Epilogue

Abstract

The three major pillars of the book are summarized. These are: (a) Schrödinger theory of electronic structure from the perspective of the individual electron. This is a new description of the theory in terms of the ‘Quantal Newtonian’ second and first laws. These laws describe the various external and internal ‘classical’ fields experienced by each electron, the sources of the fields being quantum-mechanical expectations of Hermitian operators taken with respect to the wave function. The new perspective elucidates the intrinsic self-consistent nature of the Schrödinger equation, and is explicated by examples. (b) Hohenberg-Kohn (HK), Runge-Gross (RG), and Kohn-Sham (KS) density functional theory are described. The emphasis here is on the first theorem of HK which establishes the concept of a basic variable of quantum-mechanics: gauge invariant properties, knowledge of which uniquely determines the external potentials, and thereby the wave functions of the system. There are further understandings of the respective first theorems of HK and RG arrived at via density preserving unitary transformations and corollaries. The theorems of HK are generalized to the added presence of an external uniform magnetostatic field. (The proof of the theorems is rigorous in the HK sense of the relationship between the basic variables and external potentials, but differs from the original proof of HK.) (c) Quantal-density functional theory (Q–DFT) is the principal component of the book. It is the description of the mapping from the interacting system as defined by Schrödinger theory to one of noninteracting fermions or bosons possessing the same basic variables. This mapping is in physical terms of ‘classical’ fields and quantal sources via the interacting and corresponding model system ‘Quantal Newtonian’ second and first laws. The theory is generalized to the presence of an external magnetostatic field. Examples of the application of the theory are given. Q–DFT provides the rigorous physical interpretation of the mathematical constructs of KS and other local effective potential theories, and approximations to them. It thereby provides further physical insights into such theories.