| This thesis is concerned with the first Hohenberg-Kohn (HK) theorem, its extension to the time-dependent case by Runge-Gross (RG), and the application and further extension of Quantal Density Functional theory (Q-DFT). According to the HK theorem, the ground state density rho(r) of a system uniquely determines its Hamiltonian H to within an additive constant C, and thus the physical system and its properties. In the RG extension, the density rho(rt) uniquely determines the time-dependent Hamiltonian H(t) to within a purely time-dependent function C(t), and hence the system and its properties. We prove, by construction, the corollary that degenerate {lcub}time-independent/time-dependent{rcub} Hamiltonians {lcub}H/H (t){rcub} that represent different physical systems; but which differ by a {lcub}constant C/function C( t){rcub}, and yet possess the same density {lcub}rho(r)/rho( rt){rcub}, cannot be distinguished on the basis of the HK/RG theorem.; Q-DFT is a local effective potential energy theory that maps the interacting system described by Schrodinger's equation to one of noninteracting Fermions such that the equivalent density {lcub}rho(r)/rho(r t){rcub}, energy {lcub}E/E(t){rcub}, and ionization potential are obtained. This mapping is in terms of 'classical' fields and quantal sources representative of electron correlations the model system must account for, viz. those due to the Pauli exclusion principle, Coulomb repulsion, Correlation-Kinetic and Correlation-Current-Density effects. The mapping contrasts with that of traditional Kohn-Sham (KS) DFT, which in the time-independent case, is in terms of an energy density functional for the ground state, and an energy bidensity functional for excited states, and of their functional derivatives. Here we apply time-independent Q-DFT to study the properties of the hydrogen molecule in its ground state. We further extend time-independent nondegenerate Q-DFT to degenerate states for both ground and excited states, and for both pure state and ensemble densities. This further provides a rigorous physical interpretation of the energy density and bidensity functionals, and their derivatives, of degenerate state KS-DFT. |
