Reduced scaling CASPT2 using supporting subspaces and tensor hyper-contraction.

We present a reduced scaling formulation of the state specific complete active space second-order perturbation method (CASPT2) requiring O(N4) operations and O(N2) memory for a fixed active space, where N is proportional to system size. Motivated by the properties of the Kronecker sum, we introduce the supporting subspace technique (SST), which decomposes the CASPT2 linear equations into two parts: a single-reference MP2 energy term using dressed orbitals, plus a reduced linear system with dimension scaling as O(N2). Together with Laplace quadrature, the SST allows us to reformulate CASPT2 using a MP2 energy computation and Fock builds. By further applying the tensor hyper-contraction (THC) approximation, the MP2-like term can be computed with O(N4) operations, and the remainder can be solved with O(N3) operations using the preconditioned conjugate gradient method. This is the first application of THC in the context of multi-reference methods. We also developed an efficient implementation of the method by utilizing graphical processing units and exploiting spatial sparsity in tensor operations. We benchmark the accuracy of the new method against conventional CASPT2 for reactions in the gas phase. We apply the new method to Menshutkin SN2 reactions in carbon nanotubes, demonstrating the feasibility of CASPT2 calculations with O(100) atoms.