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${\mathrm{V}}_{1\ensuremath{-}x}{\mathrm{Al}}_{x}$ is a representative example of highly resistive metallic alloys exhibiting a crossover to a negative temperature coefficient of resistivity (TCR), known as the Mooij correlation. Despite numerous proposals to explain this anomalous behavior, none have provided a satisfactory quantitative explanation thus far. In this work, we calculate the electrical conductivity using an ab initio methodology that combines the Kubo-Greenwood formalism with the coherent potential approximation (CPA). The temperature dependence of the conductivity is obtained within a CPA-based model of thermal atomic vibrations. Using this approach, we observe the crossover to the negative TCR behavior in ${\mathrm{V}}_{1\ensuremath{-}x}{\mathrm{Al}}_{x}$, with the temperature coefficient following the Mooij correlation, which matches experimental observations in the intermediate-to-high temperature range. Analysis of the results allows us to clearly identify a non-Boltzmann contribution responsible for this behavior and describe it as a function of temperature and composition.