Dynamical low rank approximation in hierarchical tensor formats for electronic structure methods

Reinhold Schneider
Technische Universität Berlin

In tensor product approximation, Hierarchical Tucker tensor format (Hackbusch) and Tensor Trains (TT) (Tyrtyshnikov) have been introduced recently offering stable and robust approximation by a low order cost . If $ \mathcal{V} = \bigotimes_ {i=1}^d \mathbb{C}^2 $, these formats are equivalent to tree tensor networks states and matrix product states (MPS) originally introduced for the treatment of quantum spin systems. Considering the electronic Schrodinger equation, we use an occupation number labeling of Slater determinants, and show that the discrete Fock space becomes isometric to d-fold tensor product of a two-dimensional Hilbert space. For the computation of an approximate ground solution this problem can be casted into an optimization problem constraint by the restriction to tensors of prescribed multi-linear ranks r. Dirac Frenkel variational principle developed in a similar fashion as for Multi-Configurational Hartree (-Fock) by observing the differential geometric structure of the novel tensor formats. This provides a variational formulation of the QC (Quantum Chemistry) DMRG (Density Renormalization Group) algorithm. We propose a dynamical low rank approximation, corresponding to the Dirac-Frenkel variational principle, for solving a constraint optimization problem. The approach can be applied to ground state calculations as well as to dynamical problems. Convergence of (Riemannian) gradient algorithms can be shown. A simple optimization method is provided by alternating direction methods, which reveals the DMRG (density matrix renormalization group) algorithm. This approach has been applied by G.C. Chan et al. and O. Legeza et al. to analyze the dissociation of diatomic molecules and to transition metal complexes, supporting that the presented approach has a certain potential to treat some strongly correlated electronic systems. Literature: 1. O.Legeza, T. Rohwedder and R. Schneider. Tensor methods in quantum chemistry to appear in in Encyclopedia of Applied and Computational Mathematics 2. C. Lubich, T. Rohwedder, R. Schneider and B. Vandereycken. Dynamical approximation of hierarchical Tucker and tensor train tensors SPP1324 Preprint (126/2012) 3. B. Khoromskij, I. Oseledets and R. Schneider. Efficient time-stepping scheme for dynamics on TT-manifolds, MIS Preprint 80/2011

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