| This dissertation consists of seven relatively independent chapters.; In Chapter 1, a general review is given about the density-functional theory and about the work I have done in last couple of years.; In Chapter 2, a formal expansion scheme for any well-behaved functional in terms of its functional derivatives is present. Several special cases, such as local functionals, the weighted-density approximation (WDA), the general gradient approximation (GGA), and so on, will be discussed.; In Chapter 3, based on the Levy-Perdew relation and the one we derive, the homogeneity property of the correlation energy density functional and its kinetic component is investigated. It is found that the correlation energy density functional can be expressed as a linear combination of homogeneous functionals of degree (1 {dollar}-{dollar} n) with n = 1, 2, 3, {dollar}...{dollar}, in coordinate scaling. With the locality assumption, a new local form is obtained.; In Chapter 4, the new local form of the correlation energy density functional is applied to atoms, ions, and simple molecules. It is observed that among the five local forms available, this one produces the best result in all cases. In some cases its results are comparable to those of the non local LYP (Lee-Yang-Parr) form.; In Chapter 5, the consequences of our functional expansion of the correlation energy density functional are investigated. One finds that the pair distribution function and the second-order density matrix can also be expanded in a similar way.; In Chapter 6, application of our functional expansion scheme is extended to the current-density functional theory (CDFT). We are able to establish the local density approximation (LDA) for the kinetic, exchange and correlation energy density functionals in CDFT. A hierarchy of equations for the kinetic, exchange and correlation energy density functionals is also obtained. The equations are generalizations of the Kugler, Nagy, and Parr-Liu-Kugler-Nagy formulations. In addition, a general formulation of the adiabatic connection is investigated in the form of the constrained-search of a coupled Hamiltonian.; Finally, in Chapter 7, four exact formulas are presented to calculate cusp values for any well-behaved functions with strong decaying properties. They are rigorously proved in terms of the Green's theorem, and then applied to calculate cusp values of the density, its gradient, and the exchange-correlation potentials of a few atomic systems. |
