## Final Report Summary - ABACUS (Ab-initio adiabatic-connection curves for density-functional analysis and construction.)

Nearly all quantum-mechanical studies are today based on density-functional theory (DFT), where the electronic energy and other properties of molecules are calculated from the electron density of the system studied rather than from the much more complicated wave function. As a result, DFT calculations are highly efficient and applicable to large molecules. However, although the energy of a molecule is known to be a function of its density (the assumption underlying DFT), the explicit form of this "density functional" is unknown. We are therefore forced to work with approximate functionals, often with limited applicability. The central problem in DFT is therefore to generate improved approximate density functionals and to extend the applicability of DFT to broader classes of system. Moreover, despite the enormous practical utility of DFT, its mathematical foundations have often been questioned. In the ABACUS project, we have both extended the applicability of DFT to new systems and addressed a number of important problems that plague the standard formulation of DFT. The tool for our work has been convex analysis; a branch of mathematics that provides an ideal framework for DFT but which so far has been little used and is not sufficiently appreciated.

We begin by first considering some of the theoretical results on DFT obtained in the ABACUS project. It is a known fact that the exact density functional is everywhere discontinuous, putting a question mark over all attempts at modelling this functional by continuous functions (as is invariably done). Using the technique of Moreau–Yosida regularisation, we have demonstrated that the discontinuous exact density functional can be modelled, to any desired accuracy, by differentiable (and therefore) continuous functions. Another long-standing question in DFT concerns the possible existence of "false solutions"- that is, unphysical densities that do not represent a real system, even if obtained with the (unknown) exact functional. We have solved this problem, demonstrating that each ground-state density in DFT represents a real physical system. The convex treatment of DFT developed in the ABACUS project is developed in the monograph "Principles of Density-Functional Theory", to appear on Wiley in 2017.

The strength and versatility of the convex framework has been further demonstrated by considering current-density-functional theory (CDFT), a generalization of DFT to molecules in magnetic fields. We have given a simple, convex formulation of CDFT (which in the previous formulations appears significantly more complicated than DFT itself) and clarified a number of longstanding questions regarding the theory.

The difficulty in developing CDFT into a practical tool in chemistry is twofold: not only is little known about the behaviour of the density functional in magnetic fields - little is in fact also known about the true behaviour of molecules in magnetic fields. To develop CDFT in magnetic fields, it was therefore also necessary to develop highly accurate wave-function methods for molecules in magnetic fields; being systematically improvable towards the exact solution, these methods can be used to benchmark CDFT in magnetic fields. Surprisingly, calculations using these accurate methods revealed the existence of a new chemical bonding mechanism, paramagnetic bonding, in strong magnetic fields. In a sufficiently strong magnetic field, anti-bonding orbitals are stabilized by the induced currents, giving rise to zero-bond-order bonds. As a result, H2 in its dissociative lowest triplet state become bonded and rare-gas molecules such as He2 and Ne2 exist. Paramagnetic bonding occurs under conditions that do not exist on the Earth but are encountered on certain stellar objects such as white dwarf stars. Moreover, in crystal impurities, magnetic effects are greatly enhanced, opening up the possibility of observing and manipulating paramagnetic bonding under laboratory conditions.

Having developed wave-function methods for accurate calculations of molecules in magnetic fields, we implemented and benchmarked CDFT in magnetic fields. Our first studies, using a functional (VRG) developed based on the behaviour of the uniform electron gas in a magnetic field, were discouraging, giving a poor description of the molecules in magnetic fields. Subsequently, adapting meta-GGA density-functionals to magnetic fields, a better behaviour was observed, with a good prediction of bond distances and dissociation energies of rare-gas dimers in magnetic fields. We have thus established a good framework for molecular calculations in magnetic fields. Given the low cost of CDFT calculations relative to high-level many-body methods, we are now in position to study large molecules in magnetic fields.

We begin by first considering some of the theoretical results on DFT obtained in the ABACUS project. It is a known fact that the exact density functional is everywhere discontinuous, putting a question mark over all attempts at modelling this functional by continuous functions (as is invariably done). Using the technique of Moreau–Yosida regularisation, we have demonstrated that the discontinuous exact density functional can be modelled, to any desired accuracy, by differentiable (and therefore) continuous functions. Another long-standing question in DFT concerns the possible existence of "false solutions"- that is, unphysical densities that do not represent a real system, even if obtained with the (unknown) exact functional. We have solved this problem, demonstrating that each ground-state density in DFT represents a real physical system. The convex treatment of DFT developed in the ABACUS project is developed in the monograph "Principles of Density-Functional Theory", to appear on Wiley in 2017.

The strength and versatility of the convex framework has been further demonstrated by considering current-density-functional theory (CDFT), a generalization of DFT to molecules in magnetic fields. We have given a simple, convex formulation of CDFT (which in the previous formulations appears significantly more complicated than DFT itself) and clarified a number of longstanding questions regarding the theory.

The difficulty in developing CDFT into a practical tool in chemistry is twofold: not only is little known about the behaviour of the density functional in magnetic fields - little is in fact also known about the true behaviour of molecules in magnetic fields. To develop CDFT in magnetic fields, it was therefore also necessary to develop highly accurate wave-function methods for molecules in magnetic fields; being systematically improvable towards the exact solution, these methods can be used to benchmark CDFT in magnetic fields. Surprisingly, calculations using these accurate methods revealed the existence of a new chemical bonding mechanism, paramagnetic bonding, in strong magnetic fields. In a sufficiently strong magnetic field, anti-bonding orbitals are stabilized by the induced currents, giving rise to zero-bond-order bonds. As a result, H2 in its dissociative lowest triplet state become bonded and rare-gas molecules such as He2 and Ne2 exist. Paramagnetic bonding occurs under conditions that do not exist on the Earth but are encountered on certain stellar objects such as white dwarf stars. Moreover, in crystal impurities, magnetic effects are greatly enhanced, opening up the possibility of observing and manipulating paramagnetic bonding under laboratory conditions.

Having developed wave-function methods for accurate calculations of molecules in magnetic fields, we implemented and benchmarked CDFT in magnetic fields. Our first studies, using a functional (VRG) developed based on the behaviour of the uniform electron gas in a magnetic field, were discouraging, giving a poor description of the molecules in magnetic fields. Subsequently, adapting meta-GGA density-functionals to magnetic fields, a better behaviour was observed, with a good prediction of bond distances and dissociation energies of rare-gas dimers in magnetic fields. We have thus established a good framework for molecular calculations in magnetic fields. Given the low cost of CDFT calculations relative to high-level many-body methods, we are now in position to study large molecules in magnetic fields.