A theory of electronic structure for coarse-grained resolutions

We present a stochastic diabatic projection method for constructing electronic structure models that operate at coarse-grained (CG) molecular resolutions. By minimizing the relative entropy of joint electronic-nuclear distributions between the fine-grained (FG) and CG resolutions, we derive a stochastic effective Hamiltonian that decomposes CG configuration dependent electronic structure into deterministic means and residual fluctuations. This framework bridges downfolding and molecular coarse-graining theory, enabling the systematic reproduction of electronic structure renormalized over eliminated nuclear and electronic degrees of freedom.