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A theoretical scheme for the treatment of an open molecular system withelectrons and nuclei is proposed. The idea is based on the Grand Canonicaldescription of a quantum region embedded in a classical reservoir of molecules.Electronic properties of the quantum region are calculated at constantelectronic chemical potential equal to that of the corresponding (large) bulksystem treated at full quantum level. Instead, the exchange of moleculesbetween the quantum region and the classical environment occurs at the chemicalpotential of the macroscopic thermodynamic conditions. T he Grand CanonicalAdaptive Resolution Scheme is proposed for the treatment of the classicalenvironment; such an approach can treat the exchange of molecules according tofirst principles of statistical mechanics and thermodynamic. The overall schemeis build on the basis of physical consistency, with the correspondingdefinition of numerical criteria of control of the approximations implied bythe coupling. Given the wide range of expertise required, this work has theintention of providing guiding principles for the construction of a wellfounded computational protocol for actual multiscale simulations from theelectronic to the mesoscopic scale.

Grand Canonical Adaptive Resolution Simulation for Molecules with Electrons: A Theoretical Framework based on Physical Consistency