Efficient Electronic Structure Theory via Hierarchical Scale-Adaptive Coupled-Cluster Formalism: I. Theory and Computational Complexity Analysis

摘要

A novel reduced-scaling, general-order coupled-cluster approach is formulatedby exploiting hierarchical representations of many-body tensors, combined withthe recently suggested formalism of scale-adaptive tensor algebra. Inspired bythe hierarchical techniques from the renormalization group approach,H/H2-matrix algebra and fast multipole method, the computational scalingreduction in our formalism is achieved via coarsening of quantum many-bodyinteractions at larger interaction scales, thus imposing a hierarchicalstructure on many-body tensors of coupled-cluster theory. In our approach, theinteraction scale can be defined on any appropriate Euclidean domain (spatialdomain, momentum-space domain, energy domain, etc.). We show that thehierarchically resolved many-body tensors reduce the storage requirements toO(N), where N is the number of simulated quantum particles. Subsequently, weprove that any connected many-body diagram with arbitrary-order tensors, e.g.,an arbitrary coupled-cluster diagram, can be evaluated in O(NlogN)floating-point operations. On top of that, we elaborate an additionalapproximation to further reduce the computational complexity of higher-ordercoupled-cluster equations, i.e., equations involving higher than doubleexcitations, which otherwise would introduce a large prefactor into formalO(NlogN) scaling.