Recent experimental studies on Fe substituted spinel CoCr$_{2}$O$_{4}$ have discovered multiple functional properties in the system such as temperature and composition dependent magnetic compensation, tunable exchange bias and magnetostriction. These properties are attributed to the renormalisation of the inter-atomic magnetic exchange interactions arising due to the non-regular site occupancies of the magnetic cations in the system. In this work, we perform {it ab initio} electronic structure calculations by DFT+U method and combine with a generalised thermodynamic model to compute the site occupancy patterns of the magnetic cations, the structural properties and the magnetic exchange interactions of Co$left(Cr_{1-x}Fe_{x} right)_{2}$O$_{4}$ for the entire composition range $0<x<1$. We find that the substituting Fe atoms prefer to occupy the tetrahedral sites of the spinel structure for the entire range of $x$, in agreement with the experimental inferences. Our results on the variations of the structural parameters with compositions agree very well with the experiments. By computing the variations of the various inter-atomic magnetic exchange interactions, we provide a microscopic picture of the evolution of a collinear structure from a non-collinear one due to substitution of Fe in CoCr$_{2}$O$_{4}$. The computed results are analysed in terms of the elements of the crystal field theory, and the features in the atoms and orbital-projected densities of states. The results and analysis presented in this work is the first comprehensive study on this system which would help understanding the complexities associated with the site occupancies, the electronic structures and the magnetic interactions in this multi-functional material.
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