Molecular spin frustration

Electrons have an intrinsic property known as spin. A standard assumption in quantum chemical modelling is that there is a common spin axis that each electron spin is either parallel or anti-parallel to. On an abstract level, this means that the system has a particular spin symmetry. When the system is perturbed, e.g. by a laser pulse, it might jump to a different state. Only transitions to states that precisely match the symmetry of the original state and the symmetry of the perturbation are spin-symmetry allowed. This standard assumption is accurate in many situations. It breaks down, however, for molecules containing so heavy atoms that relativistic effects become important and for molecules in non-uniform magnetic fields. The system is spin frustrated in the sense that it would require energy for the spins to align to a common axis. Instead, spins align to local axes to reach a compromise between magnetic and non-magnetic forces. For molecular systems, the effect of magnetic fields on spin symmetry breaking were largely unexplored. This project has developed computational methods that completely do away with the standard limitations that prevent a description of molecules in magnetic fields. The methods have been used to explore how spin-symmetry broken ground states, i.e. the stable states with the lowest energy, can be tuned by magnetic fields. Unexpectedly, we found that the response from local spin density and other degrees of freedom is independent in the sense that the energy contributions are additive. We were able to show that this independence between the response in spin- and other degrees of freedom is a general feature as long as two conditions are satisfied: the applied, non-uniform magnetic field is weak and the ground state is close to a singlet state (where all electrons come pairs with opposite spins). Also computational methods that allow us to study transitions to excited states have been implemented. These methods rely on first computing the ground state and then describing the changes needed to go from the ground to an excited state. Using the new methods, we are able to study how the energy spectrum of molecules are affected by magnetic fields. For example, the ground state in the absence of magnetic fields can be "pushed up" in the spectrum by strong applied magnetic field, and another state "pulled down" and become the ground state instead. In this way, the ground state can discontinuously change character as a function of the applied field. We are also able to study the intensity of transitions between the ground state and excited states, and how they vary with the applied field. In particular, some transitions are forbidden due to incompatible spin symmetries in the ground and excited state. A non-uniform field breaks the spin symmetry and enables its intensity to become non-zero. This phenomenon has been studied in several molecules. We have also used the computational techniques to address more fundamental questions about a computational framework called density-functional theory (DFT), where it is exploited that all ground-state properties of a system are in principle determined by the electron density, spin density and current density. We have compared more accurate models to DFT models. Additionally, some mathematical results related to the foundations of density-functional models for magnetic fields have also emerged form the work within the project: (1) A relationship between noncollinear spin densities and the kinetic energy density. (2) A new way to incorporate magnetic-field effects in DFT. (3) A new way to incorporate the orbital- and spin-effects via the physical current density in DFT. (4) A careful mathematical analysis of the algorithm used to determine the ground state in DFT.

Outcomes: * I have established myself as an independent researchers with several publications in this capacity. * I have research results that further research and future funding applications can build on. * The person I hired as a postdoc on the project has advanced her career with publications in a new (to her) field, and she successfully applied for a Marie Curie grant under my supervision. Impacts: * Developed and implemented computational methods for the study of noncollinear spin in molecules. These methods are the first of their kind. * Deeper understanding of the relationship between noncollinear spin density and kinetic energy density. This is likely to lead to improved practical density-functional approximations in the future. * Deeper understanding of density-functional theory for molecules in strong magnetic fields. * Deeper understanding of the role of the current density and noncollinear spin density in density-functional theory.

An external non-uniform magnetic field is capable of breaking both spin-symmetry and time reversal symmetry. Alternatively, spin-orbit coupling in heavy elements combined with an external uniform magnetic field is also capable of the same symmetry-breaking. There will then no longer be a single, global spin quantization axis, and the system is said to be spin frustrated. Another consequence is that transitions that would otherwise have been symmetry-forbidden may now be allowed, i.e. a non-zero transition moment may be induced. This opens up for modulation of transitions to excited states using magnetic fields. The primary objective is to use ab initio calculations to study the magnetic field-modulation of ground state spin frustration and transitions to excited states. To achieve this, new computational methods will be implemented: * For molecules without heavy atoms, a General Hartree-Fock (HF) and DFT method for non-uniform magnetic fields will be develop. Important technical point are the direct, non-perturbative inclusion of magnetic fields and the use of London atomic orbitals (LAOs). * For molecules with heavy atoms, a 4-component relativistic HF and DFT method will be developed. Important technical points are again non-perturbative treatment of magnetic fields and LAOs as well as the use of Restricted Magnetic Balance. * A 2-component relativistic method (X2C) will also be developed to enable studies on larger molecules. * Both 2- and 4-component linear response/RPA methods (for non-magnetic properties) will be developed and used to study magnetic field effects on transition moments. Magnetic field-variation of several quantities related to spin frustration will be studied. In particular, the total S^2 of the ground state provides a fairly direct measure of spin-mixing. Also, anapole moments will be studied, since spin contributions have previously only been addressed using model Hamiltonians.