Developing Thermal Density Functional Theory Using the Asymmetric Hubbard Dimer

In this dissertation, I introduce both ground-state and thermal density functional theory. Throughout I use the asymmetric two-site Hubbard model, called the Hubbard dimer for short, to better understand and/or develop these theories. This model is used because it can be solved analytically and it contains all the necessary physics while still being conceptually simple enough to tease apart the various aspects of density functional theory. Ground-statedensity functional theory has seen broad use in many disciplines including physics, chemistry, geology, and material science and has led to a number of important physical and technological successes. In the first two chapters I elucidate the behavior of the ground-state theory using the Hubbard dimer. The simplicity of the model allows me to showcase aspects of the theory that are common points of confusion within the electronic structure community, e.g. the fundamental gap problem. The next two chapters focus on thermal density functional theory which has been coming to prominence as the study of warm dense matter has becomea growing interest at the national laboratories and in the astronomical body community. The Hubbard dimer allows me to do the first ever exact thermal density functional theory calculation. In this work I am better able to understand the approximations used in thermal density functional theory and can point to why they succeed and fail. This also allows me to illustrate old conditions and derive new ones. I conclude with an overview of the work and a few different directions in which the asymmetric Hubbard dimer could be used further