1089-7674

Warm dense matter is one of the most active frontiers in plasma physics due to its relevance for denseastrophysical objects as well as for novel laboratory experiments in which matter is being strongly compressede.g. by high-power lasers. Its description is theoretically very challenging as it contains correlated quantumelectrons at nite temperature|a system that cannot be accurately modeled by standard analytical or groundstate approaches. Recently several breakthroughs have been achieved in the eld of fermionic quantum MonteCarlo simulations. First, it was shown that exact simulations of a nite model system (30 : : : 100 electrons)is possible that avoid any simplifying approximations such as xed nodes [Schoof et al., Phys. Rev. Lett.115, 130402 (2015)]. Second, a novel way to accurately extrapolate these results to the thermodynamic limitwas reported by Dornheim et al. [Phys. Rev. Lett. 117, 156403 (2016)]. As a result, now thermodynamicresults for the warm dense electron gas are available that have an unprecedented accuracy on the order of0:1%. Here we present an overview on these results and discuss limitations and future directions.

Azadi S, Drummond ND, Foulkes WMC, 2017, Nature of the metallization transition in solid hydrogen, *Physical Review. B, Condensed Matter*, Vol: 95, ISSN: 0163-1829

We present an accurate study of the static-nucleus electronic energy band gap of solid molecular hydrogen at high pressure. The excitonic and quasiparticle gaps of the C2/c, Pc, Pbcn, and P63/mstructures at pressures of 250, 300, and 350 GPa are calculated using the fixed-node diffusion quantum Monte Carlo (DMC) method. The difference between the mean-field and many-body band gaps at the same density is found to be almost independent of system size and can therefore be applied as a scissor correction to the mean-field gap of an infinite system to obtain an estimate of the many-body gap in the thermodynamic limit. By comparing our static-nucleus DMC energy gaps with available experimental results, we demonstrate the important role played by nuclear quantum effects in the electronic structure of solid hydrogen.

Dornheim T, Groth S, Sjostrom T,
et al., 2016, Ab initio quantum Monte Carlo simulation of the warm dense electron gas in the thermodynamic limit, *Physical Review Letters*, Vol: 117, ISSN: 1079-7114

We perform ab initio quantum Monte Carlo (QMC) simulations of the warm dense uniform electrongas in the thermodynamic limit. By combining QMC data with linear response theory we are able toremove finite-size errors from the potential energy over the entire warm dense regime, overcoming thedeficiencies of the existing finite-size corrections by Brown et al. [PRL 110, 146405 (2013)]. Extensivenew QMC results for up to N = 1000 electrons enable us to compute the potential energy V and theexchange-correlation free energy Fxc of the macroscopic electron gas with an unprecedented accuracyof |∆V |/|V |, |∆Fxc|/|F|xc ∼ 10−3. A comparison of our new data to the recent parametrization ofFxc by Karasiev et al. [PRL 112, 076403 (2014)] reveals significant deviations to the latter.

Foulkes WMC, 2016, Tight-Binding Models and Coulomb Interactions for s, p, and d Electrons, Quantum Materials: Experiments and Theory, Editors: Pavarini, Koch, van den Brink, Sawatzky, Jülich, Germany, Publisher: Forschungszentrum Jülich GmbH, Pages: 3.1-3.42, ISBN: 978-3-95806-159-0

Malone FD, Blunt NS, Brown EW,
et al., 2016, Accurate exchange-correlation energies for the warm dense electron gas, *Physical Review Letters*, Vol: 117, ISSN: 1079-7114

The density matrix quantum Monte Carlo (DMQMC) method is used to sample exact-on-average N-body density matrices for uniform electron gas systems of up to 10124 matrix elements via a stochastic solution of the Bloch equation. The results of these calculations resolve a current debate over the accuracy of the data used to parametrize finite-temperature density functionals. Exchange-correlation energies calculated using the real-space restricted path-integral formalism and the k-space configuration path-integral formalism disagree by up to ∼10% at certain reduced temperatures T/TF≤0.5 and densities rs≤1. Our calculations confirm the accuracy of the configuration path-integral Monte Carlo results available at high density and bridge the gap to lower densities, providing trustworthy data in the regime typical of planetary interiors and solids subject to laser irradiation. We demonstrate that the DMQMC method can calculate free energies directly and present exact free energies for T/TF≥1 and rs≤2.

Horsfield AP, Lim A, Foulkes WMC,
et al., 2016, Adiabatic perturbation theory of electronic stopping in insulators, *Physical Review B*, Vol: 93, Pages: 1-1, ISSN: 2469-9950

A model able to explain the complicated structure of electronic stopping at low velocities in insulating materials is presented. It is shown to be in good agreement with results obtained from time-dependent density-functional theory for the stopping of a channeling Si atom in a Si crystal. If we define the repeat frequency f=v/λ, where λ is the periodic repeat length of the crystal along the direction the channeling atom is traveling, and v is the velocity of the channeling atom, we find that electrons experience a perturbing force that varies in time at integer multiples l of f. This enables electronic excitations at low atom velocity, but their contributions diminish rapidly with increasing values of l. The expressions for stopping power are derived using adiabatic perturbation theory for many-electron systems, and they are then specialized to the case of independent electrons. A simple model for the nonadiabatic matrix elements is described, along with the procedure for determining its parameters.

Heuer AH, Azar MZ, Guhl H,
et al., 2016, The band structure of polycrystalline Al2O3 and its influence on transport phenomena, *Journal of the American Ceramic Society*, Vol: 99, Pages: 733-747, ISSN: 1551-2916

Coury MEA, Dudarev SL, Foulkes WMC,
et al., 2016, Hubbard-like Hamiltonians for interacting electrons in s, p, and d orbitals, *Physical Review B*, Vol: 93, ISSN: 1550-235X

Hubbard-like Hamiltonians are widely used to describe on-site Coulomb interactions in magnetic and strongly-correlated solids, but there is much confusion in the literature about the form these Hamiltonians should take for shells of p and d orbitals. This paper derives the most general s,p, and d orbital Hubbard-like Hamiltonians consistent with the relevant symmetries, and presents them in ways convenient for practical calculations. We use the full configuration interaction method to study p and d orbital dimers and compare results obtained using the correct Hamiltonian and the collinear and vector Stoner Hamiltonians. The Stoner Hamiltonians can fail to describe properly the nature of the ground state, the time evolution of excited states, and the electronic heat capacity.

Lim A, Foulkes WM, Horsfield AP,
et al., 2016, Electron elevator: excitations across the band gap via a dynamical gap state, *Physical Review Letters*, Vol: 116, Pages: 1-1, ISSN: 0031-9007

We use time-dependent density functional theory to study self-irradiated Si. We calculate the electronic stopping power of Si in Si by evaluating the energy transferred to the electrons per unit path length by an ion of kinetic energy from 1 eV to 100 keV moving through the host. Electronic stopping is found to be significant below the threshold velocity normally identified with transitions across the band gap. A structured crossover at low velocity exists in place of a hard threshold. An analysis of the time dependence of the transition rates using coupled linear rate equations enables one of the excitation mechanisms to be clearly identified: a defect state induced in the gap by the moving ion acts like an elevator and carries electrons across the band gap.

Spencer JS, Blunt NS, Vigor WA,
et al., 2015, Open-source development experiences in scientific software: the HANDE quantum Monte Carlo project, *Journal of Open Research Software*, Vol: 3, ISSN: 2049-9647

The HANDE quantum Monte Carlo project offers accessible stochastic algorithmsfor general use for scientists in the field of quantum chemistry. HANDE is anambitious and general high-performance code developed by ageographically-dispersed team with a variety of backgrounds in computationalscience. In the course of preparing a public, open-source release, we havetaken this opportunity to step back and look at what we have done and what wehope to do in the future. We pay particular attention to development processes,the approach taken to train students joining the project, and how a flathierarchical structure aids communication

Edmunds DM, Tangney P, Vvedensky DD,
et al., 2015, Free-energy coarse-grained potential for C60, *Journal of Chemical Physics*, Vol: 143, Pages: 164509-1-164509-4, ISSN: 0021-9606

We propose a new deformable free energy method for generating a free-energy coarse-graining potential for C60. Potentials generated from this approach exhibit a strong temperature dependence and produce excellent agreement with benchmark fully atomistic molecular dynamics simulations. Parameter sets for analytical fits to this potential are provided at four different temperatures.

Guhl H, Lee HS, Tangney P,
et al., 2015, Structural and electronic properties of sigma7 grain boundaries in alpha-Al2O3, *Acta Materialia*, Vol: 99, Pages: 16-28, ISSN: 1359-6454

Applying simulated annealing with a classical potential followed by screening of low-energy structures with density functional theory, we examined the atomic and electronic structures of the View the MathML source and View the MathML source symmetric tilt grain boundaries in α-Al2O3. The lowest energy View the MathML source boundary exhibits a pronounced pattern of alternating columns of exclusively four- or fivefold coordinated Al atoms, with a grain boundary energy of 1.84 Jm−2. For the View the MathML source boundary, numerous structures were found with energy just below 2.11 Jm−2. Furthermore, by analysing the full set of candidate structures generated by simulated annealing for the two grain boundaries, we find that the number of fivefold coordinated Al atoms tends to increase with grain boundary energy, which we can also correlate with the behaviour of the electronic density of states. On the other hand, we find no systematic trend with energy that might be expected for other quantities, notably the excess volume of the interface. We compare simulated high-resolution transmission electron microscope (HRTEM) images of the lowest energy calculated structures with experimental images. The disparate structural and electronic features of these two boundaries suggest reasons for their very different oxygen diffusion coefficients that have been observed experimentally.

Malone FD, Blunt NS, Shepherd JJ,
et al., 2015, Interaction picture density matrix quantum Monte Carlo, *Journal of Chemical Physics*, Vol: 143, ISSN: 1089-7690

The recently developed density matrix quantum Monte Carlo (DMQMC) algorithm stochastically samplesthe N-body thermal density matrix and hence provides access to exact properties of many-particle quantumsystems at arbitrary temperatures. We demonstrate that moving to the interaction picture provides substan-tial benefits when applying DMQMC to interacting fermions. In this first study, we focus on a system ofmuch recent interest: the uniform electron gas in the warm dense regime. The basis set incompleteness errorat finite temperature is investigated and extrapolated via a simple Monte Carlo sampling procedure. Finally,we provide benchmark calculations for a four-electron system, comparing our results to previous work wherepossible.

Azadi S, Foulkes WMC, 2015, Systematic study of finite-size effects in quantum Monte Carlo calculations of real metallic systems, *Journal of Chemical Physics*, Vol: 143, ISSN: 1089-7690

We present a systematic and comprehensive study of finite-size effects in diffusion quantum Monte Carlo calculations of metals. Several previously introduced schemes for correcting finite-size errors are compared for accuracy and efficiency, and practical improvements are introduced. In particular, we test a simple but efficient method of finite-size correction based on an accurate combination of twist averaging and density functional theory. Our diffusion quantum Monte Carlo results for lithium and aluminum, as examples of metallic systems, demonstrate excellent agreement between all of the approaches considered.

Drummond ND, Needs RJ, Sorouri A,
et al., 2014, Erratum: Finite-size errors in continuum quantum Monte Carlo calculations (Physical Review B - Condensed Matter and Materials Physics (2008) 78 (125106)), *Physical Review B - Condensed Matter and Materials Physics*, Vol: 90, ISSN: 1098-0121

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Blunt NS, Rogers TW, Spencer JS,
et al., 2014, Density-matrix quantum Monte Carlo method, *Physical Review B*, Vol: 89, ISSN: 2469-9950

We present a quantum Monte Carlo method capable of sampling the full density matrix of a many-particle system at finite temperature. This allows arbitrary reduced density matrix elements and expectation values of complicated nonlocal observables to be evaluated easily. The method resembles full configuration interaction quantum Monte Carlo but works in the space of many-particle operators instead of the space of many-particle wave functions. One simulation provides the density matrix at all temperatures simultaneously, from T=∞ to T=0, allowing the temperature dependence of expectation values to be studied. The direct sampling of the density matrix also allows the calculation of some previously inaccessible entanglement measures. We explain the theory underlying the method, describe the algorithm, and introduce an importance-sampling procedure to improve the stochastic efficiency. To demonstrate the potential of our approach, the energy and staggered magnetization of the isotropic antiferromagnetic Heisenberg model on small lattices, the concurrence of one-dimensional spin rings, and the Renyi S2 entanglement entropy of various sublattices of the 6×6 Heisenberg model are calculated. The nature of the sign problem in the method is also investigated.

Azadi S, 2014, Dissociation of High-Pressure Solid Molecular Hydrogen: A Quantum Monte Carlo and Anharmonic Vibrational Study, *Phys. Rev. Lett*, Vol: 112

Azadi S, Foulkes WMC, Kuehne TD, 2013, Quantum Monte Carlo study of high pressure solid molecular hydrogen, *NEW JOURNAL OF PHYSICS*, Vol: 15, ISSN: 1367-2630

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- Citations: 40

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