Recursive formulation of the multiconfigurational time-dependent Hartree method for fermions, bosons and mixtures thereof in terms of one-body density operators

Ofir E. Alon, Alexej I. Streltsov, Kaspar Sakmann, Axel U.J. Lode, Julian Grond, Lorenz S. Cederbaum

Research output: Contribution to journalArticlepeer-review


The multiconfigurational time-dependent Hartree method (MCTDH) [H.-D. Meyer, U. Manthe, L.S. Cederbaum, Chem. Phys. Lett. 165, 73 (1990); U. Manthe, H.-D. Meyer, L.S. Cederbaum, J. Chem. Phys. 97, 3199 (1992)] is celebrating nowadays entering its third decade of tackling numerically-exactly a broad range of correlated multi-dimensional non-equilibrium quantum dynamical systems. Taking in recent years particles' statistics explicitly into account, within the MCTDH for fermions (MCTDHF) and for bosons (MCTDHB), has opened up further opportunities to treat larger systems of interacting identical particles, primarily in laser-atom and cold-atom physics. With the increase of experimental capabilities to simultaneously trap mixtures of two, three, and possibly even multiple kinds of interacting composite identical particles together, we set up the stage in the present work and specify the MCTDH method for such cases. Explicitly, the MCTDH method for systems with three kinds of identical particles interacting via all combinations of two- and three-body forces is presented, and the resulting equations-of-motion are briefly discussed. All four possible mixtures (Fermi-Fermi-Fermi, Bose-Fermi-Fermi, Bose-Bose-Fermi and Bose-Bose-Bose) are presented in a unified manner. Particular attention is paid to represent the coefficients' part of the equations-of-motion in a compact recursive form in terms of one-body density operators only. The recursion utilizes the recently proposed Combinadic-based mapping for fermionic and bosonic operators in Fock space [A.I. Streltsov, O.E. Alon, L.S. Cederbaum, Phys. Rev. A 81, 022124 (2010)], successfully applied and implemented within MCTDHB. Our work sheds new light on the representation of the coefficients' part in MCTDHF and MCTDHB without resorting to the matrix elements of the many-body Hamiltonian with respect to the time-dependent configurations. It suggests a recipe for efficient implementation of the schemes derived here for mixtures which is suitable for parallelization.

Original languageEnglish
Pages (from-to)2-14
Number of pages13
JournalChemical Physics
StatePublished - 5 Jun 2012

Bibliographical note

Funding Information:
The paper is dedicated to Professor Debashis Mukherjee, a dear colleague and friend, on the occasion of his 65th birthday. We are grateful to Hans-Dieter Meyer for multiple and continuous discussions on MCTDH, and acknowledge financial support by the DFG.


  • Dirac-Frenkel variational principle
  • Fock-space mapping of bosonic and fermionic operators
  • MCTDH for Bose-Bose-Bose mixtures (MCTDH-BBB)
  • MCTDH for Bose-Bose-Fermi mixtures (MCTDH-BBF)
  • MCTDH for Bose-Fermi-Fermi mixtures (MCTDH-BFF)
  • MCTDH for Fermi-Fermi-Fermi mixtures (MCTDH-FFF)
  • MCTDH for bosons (MCTDHB)
  • MCTDH for fermions (MCTDHF)
  • Multiconfigurational time-dependent Hartree (MCTDH)
  • Reduced density matrices
  • Time-dependent many-particle Schrödinger equation

ASJC Scopus subject areas

  • General Physics and Astronomy
  • Physical and Theoretical Chemistry


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