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

TL;DR

A canonical polyadic decomposition (CPD) based low-level solver is incorporated as a means to accelerate the environment-level solver for the recently developed MPCC embedding framework and benchmarks on representative chemical environments are provided.

Abstract

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.

Figures (10)

• Figure 1: (a) Graphical representation of the four-index tensor $G_{ij}^{ab}$ decomposed using the DF approximation. (b) Representation of a CPD approximation TEI tensor where the CPD is applied to each DF TEI tensor.
• Figure 2: (a) LL energy and (b) LL energy error per non-hydrogen atom, both reported as a function of LL iteration, for a 6-water cluster in the TZ/TZ-RI basis.
• Figure 3: $L_2$ relative percent error in $\Omega$ for a 6-water cluster and hexane molecule in the TZ/TZ-RI basis. In (a) only the rank of the CPD approximation of $J^Q_{ab}$ is modified and in (b) the ranks of the CPD approximation of both $J^Q_{ab}$ and $J^Q_{ai}$ are modified simultaneously.
• Figure 4: (a) HL energy and (b) HL energy error per non-hydrogen atom at each HL iteration during the second macro-iteration of the MPCC procedure for a 6-water cluster in the TZ/TZ-RI basis.
• Figure 5: (a) MPCC energy and (b) MPCC energy error per non-hydrogen atom at each MPCC iteration for a 6-water cluster in the TZ/TZ-RI basis.