We give a detailed presentation of the theory and numerical implementation of an expression for the adiabatic energy flux in extended systems, derived from density-functional theory. This expression can be used to estimate the heat conductivity from equilibrium ab initio molecular dynamics, using the Green-Kubo linear response theory of transport coefficients. Our expression is implemented in an open-source component of the QE suite of computer codes for quantum mechanical materials modelling, which is being made publicly available.
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