Given a partition of a large system into an active quantum mechanical (QM) region and its environment, we present a simple way of embedding the QM region into an effective electrostatic potential representing the environment. This potential is generated by partitioning the environment into well defined fragments, and assigning each one a set of electrostatic multipoles, which can then be used to build up the electrostatic potential. We show that, providing the fragments and the projection scheme for the multipoles are chosen properly, this leads to an effective electrostatic embedding of the active QM region which is of equal quality as a full QM calculation. We coupled our formalism to the DFT code BigDFT, which uses a minimal set of localized in-situ optimized basis functions; this property eases the fragment definition while still describing the electronic structure with great precision. Thanks to the linear scaling capabilities of BigDFT, we can compare the modeling of the electrostatic embedding with results coming from unbiased full QM calculations of the entire system. This enables a reliable and controllable setup of an effective coarse-graining approach, coupling together different levels of description, which yields a considerable reduction in the degrees of freedom and thus paves the way towards efficient QM/QM and QM/MM methods for the treatment of very large systems.
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