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Tensor Product Approximation (DMRG) and Coupled Cluster Method in Quantum Chemistry

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Many-Electron Approaches in Physics, Chemistry and Mathematics

Part of the book series: Mathematical Physics Studies ((MPST))

Abstract

We present the Coupled Cluster (CC) method and the Density matrix Renormalization Group (DMRG) method in a unified way, from the perspective of recent developments in tensor product approximation. We present an introduction into recently developed hierarchical tensor representations, in particular tensor trains which are matrix product states in physics language. The discrete equations of full CI approximation applied to the electronic Schrödinger equation is casted into a tensorial framework in form of the second quantization. A further approximation is performed afterwards by tensor approximation within a hierarchical format or equivalently a tree tensor network. We establish the (differential) geometry of low rank hierarchical tensors and apply the Driac Frenkel principle to reduce the original high-dimensional problem to low dimensions. The DMRG algorithm is established as an optimization method in this format with alternating directional search. We briefly introduce the CC method and refer to our theoretical results. We compare this approach in the present discrete formulation with the CC method and its underlying exponential parametrization.

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Authors and Affiliations

  1. Wigner Research Centre for Physics, Budapest, P.O. Box 49, Hungary

    Örs Legeza & Szilárd Szalay

  2. Humboldt Universität Berlin Fakultät II–Mathematisch-Naturwissenschaftliche Fakultät, Rudower Chaussee 25, Berlin, Germany

    Thorsten Rohwedder

  3. Technische Universität Berlin Fakultät II–Mathematik und Naturwissenschaften Institut für Mathematik, Strasse des 17. Juni 136, Berlin, Germany

    Reinhold Schneider

Authors
  1. Örs Legeza
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  2. Thorsten Rohwedder
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  3. Reinhold Schneider
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  4. Szilárd Szalay
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Corresponding author

Correspondence to Reinhold Schneider .

Editor information

Editors and Affiliations

  1. Institut für Analysis Und Algebra Carl-Friedrich-Gauss-Fakultät, Technische Universität Braunschweig, Braunschweig, Germany

    Volker Bach

  2. Institute for Mathematics, Freie Universität Berlin, Berlin, Germany

    Luigi Delle Site

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Legeza, Ö., Rohwedder, T., Schneider, R., Szalay, S. (2014). Tensor Product Approximation (DMRG) and Coupled Cluster Method in Quantum Chemistry. In: Bach, V., Delle Site, L. (eds) Many-Electron Approaches in Physics, Chemistry and Mathematics. Mathematical Physics Studies. Springer, Cham. https://doi.org/10.1007/978-3-319-06379-9_3

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