## Low rank factorization of Coulomb integrals

### Reducing complexity

One of the most expensive steps in a canonical coupled cluster calculation
is the particle-particle contraction of the amplitudes with the
Coulomb integrals scaling as $\mathcal O(N^6)$.
For larger systems this part of the integrals cannot even
be kept in memory and needs to be computed on the fly.

When factorizing the Coulomb integrals into a product
of matrizes of which only 2 are distinct, the complexity of above
contraction reduces to $\mathcal O(N^5)$.
The factorization ansatz is motivated from the real space
definition of the integrals and it is known under the name
Tensor Hyper Contraction. The
factorization is, however, difficult to compute despite its comparatively
low formal complexity of $\mathcal O(N^4)$. We employ a modified
regularized alternating least squares fit to determine the factorization,
lowering the overall computation times of CCSD by an order of magnitude.