Tensor decomposition in post-Hartree-Fock methods. I. Two-electron integrals and MP2

A new approximation for post-Hartree-Fock (HF) methods is presented applying tensor decomposition techniques in the canonical product tensor format. In this ansatz, multidimensional tensors like integrals or wavefunction parameters are processed as an expansion in one-dimensional representing vectors. This approach has the potential to decrease the computational effort and the storage requirements of conventional algorithms drastically while allowing for rigorous truncation and error estimation. For post-HF ab initio methods, for example, storage is reduced to O(d . R . n) with d being the number of dimensions of the full tensor, R being the expansion length (rank) of the tensor decomposition, and n being the number of entries in each dimension (i.e., the orbital index). If all tensors are expressed in the canonical format, the computational effort for any subsequent tensor contraction can be reduced to O(R-2 . n). We discuss details of the implementation, especially the decomposition of the two-electron integrals, the AO-MO transformation, the Moller-Plesset perturbation theory (MP2) energy expression and the perspective for coupled cluster methods. An algorithm for rank reduction is presented that parallelizes trivially. For a set of representative examples, the scaling of the decomposition rank with system and basis set size is found to be O(N-1.8) for the AO integrals, O(N-1.4) for the MO integrals, and O(N-1.2) for the MP2 t(2)-amplitudes (N denotes a measure of system size) if the upper bound of the error in the l(2)-norm is chosen as epsilon = 10(-2). This leads to an error in the MP2 energy in the order of mHartree. (C) 2011 American Institute of Physics. [doi:10.1063/1.3514201]