For this binary alloy, besides the two forms of FeSi\({}_{2}\), it is worth noting the observation of the cubic fluorite \(\gamma\)-FeSi\({}_{2}\) phase (C1, CaF\({}_{2}\)-type, space group, cF12, Fm-3m, #225) that exists in epitaxially stabilized thin-film configuration [15] or as small coherent precipitates [16; 17] due to the small misfit to the silicon substrate lattice parameter. This phase will be re-examined in Section 2. These examples illustrate the fact that, for many TM-X alloys where TM is a \(d\)-like transition metal element and X is an \(sp\)-like element (e.g., Al, Ge, Si, Zn, etc. ), there is formation of complex phases, especially at X-rich compositions, in contrast to the situation in TM-rich compositions, where simple (ordered) phases tend to form. This would also be the case for, e.g., Cu-Ge, Fe-Ge, and Ti-Al, among many others. Therefore, the question becomes the following: Why is it that, although the pure elements stabilize in a simple structure such as fcc, bcc, or hcp, complex phases form when these elements become constituents of an alloy? Here, we will focus on the following question: Starting from simple, structure-based pure elements that make up an alloy at a given composition, what is the pathway to a structural transformation that leads to the formation of a complex structure at thermodynamic equilibrium around a specific composition? FeSi\({}_{2}\) will be used as a template. First-principles calculations were used to investigate the stability and the structural, elastic, and electronic properties of \(\beta\)-FeSi\({}_{2}\)[18; 19; 20; 21; 22], \(\gamma\)-FeSi\({}_{2}\)[21; 23], and \(\alpha\)-FeSi\({}_{2}\)[21]. The first-principles approach and the Debye model were combined to predict the vibrational thermodynamic contribution for Fe-Si compounds [24]. In a semi-empirical tight-binding scheme, the total energy and elastic constants of \(\gamma\)-FeSi\({}_{2}\) were calculated [25]. This scheme was also employed to (1) investigate the transformation of the electronic structure when the original calcium-fluorite structure is distorted into the orthorhombic one [26], and (2) analyze the leading role of the electron-phonon interaction in promoting the structural distortion from fluorite (\(\gamma\)-) to orthorhombic \(\beta\)-FeSi\({}_{2}\)[27]. Semi-empirical tight-binding molecular dynamics simulations were carried out to investigate the \(\gamma\)-to-\(\beta\) structural transformation [23; 24], confirming that the driving force leading to the metal-semiconductor structural phase transition originates from a local Jahn-Teller distortion [28; 29]. The literature data were analyzed, and it was concluded that the \(\gamma\)-to-\(\beta\) transformation path, the behavior of the FeSi\({}_{2}\) phases at high temperatures, their stability, and their thermodynamic and mechanical properties were not yet investigated in the framework of the same first-principles approach. To fill this gap, we will use first-principles calculations that include structural optimization and molecular dynamics to study FeSi\({}_{2}\). This will enable us to explain the origin of the intermediate phases that arise during first-principles molecular dynamics (FPMD) simulations of the \(\gamma\) and \(\beta\) phases, to suggest possible mechanisms of the \(\gamma\rightarrow\beta\) and \(\beta\rightarrow\alpha\) transformations and to analyze the thermodynamic and mechanical properties of FeSi\({}_{2}\) phases. The paper is organized as follows. In Section 2, we revisit the crystallography of the two polytype forms of FeSi\({}_{2}\), and define simpler forms (approximants) of these two compounds with their relationships to other possible related phases. In Section 3, we present the details of first-principles calculations. In Section 4, the structural, mechanical, thermodynamic, and electronic structural properties of the \(\alpha\), \(\beta\), and \(\gamma\) forms of FeSi\({}_{2}\) are presented. Finally, in Section 5, a summary of the present study is discussed.