# Tensor Hyper-contraction

The electron repulsion integral (ERI) tensor is a demanding source of computational complexity in many ab initio methods. In the THC approximation, we reduce this fourth-order tensor to a product of five second-order tensors. This new approximation allows ab initio methods to be evaluated with reduced scaling. We have shown that this reduces the scaling to O(N4) for MP2, MP3, CC2, EOM-CC2, and CCSD. Other efforts to increase computational efficiency include the use of the Hubbard type correction to minimal basis sets.

### Related Publications

• Hohenstein, Edward G. and Kokkila, Sara I. L. and Parrish, Robert M. and Martinez, Todd J. , Quartic scaling second-order approximate coupled cluster singles and doubles via tensor hypercontraction: THC-CC2 , Journal of Chemical Physics , 138 , , 2013. [link]
• Hohenstein, Edward G. and Kokkila, Sara I. L. and Parrish, Robert M. and Martinez, Todd J. , Tensor Hypercontraction Equation-of-Motion Second-Order Approximate Coupled Cluster: Electronic Excitation Energies in O(N-4) Time , Journal of Physical Chemistry B , 117 , 12972-12978 , 2013. [link]
• Hohenstein, Edward G. and Parrish, Robert M. and Martinez, Todd J. , Tensor hypercontraction density fitting. I. Quartic scaling second- and third-order Moller-Plesset perturbation theory , Journal of Chemical Physics , 137 , , 2012. [link]
• Hohenstein, Edward G. and Parrish, Robert M. and Sherrill, C. David and Martinez, Todd J. , Communication: Tensor hypercontraction. III. Least-squares tensor hypercontraction for the determination of correlated wavefunctions , Journal of Chemical Physics , 137 , , 2012. [link]
• Parrish, Robert M. and Hohenstein, Edward G. and Martinez, Todd J. and Sherrill, C. David , Tensor hypercontraction. II. Least-squares renormalization , Journal of Chemical Physics , 137 , , 2012. [link]
• Parrish, Robert M. and Hohenstein, Edward G. and Martinez, Todd J. and Sherrill, C. David , Discrete variable representation in electronic structure theory: Quadrature grids for least-squares tensor hypercontraction , Journal of Chemical Physics , 138 , , 2013. [link]
• Parrish, Robert M. and Hohenstein, Edward G. and Schunck, Nicolas F. and Sherrill, C. David and Martinez, Todd J. , Exact Tensor Hypercontraction: A Universal Technique for the Resolution of Matrix Elements of Local Finite-Range N-Body Potentials in Many-Body Quantum Problems , Physical Review Letters , 111 , , 2013. [link]
• Parrish, Robert M. and Sherrill, C. David and Hohenstein, Edward G. and Kokkila, Sara I. L. and Martinez, Todd J. , Communication: Acceleration of coupled cluster singles and doubles via orbital-weighted least-squares tensor hypercontraction , Journal of Chemical Physics , 140 , , 2014. [link]
• Sara I. L. Kokkila Schumacher, Edward G. Hohenstein, Robert M. Parrish, Lee-Ping Wang, and Todd J. Martínez, Tensor Hypercontraction Second-Order Møller–Plesset Perturbation Theory: Grid Optimization and Reaction Energies, J. Chem. Theory Comput., 11,3042–3052, 2015. [link]
• Robert M. Parrish, Edward G. Hohenstein, and Todd J. Martinez, Comment on “Positive semidefinite tensor factorizations of the two-electron integral matrix for low-scaling ab initio electronic structure” [J. Chem. Phys. 143, 064103 (2015)], J. Chem. Phys., 145,, 2016. [link]
• Song, Chenchen and Martinez, Todd J. , Atomic orbital-based SOS-MP2 with tensor hypercontraction. II. Local tensor hypercontraction , Journal of Chemical Physics , 146 , NULL , 2017. [link]
• Song, Chenchen and Martinez, Todd J. , Atomic orbital-based SOS-MP2 with tensor hypercontraction. I. GPU-based tensor construction and exploiting sparsity , The Journal of Chemical Physics , 144 , 174111 , 2016. [link]
• Chenchen Song, Todd J Martínez, Analytical gradients for tensor hyper-contracted MP2 and SOS-MP2 on graphical processing units, The Journal of Chemical Physics, 147,161723, 2017. [link]