A CPD-enabled low-scaling environment solver in a coupled cluster based static quantum embedding theory

We incorporate a canonical polyadic decomposition (CPD) based low-level solver as a means to accelerate the environment-level solver for the recently developed MPCC embedding framework. Using CPD, we both factorize the three dominant order-three density-fitting two-electron integral (DF TEI) tensors and develop a novel formulation that reduces the storage complexity of the low-level solver from ${O}(N^3)$ to $O(NR)$, where $R$ is the CPD rank, and the computational scaling of the most time-consuming contractions from ${O}(N^4)$ to ${O}(NR^2)$. We provide benchmarks on representative chemical environments, namely water clusters $\mathrm{{(H_2O)_n}}$ with $n = 1$ to $6$ and linear alkane chains $\mathrm{{C_nH_{2n+2}}}$ with $n = 1$ to $6$. For both test sets, using the CPD-compressed DF TEI tensors reproduces the DF reference convergence behavior of the low-level solver, the subsequent high-level step, and the fully self-consistent MPCC iterations, while introducing only small, rank-controlled shifts in absolute energies. At a fixed tolerance in the absolute MPCC energy, the CP ranks required for these tensor approximations increase linearly with system size. Chemically relevant energy differences are likewise preserved, as demonstrated for water-cluster dissociation energies and in a proof-of-concept embedding calculation of methane in a four-water cluster.