## Periodic Reporting for period 1 - WFNQMC (Development of a Novel Computational Toolbox for Stochastic Electronic Structure in Chemistry and Condensed Matter)

Reporting period: 2018-02-01 to 2019-07-31

The interactions between electrons and nuclei provide the glue that binds together all materials and chemicals. The specifics of these interactions give rise to the vast array of different properties around us, ranging from metallic behaviour, to catalytic biomolecules or magnetic interactions. However, while it is now common to design large-scale engineering projects from buildings to planes with accurate computational simulation, it is curious that the techniques for microscopic simulations of the interactions of these most fundamental particles is still very much lacking.

As an often used and particularly impressive example, the property of ‘high-temperature superconductivity allows for the amazing ability to levitate a ceramic material above a magnet when sufficiently cooled. It is the coordinated collective motion of the electrons inside the bulk of the material which allow this property to emerge, and despite decades of research, a consensus among scientists about the mechanism behind this emergent behaviour is still lacking. Would an understanding and computational description of these materials and this emergent property enable us to design materials with improved properties, or even for example unlock the mechanisms behind some of the most important biological enzymes that enable life?

The main difficulty in solving these problems comes from the fact that we have many particles (generally electrons), all repelling each other. Therefore, precisely determining the motion of any one particle will depend simultaneously on all other particles. This is what we mean when we say that the motions of all the particles are 'correlated'. The first step is generally to approximate the motion of all other particles by some averaged distribution of the particles you want to move through, but this isn't generally sufficiently accurate. For instance, imagine trying to navigate through a crowd, knowing only what the average position of all the other people in the crowd - you'd end up bumping into a lot of people, and it would certainly not be an optimal route. This problem is a very general problem in many situations, known as the 'many-body' problem, and crops up when trying to model the motion of people through crowds, the optimal delivery route of your online shopping delivery van, or indeed the motion of electrons through a material. Unfortunately, the exact solution to these problems requires exponentially increasing computational effort as you increase the number of interacting particles, and so to see large-scale behaviour such as the emergence of superconductivity requires far too many particles for even the largest supercomputer to even approach solving.

While this is a huge problem in itself, when working with electrons we also have to work within the strange world of quantum mechanics. Unlike the examples of the delivery van or person in a crowd, electrons cannot be thought of solely as particles, but also have a wave-like character which has to be considered, giving rise to the electron distributions. This means that if the path of two electrons is the same, they can not only reinforce each other, but also potentially cancel each other out. Strange effects like swapping the position of two electrons giving rise to a negative distribution of the electrons, which has no analog in traditional distributions, still occupies philosophers and physicists alike in trying to conceptially rationalise this behaviour, but still must be taken into account in our calculations to observe the correct behaviour. The exact solution to the quantum many-body problem therefore does not give at the end a single optimal path for the electrons which needs to be considered (as there is for the delivery van), but instead an infinite number of paths, all of which have a positive or negative probability amplitude associated with them. Once we have these, then in principle, all properties of the system can be extracted.

The aim of this project is to make a leap in our ability to computationally and explicitly model the nature of the electrons in a system, and therefore being able to predict the emergent properties of the system, without prior information from experimental observations. This would allow a way to inform and guide experimental details, especially important in the domain of strongly correlated electrons. The other overarching aim is to bring together the domains of condensed matter theory (dealing with extended, bulk systems), and quantum chemistry (dealing with the electronic interactions in molecules) under a framework which can treat the electronic effects on both these types of systems. To do this, we will develop a toolbox of methods, exploiting stochastic (random) sampling of this wavefunction, as well as its combination with other tools at our disposal, including perturbation theory, local embedding, and entanglement structure, in order to advance the algorithms and computational methods which we can use to understand the materials and molecules of fundamental importance in the world around us all.

As an often used and particularly impressive example, the property of ‘high-temperature superconductivity allows for the amazing ability to levitate a ceramic material above a magnet when sufficiently cooled. It is the coordinated collective motion of the electrons inside the bulk of the material which allow this property to emerge, and despite decades of research, a consensus among scientists about the mechanism behind this emergent behaviour is still lacking. Would an understanding and computational description of these materials and this emergent property enable us to design materials with improved properties, or even for example unlock the mechanisms behind some of the most important biological enzymes that enable life?

The main difficulty in solving these problems comes from the fact that we have many particles (generally electrons), all repelling each other. Therefore, precisely determining the motion of any one particle will depend simultaneously on all other particles. This is what we mean when we say that the motions of all the particles are 'correlated'. The first step is generally to approximate the motion of all other particles by some averaged distribution of the particles you want to move through, but this isn't generally sufficiently accurate. For instance, imagine trying to navigate through a crowd, knowing only what the average position of all the other people in the crowd - you'd end up bumping into a lot of people, and it would certainly not be an optimal route. This problem is a very general problem in many situations, known as the 'many-body' problem, and crops up when trying to model the motion of people through crowds, the optimal delivery route of your online shopping delivery van, or indeed the motion of electrons through a material. Unfortunately, the exact solution to these problems requires exponentially increasing computational effort as you increase the number of interacting particles, and so to see large-scale behaviour such as the emergence of superconductivity requires far too many particles for even the largest supercomputer to even approach solving.

While this is a huge problem in itself, when working with electrons we also have to work within the strange world of quantum mechanics. Unlike the examples of the delivery van or person in a crowd, electrons cannot be thought of solely as particles, but also have a wave-like character which has to be considered, giving rise to the electron distributions. This means that if the path of two electrons is the same, they can not only reinforce each other, but also potentially cancel each other out. Strange effects like swapping the position of two electrons giving rise to a negative distribution of the electrons, which has no analog in traditional distributions, still occupies philosophers and physicists alike in trying to conceptially rationalise this behaviour, but still must be taken into account in our calculations to observe the correct behaviour. The exact solution to the quantum many-body problem therefore does not give at the end a single optimal path for the electrons which needs to be considered (as there is for the delivery van), but instead an infinite number of paths, all of which have a positive or negative probability amplitude associated with them. Once we have these, then in principle, all properties of the system can be extracted.

The aim of this project is to make a leap in our ability to computationally and explicitly model the nature of the electrons in a system, and therefore being able to predict the emergent properties of the system, without prior information from experimental observations. This would allow a way to inform and guide experimental details, especially important in the domain of strongly correlated electrons. The other overarching aim is to bring together the domains of condensed matter theory (dealing with extended, bulk systems), and quantum chemistry (dealing with the electronic interactions in molecules) under a framework which can treat the electronic effects on both these types of systems. To do this, we will develop a toolbox of methods, exploiting stochastic (random) sampling of this wavefunction, as well as its combination with other tools at our disposal, including perturbation theory, local embedding, and entanglement structure, in order to advance the algorithms and computational methods which we can use to understand the materials and molecules of fundamental importance in the world around us all.

Within the current (1st) reporting period, we have spent most of the time focussing on the development of an overarching framework to bring together the fields of quantum chemistry and condensed matter physics. To this end, we have developed the ‘Energy-weighted density matrix embedding’ theory – a framework which allows for the embedding of a correlated electronic region inside a wider fragment. This has been applied equally to infinite systems, where the correlated fragment consists of a small part of the system, and molecules, where strong correlation centers such as transition metal atoms can be embedded in wider chemical environments. This work has been presented in three papers so far, one to Physical Review B, Phys. Rev. B 98, 235132 (2018) (aimed at the condensed matter physics community), one in the Journal of Chemical Physics, J. Chem. Phys. 151, 014115 (2019) (aimed at the quantum chemistry community), and one which has been submitted to Physical Review X (under review, available on the ArXiv:1909.07713).

Other tools developed in the project can then be considered as approaches to ‘solve’ the problem of this embedded fragment, for which we are turning to the stochastic wave function toolbox that we are developing. Extensions to the scope of this approach will allow for the embedding of larger fragments, their coupling to perturbative corrections, as well as the treatment of intrinisic relativistic effects, which we are also working on. Research on the ability to solve stochastic eigenvalue equations was written up as another publication in Physical Review B (Phys. Rev. B 98, 085118 (2018)).

Furthermore, the interest generated by the research has led to 12 invited international presentations, including talks at major conferences in Hong Kong, Europe and the US, allowing this research to be disseminated to a wide, international audience.

Other tools developed in the project can then be considered as approaches to ‘solve’ the problem of this embedded fragment, for which we are turning to the stochastic wave function toolbox that we are developing. Extensions to the scope of this approach will allow for the embedding of larger fragments, their coupling to perturbative corrections, as well as the treatment of intrinisic relativistic effects, which we are also working on. Research on the ability to solve stochastic eigenvalue equations was written up as another publication in Physical Review B (Phys. Rev. B 98, 085118 (2018)).

Furthermore, the interest generated by the research has led to 12 invited international presentations, including talks at major conferences in Hong Kong, Europe and the US, allowing this research to be disseminated to a wide, international audience.

A further PDRA and PhD student have been recruited to start in Oct ’19, with the aim of pushing the research to the next phase. This will include coupling this embedding framework to perturbative, dynamical mean-field theories, as well as the stochastic perturbation theory and novel wave function optimization. This is expected to hugely increase the scope and impact of the tools we are developing, and make them a major computational technique for a wide array of quantum many-body problems.