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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 regioninto an effective electrostatic potential representing the environment. Thispotential is generated by partitioning the environment into well definedfragments, and assigning each one a set of electrostatic multipoles, which canthen be used to build up the electrostatic potential. We show that, providingthe fragments and the projection scheme for the multipoles are chosen properly,this leads to an effective electrostatic embedding of the active QM regionwhich is of equal quality as a full QM calculation. We coupled our formalism tothe DFT code BigDFT, which uses a minimal set of localized in-situ optimizedbasis functions; this property eases the fragment definition while stilldescribing the electronic structure with great precision. Thanks to the linearscaling capabilities of BigDFT, we can compare the modeling of theelectrostatic embedding with results coming from unbiased full QM calculationsof the entire system. This enables a reliable and controllable setup of aneffective coarse-graining approach, coupling together different levels ofdescription, which yields a considerable reduction in the degrees of freedomand thus paves the way towards efficient QM/QM and QM/MM methods for thetreatment of very large systems.

Complexity Reduction in Large Quantum Systems: Reliable Electrostatic Embedding for Multiscale Approaches via Optimized Minimal Basis Functions