## Periodic Reporting for period 1 - MAGSPEC (Spectra of Molecules in Strong Magnetic Fields)

Période du rapport: 2018-02-15 au 2020-02-14

Strong magnetic fields cause exotic behaviour in the electrons of a molecule. For experiments, we are technically limited by the strength of the magnetic field we can generate, so we can only explore the domain of very strong fields theoretically. An understanding of strong field effects and development of theoretical machinery capable of handling them also feeds back into our understanding and computational accuracy of weak field behaviour which is vital not only in the scientific community and for technological advances but also in real world applications. This project, in particular, has focussed on magnetic field effects on electronic spectra of molecules where not only peak positions and intensities can be modulated but also non-uniform fields can switch on and off spin-forbidden transitions. We have developed and benchmarked a computational machinery capable of computing molecular electronic spectra in a wide range of magnetic fields. In addition, a very popular group of theoretical methods called Density Functional Theory (DFT) have been benchmarked for exotic magnetic properties which can guide the scientific community on what approximations to adopt when computing magnetic properties with DFT.

Theory and code development was undertaken towards the fulfilment of this objective. The Random Phase Approximation theory was chosen as a suitable balance of cost and accuracy and it was suitably rederived to allow complex orbitals. An implementation was made in the program package LONDON to compute peak positions and intensities. Applications were undertaken for benchmarking as well as exploring the physics of excited states in strong magnetic fields. The energies of the molecular electronic states were followed across a wide range of field strengths as well as field non-uniformities. Ground states of molecules change from the closed shell singlet to states of higher spin multiplicity when field strength is increased. Excited states are found to be more sensitive to changes in magnetic fields. Polar molecules like LiH show larger responses at weaker fields. The demonstration of switching on spin-forbidden electronic transitions was considered novel and it was possible to identify classes of excitations which are most sensitive to this spin-symmetry breaking. Computation of spectral intensities in both length and velocity gauge revealed their equivalence in the basis set limit for RPA even for complex orbitals - which was hitherto expected but not proved. This work resulted in a publication.

The relative contributions of orbital and spin effects to anapole susceptibilities were assessed both theoretically and computationally. The computations were carried out under the self-consistent field (SCF) Hartree-Fock (HF) approximation. A non-relativistic general Hartree-Fock code involving two-component orbitals was developed to enable computations in non-uniform magnetic fields which enabled assessment of spin contributions to anapole susceptibility for the first time. Spin effects on the anapole susceptibility are found to be large and have a consistent direction - they lower the second order energy for all molecules that are closed shell singlets in the absence of a field. A perturbative argument was put forward to justify trends observed in our computations. This work resulted in a publication which was chosen as Editor's Pick.

The role of electron correlation in the values of the anapole susceptibilities was studied using both wave function (HF, MP2 and coupled cluster) and density functional (DFT) approximations (LDA, KT3, TPSS, M06). A test set of 36 small molecules of varied electronic structure was employed and the susceptibility tensors computed with all the listed methods. The accuracy of the methods in computing anapole susceptibilities were assessed against the CC values. This was deemed especially important for the density functional approximations as DFT itself is not rigorous for molecules in magnetic fields. The study also prompted a proposal for generalising the definitions of diamagnetic and paramagnetic molecules. The manuscript for this work has been submitted.

In a separate study, a new ingredient for designing exchange-correlation functionals for use of DFT in magnetic fields was proposed. Instead of kinetic energy density as a scalar, a tensor , Q, was suggested which contains the usual kinetic energy density as its trace and vorticity on the off-diagonals. The natural inclusion of vorticity into the tensor form of the kinetic energy density such that functionals with Q or quantities derived from Q as ingredients would smoothly transition between computations with a field to one without suggests that the minimum level of complexity for functionals that can handle computations in magnetic fields is where the kinetic energy density is considered, ie. the so-called meta-GGA functionals. This finding fits well with our observation that TPSS, a meta-GGA functional, works very well for computing both magnetic properties as well as PES in magnetic fields. Quantities analogous to the electron localisation function (ELF) can also be derived from Q which are necessary to for designing functionals with suitable short and long rage behaviour. This work was published and selected as Editor’s Choice 2018 (72 articles out of all published in 2018), Featured article and also selected for publication as popular science article in SciLight (AIP).

The researcher participated in 5 international meetings and was an invited speaker in 2 of them. In 2 others she presented posters.

The relative contributions of orbital and spin effects to anapole susceptibilities were assessed both theoretically and computationally. The computations were carried out under the self-consistent field (SCF) Hartree-Fock (HF) approximation. A non-relativistic general Hartree-Fock code involving two-component orbitals was developed to enable computations in non-uniform magnetic fields which enabled assessment of spin contributions to anapole susceptibility for the first time. Spin effects on the anapole susceptibility are found to be large and have a consistent direction - they lower the second order energy for all molecules that are closed shell singlets in the absence of a field. A perturbative argument was put forward to justify trends observed in our computations. This work resulted in a publication which was chosen as Editor's Pick.

The role of electron correlation in the values of the anapole susceptibilities was studied using both wave function (HF, MP2 and coupled cluster) and density functional (DFT) approximations (LDA, KT3, TPSS, M06). A test set of 36 small molecules of varied electronic structure was employed and the susceptibility tensors computed with all the listed methods. The accuracy of the methods in computing anapole susceptibilities were assessed against the CC values. This was deemed especially important for the density functional approximations as DFT itself is not rigorous for molecules in magnetic fields. The study also prompted a proposal for generalising the definitions of diamagnetic and paramagnetic molecules. The manuscript for this work has been submitted.

In a separate study, a new ingredient for designing exchange-correlation functionals for use of DFT in magnetic fields was proposed. Instead of kinetic energy density as a scalar, a tensor , Q, was suggested which contains the usual kinetic energy density as its trace and vorticity on the off-diagonals. The natural inclusion of vorticity into the tensor form of the kinetic energy density such that functionals with Q or quantities derived from Q as ingredients would smoothly transition between computations with a field to one without suggests that the minimum level of complexity for functionals that can handle computations in magnetic fields is where the kinetic energy density is considered, ie. the so-called meta-GGA functionals. This finding fits well with our observation that TPSS, a meta-GGA functional, works very well for computing both magnetic properties as well as PES in magnetic fields. Quantities analogous to the electron localisation function (ELF) can also be derived from Q which are necessary to for designing functionals with suitable short and long rage behaviour. This work was published and selected as Editor’s Choice 2018 (72 articles out of all published in 2018), Featured article and also selected for publication as popular science article in SciLight (AIP).

The researcher participated in 5 international meetings and was an invited speaker in 2 of them. In 2 others she presented posters.

This project uncovered several significant scientific facts and made three distinct developments. Firstly, a low-scaling method called RPA was developed for studying molecular electronic spectra in strong magnetic fields for the first time and applications that were undertaken show promise. Secondly, the first study of spin-effects on electronic anapole susceptibilities was carried out and the balance of spin and orbital effects was explored. Thirdly, a possible new ingredient for developing DFT functionals for studying phenomena in magnetic fields was proposed and shown to have many desirable properties. Further development and application of these novel first steps is highly desirable and would help to address some of the shortcomings of the current standard practices in computational chemistry.