It is common to use the electron density to partition a molecular system into atomic regions. The necessity for such a partitioning scheme is rooted in the unquestionable role of atoms in chemistry. Nevertheless, atomic properties are not well- defined concepts within the domain of quantum mechanics, as they are not observable. This has resulted in a proliferation of different approaches to retrieve the concept of atoms in molecules (AIM) within the domain of quantum mechanics and in silico experiments based on various flavors of model theories. One of the most popular families of models is the Hirshfeld, or stockholder, partitioning methods. Hirshfeld methods do not produce sharp atomic boundaries, but instead distribute the molecular electron density at each point between all the nuclear centers constituting the molecule. The various flavors of the Hirshfeld scheme differ mainly in how the atomic shares are computed from a reference promolecular density and how the reference promolecular density is defined. We first establish the pervasiveness of the Hirshfeld portioning by extending its information-theoretic framework. This characterizes the family of f-divergence measures as necessary and sufficient for deriving Hirshfeld scheme. Then, we developed a variational version of Hirshfeld partitioning method, called Additive Variational Hirshfeld (AVH). The key idea is finding the promolecular density, expanded as a linear combination of charged and neutral spherically-averaged isolated atomic densities in their ground and/or excited states, that resembles the molecular density as much as possible. Using Kullback-Liebler divergence measure, this automatically guarantees that each atom and proatom have the same number of electrons, and that the partitioning is size consistent. The robustness of this method is confirmed by testing it on various datasets. Considering the mathematical properties and our numerical results, we believe that AVH has the potential to supplant other Hirshfeld partitioning schemes in future. / Thesis / Doctor of Philosophy (PhD)

This dissertation consists of four independent but related parts, each in a Chapter. The first part is an introductory. It serves as the background introduction and offer preparations for later parts. The second part discusses two population multivariate normal distributions with common covariance matrix. The goal for this part is to derive objective/non-informative priors for the parameterizations and use these priors to build up constructive random posteriors of the Kullback-Liebler (KL) divergence of the two multivariate normal populations, which is proportional to the distance between the two means, weighted by the common precision matrix. We use the Cholesky decomposition for re-parameterization of the precision matrix. The KL divergence is a true distance measurement for divergence between the two multivariate normal populations with common covariance matrix. Frequentist properties of the Bayesian procedure using these objective priors are studied through analytical and numerical tools. The third part considers the star-shape Gaussian graphical model, which is a special case of undirected Gaussian graphical models. It is a multivariate normal distribution where the variables are grouped into one "global" group of variable set and several "local" groups of variable set. When conditioned on the global variable set, the local variable sets are independent of each other. We adopt the Cholesky decomposition for re-parametrization of precision matrix and derive Jeffreys' prior, reference prior, and invariant priors for new parameterizations. The frequentist properties of the Bayesian procedure using these objective priors are also studied. The last part concentrates on the discussion of objective Bayesian analysis for partial correlation coefficient and its application to multivariate Gaussian models. / Ph. D.