Imperial College London

Matthew Foulkes

Faculty of Natural SciencesDepartment of Physics

Professor of Physics



wmc.foulkes Website




810Blackett LaboratorySouth Kensington Campus




Solving the Many-Electron Schrodinger Equation using Deep Neural Networks

Given access to accurate solutions of the many-electron Schrödinger equation, nearly all chemistry could be derived from first principles. Exact wavefunctions of interesting chemical systems are out of reach because they are NP-hard to compute in general, but approximations can be found using polynomially-scaling algorithms. The key challenge for many  of these algorithms is the choice of wavefunction approximation, or Ansatz, which must trade off between efficiency and accuracy.

Neural networks have shown impressive power as accurate practical function approximators and promise as a compact wavefunction Ansatz for spin systems, but problems in electronic structure require wavefunctions that obey Fermi-Dirac statistics. In this work we introduced a novel deep learning architecture, the Fermionic Neural Network, as a powerful wavefunction Ansatz for many-electron systems.

The Fermionic Neural Network is able to achieve accuracy beyond other variational Monte Carlo trial wavefunctions on a variety of atoms and small molecules. Using no data other than atomic positions and charges, we predict the dissociation curves of the nitrogen molecule and hydrogen chain, two challenging strongly-correlated systems, to significantly higher accuracy than the coupled cluster method, widely considered the gold standard for quantum chemistry. This demonstrates that deep neural networks can outperform existing ab-initio quantum chemistry methods, opening the possibility of accurate direct optimisation of wavefunctions for previously intractable molecules and solids.

Ab-Initio Solution of the Many-Electron Schrödinger Equation with Deep Neural Networks, by David Pfau, James S. Spencer, Alexander G. de G. Matthews, and W.M.C. Foulkes, arxiv:1909.02487

Warm Dense Matter

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Electrons are an elementary component of our world and determine many of the properties of solids and liquids, but despite their ubiquity, scientists have not yet been able to accurately describe the behaviour of large numbers of interacting electrons.

This is especially true at extreme temperatures and densities, such as inside planets or in stars, where the electrons form ‘warm dense matter’. Scientists have plenty of approximate models to choose from, but little idea of their accuracy or reliability.

Recently, working with groups from Kiel University, the Los Alamos National Laboratory and the Lawrence Livermore National Laboratory,  we have succeeded in describing electrons under these extreme conditions by means of accurate quantum Monte Carlo simulations.

How electrons behave on a ‘large scale’ – for example the relation between electrical voltage, resistance and current – is often easy to describe. On a microscopic level, however, the electrons in liquids and solids behave differently, according to the laws of quantum mechanics. They act like a quantum mechanical ‘gas’, which can only be understood by solving the complicated mathematical equations of quantum theory.

In the past, simulations were only able to describe the electron gas at very low temperature. Recently, however, there has been growing interest in matter under extreme conditions - ten thousand times warmer than room temperature and up to a hundred times denser than conventional solids.

Warm dense matter occurs inside planets, including the cores of Jupiter and Saturn. (The Earth’s core is nearly in the warm dense regime, too, although a little cold.) Warm dense matter can be created experimentally in a laboratory, for example by targeted shooting of solid matter with a high-intensity laser, or with a free electron laser such as the new European XFEL in Hamburg. Warm dense matter is relevant for experiments with inertial confinement fusion, where fuel pellets are put under extreme pressure. This can cause chain reactions that might one day provide a virtually unlimited source of clean energy.

Earlier theories of warm dense matter behaviour used models based on approximations that are difficult to verify. However, by using sophisticated computer simulations, we are now able to precisely solve the complex equations that describe the electron gas in the warm dense regime. Our recent simulations of the uniform electron gas allowed us to construct the first accurate local density approximation for the exchange-correlation free energy. This can be used, along with simpler simulation methods based on thermal density functional theory, to study more complicated warm dense objects.

Ab initio Exchange-Correlation Free Energy of the Uniform Electron Gas at Warm Dense Matter Conditions, by Simon Groth, Tobias Dornheim, Travis Sjostrom, Fionn D. Malone, W. M. C. Foulkes, and Michael Bonitz, Phys. Rev. Lett. 119, 135001 (2017).



James Shepherd, University of Iowa, 2019

Fionn Malone, Lawrence Livermore National Laboratory, 2019

Alex Matthews, David Pfau, James Spencer, Deepmind Ltd., 2018

James J. Shepherd, Troy van Voorhis, Department of ChemistryMassachusetts Institute of Technology, Quantum chemistry at high temperature, 2016

Derek Lee, Imperial College London, 2015

Professor Michael Bonitz, University of Kiel, 2015 - 2017

Travis Sjostrom, Los Alamos National Laboratory, 2015 - 2017

Dr T N Todorov, Queen's University Belfast; Professor Matthew Foulkes, Physics at Imperial College; Professor Mike Finnis Materials and Physics at , Imperial College; Dr Andrew Horsfield, Materials at Imperial College; Dr Daniel Mason, Physics at Imperial College; Mr Chris Race, Physics at Imperial College, Imperial College London and Queen's University Belfast, Electronic excitations during radiation damage of metals, 2005 - 2010

Professor Richard Needs, University of Cambridge, 1990

Sutton, Adrian, Physics, Imperial College London

Professor Arthur H. Heuer, Case Western Reserve University, Cleveland

Horsfield, Andrew, Materials, Imperial College London

Finnis, Mike, Physics, Imperial College London

Research Staff








Research Student Supervision

Bradley,C, Warm dense matter

Cassella,G, Solution of the many-electron Schrödinger equation with deep neural networks

Coury,MEA, Evolution of non-collinear magnetism in hot iron

Davies,P, Capacitors for energy storage

Edmunds,D, Coarse-grained molecular dynamics

Fraser,LM, Coulomb interactions and positron annihilation in many fermion systems: a Monte Carlo approach

Frensch,K, Ab initio studies of defect concentrations and diffusion in metal oxides

Gaudoin,R, Correlated many-electron wavefunctions for quantum Monte Carlo calculations of strongly inhomogeneous systems

Hine,NDM, New applications of quantum Monte Carlo

James,AJ, Solving the many electron problem with quantum Monte-Carlo methods

Khorashad,AS, Investigation of the exchange energy density functional

Le Page,J, The transfer of energy between electrons and ions in solids

Lim,AC, Ion channelling and electronic excitations in silicon

Lou,WT, Solution of the many-electron Schrödinger equation with deep neural networks

Malone,F, The density matrix and configuration interaction quantum Monte Carlo methods

Poole,T, Calculating derivatives with quantum Monte Carlo

Race,CP, The modelling of radiation damage in metals using Ehrenfest dynamics

Stedman,ML, Ground-state and finite-temperature quantum Monte Carlo simulations of many-particle systems

Sutterud,H, Solution of the many-electron Schrödinger equation with deep neural networks

Wells,T, Spin-lattice coupling in magnetic materials

Wood,B, Improving the accuracy of surface simulations

Yates,T, Effects of electronic temperature on the forces between atoms