Using Matrix-Free Tensor-Network Optimizations To Construct a Reduced-Scaling and Robust Second-Order Mo̷ller-Plesset Theory

We investigate the efficient combination of the canonical polyadic decomposition (CPD) and tensor hyper-contraction (THC) approaches. We first present a novel low-cost CPD solver that leverages a precomputed THC factorization of an order-4 tensor to efficiently optimize the order-4 CPD with $O\left(N{R}^2\right)$ scaling. With the matrix-free THC-based optimization strategy in hand, we can efficiently generate CPD factorizations of the order-4 two-electron integral tensors and develop novel electronic structure methods that take advantage of both the THC and CPD approximations. Next, we investigate the application of a combined CPD and THC approximation of the Laplace transform (LT) second-order Mo̷ller-Plesset (MP2) method. We exploit the ability to switch efficiently between the THC and CPD factorizations of the two-electron integrals to reduce the computational complexity of the LT MP2 method while preserving the accuracy of the approach. Furthermore, we take advantage of the robust fitting approximation to eliminate the leading-order error in the CPD approximated tensor networks. Finally, we show that modest values of THC and CPD rank preserve the accuracy of the LT MP2 method and that this CPD + THC LT MP2 strategy realizes a performance advantage over canonical LT MP2 in both computational wall-times and memory resource requirements.