Series expansion of a scalable Hermitian excitonic renormalization method

Utilizing the sparsity of the electronic structure problem, fragmentation methods have been researched for decades with great success, pushing the limits of ab initio quantum chemistry ever further. Recently, this set of methods has been expanded to include a fundamentally different approach called excitonic renormalization, providing promising initial results. It builds a supersystem Hamiltonian in a second-quantized-like representation from transition-density tensors of isolated fragments, contracted with biorthogonalized molecular integrals. This makes the method fully modular in terms of the quantum chemical methods applied to each fragment and enables massive truncation of the state-space required. Proof-of-principle tests have previously shown that an excitonically renormalized Hamiltonian can efficiently scale to hundreds of fragments, but the ad hoc approach to building the Hamiltonian was not scalable to larger fragments. On the other hand, initial tests of the originally proposed modular Hamiltonian build, presented here, show the accuracy to be poor on account of its non-Hermitian character. In this study, we bridge the gap between these with an operator expansion that is shown to converge rapidly, tending toward a Hermitian Hamiltonian while retaining the modularity, yielding an accurate, scalable method. The accuracy is tested here for a beryllium dimer. At distances near equilibrium and longer, the zeroth-order method is comparable to coupled-cluster singles, doubles, and perturbative triples and the first-order method is comparable to full configuration interaction (FCI). The second-order method agrees with FCI for distances well up the inner repulsive wall of the potential. Deviations occurring at shorter bond distances are discussed along with approaches to scaling to larger fragments.