Epilogue

Abstract

This book is an extension of my volume on Quantal Density Functional Theory (QDFT) of electronic structure to the development of approximation methods and the application of the time-independent theory to realistic physical systems. Timeindependent Quantal Density Functional Theory (Q-DFT) of electronic structure is the mapping from an interacting system of electrons in their ground or excited state, and in the presence of an external field $\mathcal{F}^{{\rm ext}}(r)$ such that $\mathcal{F}^{{\rm ext}}(r) = -\bigtriangledown v(r)$, to one of model noninteracting fermions with equivalent density ρ(r). (There is also a Q-DFT where the mapping is to a model system of noninteracting bosons.) The corresponding energy and ionization potential (or electron affinity) of the electrons are also thereby obtained. The description of both the interacting and model quantum-mechanical systems is in terms of “classical” fields and their quantal sources. In a manner similar to that of classical physics, this perspective makes the subject tangible. In this book I have attempted to demonstrate many different facets of time-independent Q-DFT as follows. (1) The applicability of Q-DFT to determine the electronic structure of matter. To do so I have chosen as examples disparate nonuniform electron gas systems with different symmetries: ground and excited states of atoms and ions of the Periodic Table, the anion–positron complex, the Hydrogen molecule, the metal-vacuum interface. (2) The applicability of the Q-DFT of the Density Amplitude for which the mapping is to a model system of noninteracting bosons is demonstrated for atomic examples. (3) As a consequence of the fact that within Q-DFT, the contributions of the electron correlations due to the Pauli exclusion principle, Coulomb repulsion, and Correlation-Kinetic effects are explicitly defined, a key facet of Q-DFT demonstrated is the ability to study the electronic structure of a physical system systematically as a function of these different electron correlations. (4) As such approximation schemes within Q-DFT, both ad hoc variational–self-consistent and perturbative, are devised to account for these correlations in a systematic manner. (5) In Q-DFT, there is a rigorous physical interpretation of the equations governing the different electron correlations in terms of their quantal sources and fields. These quantal sources and fields are in turn defined in terms of the true interacting and model noninteracting fermion (or boson) wave functions. The physics then allows for the determination of the exact analytical structure of many properties, and the contribution of each of the different electron correlations to that property. Hence, as demonstrated, it is possible to analytically determine properties at and near the nucleus of atoms, and in the classically forbidden region of atoms and molecules, and the far vacuum region of the metal–vacuum interface. (6) As noted in the Introduction, Q-DFT generalizes the concept of local effective potential energy theory beyond that originally understood on the basis of Slater theory, the Optimized Potentialmethod, and Kohn–Sham density functional theory. This generalization of local effective potential energy theory too is demonstrated. (7) Finally, I have included recent new understandings of the fundamental theorem of time-independent Hohenberg–Kohn density functional theory, and of its extension to the time-dependent case due to Runge-Gross.