<p>Partition density functional theory (P-DFT) is a density-based embedding method used to calculate the electronic properties of molecules through self-consistent calculations on fragments. P-DFT features a unique set of fragment densities that can be used to define formal charges and local dipoles. This dissertation is concerned mainly with establishing how the optimal fragment densities and energies of P-DFT depend on the specific methods employed during the self-consistent fragment calculations. First, we develop a procedure to perform P-DFT calculations on three-dimensional heteronuclear diatomic molecules, and we compare and contrast two different approaches to deal with non-integer electron numbers: Fractionally occupied orbitals (FOO) and ensemble averages (ENS). We find that, although both ENS and FOO methods lead to the same total energy and density, the ENS fragment densities are less distorted than those of FOO when compared to their isolated counterparts. Second, we formulate partition spin density functional theory (P-SDFT) and perform numerical calculations on closed- and open-shell diatomic molecules. We find that, for closed-shell molecules, while P-SDFT and P-DFT are equivalent for FOO, they partition the same total density of a molecule differently for ENS. For open-shell molecules, P-SDFT and P-DFT yield different sets of fragment densities for both FOO and ENS. Finally, by considering a one-electron system, we investigate the self-interaction error (SIE) produced by approximate exchange-correlation functionals and find that the molecular SIE can be attributed mainly to the non-additive Hartree-exchange-correlation energy.</p>