In this work, we investigate the role of inherent Structures it] the vitrification process of glass-forming materials by using a two-component Lennard-Jones mixture. We start with a simplified model that describes the dynamics of the atomistic system as a Poisson process consisting of a series of transitions from one potential energy minimum (inherent structure) to another, the rate of individual transitions being described by a first-order kinetic law. We investigate the validity of this model by comparing the mean square displacement resulting from atomistic molecular dynamics (MD) trajectories with the corresponding mean square displacement based on inherent structures. Furthermore, in the case of vitrification via stepwise cooling, we identify the role of the potential energy landscape in determining the properties of the resulting, glass. Interestingly, the cooling rate is not sufficient to define the resulting glass in a stepwise cooling process, because the time spent by the system at different temperatures (length of the steps) has a highly nonlinear impact on the properties of the resulting glass. In contrast to previous investigations of supercooled liquids, we focus on a range of temperatures close to and below the glass transition temperature, where the use of MD is incapable of producing equilibrated samples of the metastable supercooled state. Our aim is to develop a methodology that enables mapping the dynamics under these conditions to a coarse-grained first-order kinetic model based on the Poisson process approximation. This model can be used in order to extend our dynamical sampling ability to much broader time scales and therefore allow us to create computer glasses with cooling rates closer to those used experimentally. In a continuation to this work,(1) we provide the mathematical formulation for lifting the coarse-grained Poisson process model and reproducing the full dynamics of the atomistic system.