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Irreducible Cartesian tensor decomposition: A computational approach.

Andrea Bonvicini1

  • 1Departement of Chemistry, Theoretical Chemistry Laboratory, Unit of Theoretical and Structural Physical Chemistry, Namur Institute of Structured Matter, University of Namur, Rue de Bruxelles 61, 5000 Namur, Belgium.

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This study presents a computational method for irreducible Cartesian tensor (ICT) decomposition, simplifying symmetry analysis in physics and chemistry. The approach successfully decomposes tensors up to rank 5, advancing molecular property calculations.

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Area of Science:

  • Physics and Chemistry
  • Quantum Mechanics
  • Computational Chemistry

Background:

  • Cartesian tensors are fundamental in describing physical phenomena like molecular spectroscopy and response properties.
  • Decomposition of Cartesian tensors into irreducible parts is crucial for understanding their inherent symmetries.

Purpose of the Study:

  • To develop and present a computational method for the irreducible Cartesian tensor (ICT) decomposition of generic Cartesian tensors.
  • To apply this method for the explicit decomposition of tensors up to rank 5.

Main Methods:

  • The study reviews the matrix formulation of ICT decomposition, drawing parallels with rotational averaging.
  • A computational approach utilizing the reduced row echelon form (rref) algorithm is employed.
  • The protocol is implemented in a computer code.

Main Results:

  • The computational approach successfully re-derives known ICT decompositions for tensor ranks 2 through 4.
  • For the first time, the explicit ICT decomposition of a Cartesian tensor of rank 5 is achieved.
  • The method provides a systematic way to identify linearly independent mappings between tensor subspaces.

Conclusions:

  • The rref-based computational method offers an efficient and accessible protocol for ICT decomposition.
  • This work extends the applicability of ICT decomposition to higher-rank tensors, aiding in advanced spectroscopy and molecular property studies.