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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)

In birds, song can evolve quickly through cultural transmission and due to errors in the learning process may result in regional dialects. A lack of dialectal recognition may be a critical component of reproductive isolation through female mate preference. I investigated the role of cultural divergence in reproductive isolation in a widespread Neotropical passerine the rufous-collared sparrow (Zonotrichia capensis). In Chapter II, I investigated, the role of female preference for and male territorial response towards, local and non-local dialects in two allopatric populations. Females in both populations preferred their local dialect to the dialect of an allopatric population only 25 km away. In contrast, males showed similar territorial response to all conspecific dialects. Premating reproductive isolation based on culture may exist among the study populations. In chapter III, I investigated if cultural divergence can drive population divergence by examining dialects and variable microsatellite loci among eight populations of Z. capensis. I investigated the presence of population divergence and then identified the mechanism that may be driving the pattern. Apart from culture, a geographic barrier (the Andean ridge), elevation, and geographic distance were potential mechanisms of population divergence that I considered. All sites exhibited localized dialects, except for two closest neighboring populations. Populations exhibited genetic differentiation with support for the presence of five genetic clusters. Culture does not appear to be driving population divergence because song dialects and genetic population structure were not correlated. Populations separated by an Andean ridge did not display higher genetic or song differences than distance would predict, suggesting that the ridge is not driving population divergence. Elevation was not correlated to song or genetic differentiation. Both song divergence and microsatellite allele frequency differentiation were correlated with geographic distance suggesting a pattern of isolation by distance. Overall, geographic distance is the best predictor of population divergence in this system. Cumulatively, I found that culture might promote assortative mating via female mate choice, perhaps generating partial reproductive isolation; however, song dialect differences among contiguous populations is not currently driving population divergence in this species. / Ph. D.