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Mini-workshop on "Mathematical and numerical challenges in quantum chemistry"
Organizers: Claude Le Bris (Paris, France) & Christian Lubich (Tuebingen, Germany)


Reinhold Schneider (Berlin, Germany): "Tensor product approximation techniques, matrix product states and DMRG for quantum chemistry"
Monday 24 June 2013, 11:40 - 12:20
Amphi Darboux


Recent hierarchical tensor representations (HT Hackbusch-TT Oseledets), offer stable and robust approximation by a low order cost. These methods are already known for the computation of quantum lattice systems as matrix product states (MPS), and tree tensor network states. These methods has been applied to problems in quantum chemistry in the framework of density matrix renormalization group (DMRG). It exploits rank sparsity rather than sparsity of the full CI solution. It does not require a ponounced reference determinant and has a certain potential to treat multi-reference configurations.

For numerical computations in a variational framework, we cast the computation of an  approximate ground solution into an optimization problems constraint by the restriction to tensors of prescribed ranks r, which  form a differentiable manifold. For approximation by elements from this highly nonlinear manifold, we apply a non-linear Galerkin framework, the extension for the dynamical problems correspond to the Dirac Frenkel variational principle by describing  the differential geometric structure of the heirarchical tensor formats. We analyse the (open) manifold of such tensors and its projection onto the tangent space and  investigate the  long-time  behavior of  derived differential equations and gradient methods. We will compare the approach with CC methods, and its perfomance for describing the breaking of chemical bonds.


O.Legeza, T. Rohwedder and R. Schneider, Tensor methods in quantum chemistry to appear in  Encyclopedia of Applied and Computational Mathematics.
C. Lubich, T. Rohwedder, R. Schneider and B. Vandereycken, Dynamical approximation of hierarchical Tucker and tensor train tensors.