Publisher Summary This chapter focuses on the recent advances of quantum theoretical methods and applications. There are theoretical applications to photophysical and photochemical processes where the calculations indicate the importance of relativistic effects on the photodissociations of heavy atomic molecules. The chapter discusses variations on the Hohenberg–Kohn construction of in principle exact density functionals and the foundations and physical implications of density functional theory. It also describes important application to diagnostics in oncology and in medical developments related to hadron radiotherapy in medicine (Belkic). The full treatment of inelastic collisions between bare nuclei and hydrogen-like atoms is of fundamental importance to particle transport physics in general and heavy ions in medicine in particular. The chapter also focuses on the use of information theoretic concepts to deal with several classical issues in electronic structure such as single versus multiple bonds, hybridization, and the like.