## Periodic Reporting for period 3 - WFNQMC (Development of a Novel Computational Toolbox for Stochastic Electronic Structure in Chemistry and Condensed Matter)

Reporting period: 2021-02-01 to 2022-07-31

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. An understanding and computational description of these materials and their emergent properties may 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! 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.

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. 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 conceptually 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 amplitude associated with them.

The aim of this project is to make a leap in our ability to computationally model the nature of interacting electrons, and therefore our ability to predict the emergent properties of the system, without prior information from experiment, allowing us to guide and inform them.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 common framework. 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.

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! 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.

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. 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 conceptually 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 amplitude associated with them.

The aim of this project is to make a leap in our ability to computationally model the nature of interacting electrons, and therefore our ability to predict the emergent properties of the system, without prior information from experiment, allowing us to guide and inform them.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 common framework. 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.

The project aims to make progress in computational electronic structure theory, as applied in the domains of chemical and material science, as well as lattice models of condensed matter. Much of the work achieved has been designing frameworks for these approaches which can span these disciplines. Early work was spent devising the ‘Energy-weighted density matrix embedding’ theory – a framework which allows for the systematically improvable and rigorous embedding of a correlated electronic region inside a wider fragment. This has been applied equally to infinite systems and molecules, where strong correlation centers such as transition metal atoms can be embedded in wider chemical environments.

However, this ‘embedding’ approach is only one approach to multi-scale electronic structure which has been developed. We have also devised a computationally efficient approach to stochastic multireference perturbation theory. This can be considered an ‘energy-domain’ embedding, where part of the space is treated at a higher level, with perturbative coupling between these spaces. The method was applied to the outstanding electronic structure problems of the porphoryn subunit of haemoglobin, with these embedded active spaces able to be extended to unprecedented size and accuracy. In addition, we have also been devised a new approach for single-reference problems in quantum chemistry, by leveraging our experience gained in the compression of environments for quantum systems. In this approach we ask whether we can design fictitious environments, whose effect on systems is to modify it to mimic the effects of correlations. Our recasting of the problem in terms of a fictitious environment allows us to compress and renormalize this environment, allowing for an efficient and accurate new approach to treating electron correlations.

Finally, it has always been important for these developed methods to be accessible to the wider scientific community, in efficient and scalable implementations of these developed methods. To this end, we have also been working on a modern software package for our methods, which will facilitate their development into the future, and enable application on large scale supercomputers.The scope of these methods has also been advertised with marquee applications, including to solid atomic hydrogen for materials science, and inorganic metallic systems.

However, this ‘embedding’ approach is only one approach to multi-scale electronic structure which has been developed. We have also devised a computationally efficient approach to stochastic multireference perturbation theory. This can be considered an ‘energy-domain’ embedding, where part of the space is treated at a higher level, with perturbative coupling between these spaces. The method was applied to the outstanding electronic structure problems of the porphoryn subunit of haemoglobin, with these embedded active spaces able to be extended to unprecedented size and accuracy. In addition, we have also been devised a new approach for single-reference problems in quantum chemistry, by leveraging our experience gained in the compression of environments for quantum systems. In this approach we ask whether we can design fictitious environments, whose effect on systems is to modify it to mimic the effects of correlations. Our recasting of the problem in terms of a fictitious environment allows us to compress and renormalize this environment, allowing for an efficient and accurate new approach to treating electron correlations.

Finally, it has always been important for these developed methods to be accessible to the wider scientific community, in efficient and scalable implementations of these developed methods. To this end, we have also been working on a modern software package for our methods, which will facilitate their development into the future, and enable application on large scale supercomputers.The scope of these methods has also been advertised with marquee applications, including to solid atomic hydrogen for materials science, and inorganic metallic systems.

Continuing developments in a rigorous quantum embedding theory of correlations will continue, with an extension to mimic the effect of non-local interactions in a solid by a ficticious local polaron. Inclusion of these non-local interactions can allow drastic changes in the properties of a surface, and are key to understanding much of the catalytic features of materials. These long-range many-body interactions have been a constant challenge for computational methods, and a rigorous framework for treating these would be a profound step forward. The development of this framework will go hand-in-hand with progress in another work package, whereby the stochastic wave function approaches will be extended to treat coupled electron-boson systems, and will eventually be used as a ‘solver’ within the framework described above. Additionally, we are near completion on a rigorous stochastic approach to treating the Dirac equation, rather than the Schrödinger equation, allowing for all orders of relativistic effects to be included in an efficient Monte Carlo method. Combining these tools will ensure that they can be a major computational tool set for a wide array of quantum many-body problems.