Exact Conditions in Density Functional Theory and their application to Hubbard type Corrective Functionals

Abstract

The work of this thesis falls within the topic of Kohn Sham (KS) and Generalised Kohn Sham (GKS) theory, the practical implementation of which are the most popular tools for predicting the electronic properties of atoms, molecules, materials and nano-structures. The importance of these methods within the fields of physics, chemistry and material science is exemplified by the vast computational resources expended annually running such calculations on High Performance Computing (HPC) systems, approximately 30% of the total core hours, based on Irelands national HPC system. It is also worth noting that the most cited papers across the disciplines of physics and chemistry pertain to this very topic. Although a formally exact theory, practical KS and GKS calculations require the use of an approximate Exchange Correlation functional. Over the past sixty years an intense and unbroken effort has been made to both improve the underlying exchange correlation approximations, but also through the development of correction schemes to supplement standard approximations, such as van der Waals, self interaction and Hubbard type corrective functionals. Of particular interest is the development of improved methods for modeling specific types of materials where conventional exchange correlation approximations qualitatively fail, such as transition metal and lanthanide oxides. This qualitative failure of conventional exchange correlation functionals is best exemplified by their spurious prediction of many of the aforementioned materials as being metallic when they are known experimentally to be insulating. The reliable modeling of transition metal oxides is a matter of particularly pressing concern as they include many materials of technological and scientific interest, such as cathode materials for lithium ion batteries, heterogeneous catalysts for hydrogen production, high-temperature superconducting oxides and inorganic solid state electrolytes to name but a few. Hubbard type corrective functionals have shown much promise at correcting for the deficiencies of conventional exchange correlation functionals in the modeling of transition metal and lanthanide oxides. Traditionally these corrective functionals are derived by employing an electron-electron interaction term inspired by the Hubbard model, however this electron-electron interaction is already accounted for to a less favorable extent by the conventional exchange correlation functional which the Hubbard functional is designed to supplement. This necessitates the use of a double counting correction scheme. Over the decades several double counting schemes have been derived but crucially, computed material properties have been shown to depend strongly on the choice of double counting scheme. The aim of this thesis is to dispense with the need to invoke a double counting correction term and instead derive a new Hubbard type corrective functional inspired solely by exact conditions, i.e. known physical properties of the exact exchange correlation functional. It is with this in mind that, in chapters three and four, new exact conditions of the exchange correlation functional are derived, namely the convexity condition and tilted plane condition. The convexity condition is a long assumed, but until now, unproven exact condition of the exchange correlation functional, which states that the total energy of a finite electronic system should be convex with respect to electron count. Proving this exact condition is an important result not only from a purely academic perspective but also because it lifts a standing assumption in the original proof of the piecewise linearity condition with respect to electron count by Perdew et al. [1]. This piecewise linearity condition is itself widely used to motivate the mathematical form of many Hubbard type corrective functionals and will be invoked in the derivation of our new Hubbard functional in later chapters of the thesis. In passing, we note that the piecewise linearity condition is also employed to justify the use of the KS and GKS gap as a method of evaluating the fundamental bandgap of an electronic system. Thus, proving the convexity condition places the aforesaid on a firmer theoretical footing. In chapter four, another exact condition of the exchange correlation functional, namely the tilted plane condition is derived. This exact condition defines the shape of the total electronic energy surface with respect to total electron count N and total magnetization M, including non-integer values of N and M. This exact condition is a generalization of the well known flat plane condition, which defines the total electronic energy surface only for a limited range in values of M. Again, deriving this exact condition is interesting not only from a purely academic standpoint but also because a plethora of density functional approximations and corrective functionals have been designed based on the flat plane condition. Hence, the tilted plane condition could help stimulate the extension and generalization of many of these methods. In particular for our purposes, knowledge of this complete total energy surface will be employed in later chapters of the thesis to derive a new class of Hubbard-type functional. In chapters five and six, we do exactly that and develop a Hubbard-type functional named BLOR after the authors, and its many body generalisation mBLOR, based on our understanding of the flat and tilted plane conditions. To be more specific, these Hubbard type functionals are solely derived based on the differing behaviours between the exact and approximate exchange correlation functionals at non-integer values of subspace electron count and magnetization. We benchmark our newly derived Hubbard-type functional using stretched homo-nuclear s and p block molecules and find that our corrective functional yields considerable improvements in the total energy compared to standard Hubbard-type functionals. Finally, in chapter seven, we test the mBLOR functional on MnO, an archetypal transition metal oxide system where standard local and semi-local exchange correlation approximations benefit from being supplemented by a Hubbard type corrective functional. With the inclusion of an additional stabilisation term, the mBLOR functional yields an energy of cohesion for MnO in very close agreement with the Diffusion Monte Carlo reference value. Further testing of the mBLOR functional on transition metal oxide systems is needed. Nevertheless, this is a very promising preliminary result.