Hierarchical Tensor Approximation, DMRG and Combination with a Multi-Reference Coupled Cluster Method

Reinhold Schneider
Technische Universität Berlin
Institut für Mathematik, FG Modellierung, Simulation & Optimieru

Hierarchical tensor approximation (HT-Hackbusch) and tensor trains (TT-Oseledets) introduced for high-dimensional approximation are closely related to tensor network (TNS) and matrix product states (MPS) in quantum physics and partially related to Hidden Markov models. In this framework the Schrdinger equation is formulated in second quantization in a discrete Fock space, and usually solved by alternating search algorithms like DMRG (density matrix renormalization group) [1]. This provides high quality approximation of the full CI solution even in presence of strong correlation phenomena. However this tensor low rank approximation is not invariant under unitary transformations of orbital basis functions. Optimization of basis function together with low rank tensor approximation improves the DMRG calculations further. Nevertheless this approach (DMRG) became too expensive for computing dynamical correlations. We are analysing an approach which combines this approach (e.g. DMRG) and MR-SCF-CAS (multi-reference complete active space) with the tailored multi-reference coupled cluster (MRCC) ansatz, which accounts for the dynamical correlation not captured by DMRG. (Alternative approaches for extended systems are dynamical mean field theory and/or density matrix embedding, etc.) The cost of the present MR-CC calculation is not larger
that for standard single double coupled cluster calculations, but strong correlation effects, degenerate and nearly degenerate states can be computed as well. The convergence analysis is based on earlier analysis of CC approximation [2].

1) S. Szalay, M. Pfeffer, V Murg, G. Barcza, F. Verstraete and R. Schneider, Tensor product methods and entanglement optimization for ab initio quantum chemistry, International Journal of Quantum Chemistry , 115 (19), 1342-1391, 2015.
2) T. Rohwedder and R. Schneider, Error estimates for the coupled cluster method, ESAIM: Mathematical Modeling and Numerical Analysis, 47, 15531582, 2013.

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