Extending density-functional theory to molecules in magnetic fields - Furness, J.W. (2017)
The density-functional theory (DFT) of Hohenberg and Kohn [1] has become the Swiss army knife of the quantum chemist, able to tackle nearly all electronic structure problems with impressive accuracy and efficiency, and it is common to see DFT calculations reported alongside experimental observations in the wider chemical literature. Some systems remain problematic for DFT however and, as DFT was constructed to simulate the systems commonly encountered in the chemistry lab, magnetic fields and the electron currents they induce are completely absent from the theory. Instead, DFT must be generalised for use in a magnetic field by including these currents directly, becoming current-density-functional theory (CDFT) [2]. By making this generalisation it is hoped that the practical utility of standard DFT can be leveraged to investigate molecules in both weak and strong magnetic fields. Whilst the study of molecular systems in strong magnetic fields is clearly of astrochemical and academic interest [3–7], it is reasonable to ask what place such calculations have with regards to laboratory problems, and what the eventual goal of CDFT is. To answer this one should look, for example, to the laboratory field semi-conductor systems that have been shown to be good analogues of molecules in strong fields [8,9]. Empirical studies of such systems show that a proper description of the magnetic effects is likely to be essential for their accurate simulation and, with the large size of these systems, the low cost correlated description of CDFT is likely to be instrumental in understanding their electronic structure. The theoretical framework for CDFT has long been established, only very recently however has a computer code, London, been available to actually perform self-consistent molecular CDFT calculations in strong magnetic fields [10, 11]. As a result of this absence, the functionals and approximations that comprise CDFT remain relatively untested. This thesis aims to utilise the London code to address this absence, first establishing the accuracy of existing CDFT functionals before developing the ideas to investigate the effect of a strong magnetic field on various aspects of electronic structure. The thesis begins by establishing the necessary theoretical foundations of DFT in Chapter 1 and its extension to CDFT in Chapter 2. The accuracy of a number of CDFT functionals is tested in Chapter 3 for predicting both the weak field magnetic phenomena encountered on earth, and the more exotic physics found in strong fields. The course of this study reveals a modified implementation of the meta-generalised gradient approximation (mGGA) class of functionals to be particularly well suited to strong field CDFT, providing a promising route to high accuracy descriptions of strong field phenomena. Using the modified mGGA functionals, Chapter 4 makes use of the interpretive powers of CDFT and the Kohn–Sham orbitals in order to develop a qualitative understanding of strong field electronic structure, and the origins of a new bonding mechanism, termed perpendicular paramagnetic bonding. This exploration of strong field phenomena provides a rationalisation for the the successes of the mGGA functionals, highlighting the importance of self-interaction error in strong magnetic fields. The presence of a strong magnetic field has a complicated effect on the excited electronic states, with important consequences for molecular structure in a field. The maximum overlap method (MOM) of Gilbert et al. [12] is employed in Chapter 5 in the context of strong field Hartree–Fock theory to facilitate the study of excited states in magnetic fields. The MOM method is shown to give an excellent agreement with high accuracy calculations and paves the way for a MOM-CDFT program capable of giving an accurate, correlated description of excited states in a field. The rest of the thesis moves away from directly dealing with the complications of a magnetic field, and instead considers the optimised effective potential (OEP) method in DFT as a means to fully orbital dependent functionals. This investigation is made with an eye to the future extension of CDFT to incorporate the OEP method, enabling the use of orbital dependent functionals in the CDFT description of molecules in magnetic fields. Such functionals are expected to be an effective route to higher accuracy calculations, as the currents induced by the magnetic field are inherently orbital dependent quantities themselves. The core framework of magnetic field free OEP is explored in Chapter 6 and a previously unrecognised connection between the recently proposed variational principle of Gidopolous [13] and the Lieb functional [14] is established. The optimised effective potential problem is notoriously ill behaved in a finite basis representation and the problem of regularising its equations to give physical potentials remains unsolved, despite considerable effort. Previous regularisation schemes are severely limited by their requirement for a user defined regularisation strength parameter, with no a priori guide towards the correct choice. This requirement prevents the OEP method from being used as a black box method and restricts the basis set combinations that can be considered. Chapter 7 addresses the absence of a satisfactory regularisation solution by first thoroughly studying the general ill-posed problem and the classical techniques employed in its regularisation. From this base, a fully automatic regularisation scheme is constructed to automatically tune regularisation strength from measures that are internal to the OEP. In doing so, an optimal regularisation is applied throughout the calculation, removing the need for a manually set parameter. The automatic scheme is then augmented with information from the potential basis set and shown to produce a physically meaningful potential from a single OEP calculation, regardless of the basis sets chosen. Finally, Chapter 8 summarises the work and discusses future directions for the continued development of the field.