Tensor Hypercontraction Form of the Perturbative Triples Energy in Coupled-Cluster Theory

We present the working equations for a reduced-scaling
method of
evaluating the perturbative triples (T) energy in coupled-cluster
theory, through the tensor hypercontraction (THC) of the triples amplitudes
(tijkabc). Through our method,
we can reduce the scaling of the (T) energy from the traditional O(N7) to a more modest O(N5). We also discuss implementation details
to aid future research, development, and software realization of this
method. Additionally, we show that this method yields submillihartree
(mEh) differences from CCSD­(T) when evaluating absolute energies and
sub-0.1 kcal/mol energy differences when evaluating relative energies.
Finally, we demonstrate that this method converges to the true CCSD­(T)
energy through the systematic increasing of the rank or eigenvalue
tolerance of the orthogonal projector, as well as exhibiting sublinear
to linear error growth with respect to system size.