The creation of new materials is absolutely essential for developing new technologies. However, experimental efforts toward the material discovery are usually based on trial-and-error approach and thus require a huge amount of time and money. Alternatively, computational predictions can now provide a more systematic, rapid, inexpensive, and reliable method for the design of novel materials with properties suitable for new technologies. This dissertation describes the technique of theoretical predictions and presents the results on the successfully predicted and already produced (in some cases) unusual molecules, clusters, nanoparticles, and solids. The major part of scientific efforts in this dissertation was devoted to rationalizing of size- and composition-dependent properties of the materials based on understanding of their electronic structure and chemical bonding. It was shown that understanding relations between bonding and geometric structure, bonding and stability, and bonding and reactivity is an important step toward rational design of new, yet unknown materials with unusual properties. Our findings led to the discovery of the first simplest inorganic double helix structures, which can be used in the design of novel molecular devices. A significant part of this work also deals with the pseudo John-Teller effect, which potentially can be a powerful tool for rationalizing and predicting molecular and solid state structures, their deformations, transformations, and properties. Therefore, the works on the pseudo Jahn-Teller effect presented in this dissertation can be considered the steps toward further generalization and elevation of the pseudo Jahn-Teller effect to a higher level of understanding of the origin of molecular and solid state properties.