Cooperativity is key in defining the shape (i.e., gradual, abrupt, or hysteretic) of thermally driven spin transitions in magnetic switches. Despite its importance, there is very little information on its atomistic origin, which hinders the rational design of materials displaying a bistability region (i.e., hysteresis). In this paper, we investigate the spin transition of two solvatomorphs of [Fe(2-pic)(3)]Cl-2, an Fe(II)-complex displaying thermal spin crossover (SCO) from a low-spin (LS) to a high-spin (HS) state with either gradual or abrupt two-step character. To do it, we apply a novel computational protocol to study the cooperativity of SCO compounds from DFT calculations, which does not rely on a priori assumptions on the studied system. The distinct shape of the spin transition is successfully captured, and the atomistic origin of cooperativity is traced back to geometrical distortions of the Fe-N-6 core in case of the solvatomorph exhibiting an abrupt transition. According to our calculations, HS and LS molecules contribute differently to cooperativity, which results in a complex energetic evolution of the spin transition that cannot be described by the common Slichter-Drickamer model. The present work opens new avenues for the study of cooperativity of SCO systems having a chemically oriented perspective and demonstrates that quantum chemistry calculations can discriminate the shape of a spin transition, while providing insight into the atomistic underlying factors that contribute to its cooperative behavior.