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We present a multi-scale approach to efficiently embed an ab initiocorrelated chemical fragment described by its energy-weighted density matrices,and entangled with a wider mean-field many-electron system. This approach,first presented in Phys. Rev. B, 98, 235132 (2018), is here extended to accountfor realistic long-range interactions and broken symmetry states. The schemeallows for a systematically improvable description in the range of correlatedfluctuations out of the fragment into the system, via a self-consistentoptimization of a coupled auxiliary mean-field system. It is discussed that themethod has rigorous limits equivalent to existing quantum embedding approachesof both dynamical mean-field theory, as well as density matrix embeddingtheory, to which this method is compared, and the importance of thesecorrelated fluctuations is demonstrated. We derive a self-consistent localenergy functional within the scheme, and demonstrate the approach for Hydrogenrings, where quantitative accuracy is achieved despite only a single atom beingexplicitly treated.

Energy-weighted density matrix embedding of open correlated chemical fragments