Understanding structural phase transformations in solid-state materials is of great scientific and technological interest. These phenomena are governed by electronic degrees of freedom and thus, in principle, can be described with quantum mechanics alone. However, any realistic material has multiple length and time scales and access to these scales is a formidable task to deal with using quantum mechanics. On the contrary, for the continuum regime, empirical constitutive models have severe difficulties capturing material properties that ultimately arise from (sub-)atomic effects, and further, must be fit to experimental data. Thus, in order to obtain a predictive, complete understanding of a material subjected to different external loading parameters, the two regimes- atomistic and continuum-must be coupled. The present work formulates a coupled quantum-continuum model for scanning phase-space in order to determine the material's stable structure at any given pressure. This is accomplished by coupling quantumbased Density Functional Theory (DFT) calculations (employing periodic boundary conditions) with Branch- Following and Bifurcation (BFB) techniques. BFB is capable of mapping out equilibrium paths (stable and unstable) as a function of the applied pressure and ultimately creates "bifurcation maps" that identify the material's stable structures and connections between them, including: soft deformation directions, transition states, transformation mechanisms, etc.. This study shows that the coupled DFT-BFB methodology is capable of efficiently mapping out equilibrium paths. This includes the identification of stable and unstable pressure ranges and the identification of the deformation modes that first become soft-resulting in the structure's loss of stability. Example computations are provided for iron and silicon. The results obtained so far indicate that the new DFT-BFB methodology has the potential to provide a significant new insight on the mechanisms that drive structural phase transitions in a wide range of technologically important materials.