Tensor Decomposition in post-Hartree Fock Methods.

Udo Benedikt, Alexander Auer, Mike Espig, and Wolfgang Hackbusch

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Submission date: 26. Oct. 2010
Pages: 39
published in: The journal of chemical physics, 134 (2011) 5, art-no. 054118 
DOI number (of the published article): 10.1063/1.3514201
Bibtex
with the following different title: Tensor decomposition in post-Hartree Fock methods. Pt. 1 : Two-electron integrals and MP2
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Abstract:

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 of 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(R² ⋅ n). We discuss details of the implementation, especially the decomposition of the two electron integrals, the AO-MO transformation, the 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 t₂-amplitudes (N denotes a measure of system size) if the upper bound of the error in the l₂-norm is chosen as ε = 10^-2. This leads to an error in the MP2 energy in the order of mHartree.