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The Journal of Chemical Physics

Unified view on multiconfigurational time propagation for systems consisting of identical particles

Ofir E. Alon, Alexej I. Streltsov, L. S. Cederbaum

Published Oct 16, 2007• 158 citations
Publisher

Abstract

We show that the successful and formally exact multiconfigurational time-dependent Hartree method (MCTDH) takes on a unified and compact form when specified for systems of identical particles (MCTDHF for fermions MCTDHB for bosons). In particular the equations of motion for the orbitals depend explicitly and solely on the reduced one- and two-body density matrices of the system's many-particle wave function. We point out that this appealing representation of the equations of motion opens up further possibilities for approximate propagation schemes.

DOI
10.1063/1.2771159

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We show that the successful and formally exact multiconfigurational time-dependent Hartree method (MCTDH) takes on a unified and compact form when specified for systems of identical particles (MCTDHF for fermions MCTDHB for bosons). In particular the equations of motion for the orbitals depend explicitly and solely on the reduced one- and two-body density matrices of the system's many-particle wave function. We point out that this appealing representation of the equations of motion opens up further possibilities for approximate propagation schemes.
We specify the formally exact multiconfigurational time-dependent Hartree method originally developed for systems of distinguishable degrees of freedom to mixtures consisting of two types of identical particles. All three cases, Fermi-Fermi, Bose-Bose, and Bose-Fermi mixtures, are treated on an equal footing making explicit use of the reduced one- and two-body density matrices of the mixture. The theory naturally contains as specific cases the versions of the multiconfigurational time-dependent Hartree method for single-species fermions and bosons. Explicit and compact equations of motion are derived and their properties and usage are briefly discussed.
The evolution of Bose-Einstein condensates is amply described by the time-dependent Gross-Pitaevskii mean-field theory which assumes all bosons to reside in a single time-dependent one-particle state throughout the propagation process. In this work, we go beyond mean field and develop an essentially exact many-body theory for the propagation of the time-dependent Schr\"odinger equation of NNN interacting identical bosons. In our theory, the time-dependent many-boson wave function is written as a sum of permanents assembled from orthogonal one-particle functions, or orbitals, where both the expansion coefficients and the permanents (orbitals) themselves are time-dependent and fully determined according to a standard time-dependent variational principle. By employing either the usual Lagrangian formulation or the Dirac-Frenkel variational principle we arrive at two sets of coupled equations of motion, one for the orbitals and one for the expansion coefficients. The first set comprises of first-order differential equations in time and nonlinear integrodifferential equations in position space, whereas the second set consists of first-order differential equations with time-dependent coefficients. We call our theory multiconfigurational time-dependent Hartree for bosons, or MCTDHB(M)\text{MCTDHB}(M)MCTDHB(M), where MMM specifies the number of time-dependent orbitals used to construct the permanents. Numerical implementation of the theory is reported and illustrative numerical examples of many-body dynamics of trapped Bose-Einstein condensates are provided and discussed. The convergence of the method with a growing number MMM of orbitals is demonstrated in a specific example of four interacting bosons in a double well.
In this Colloquium, the wavefunction-based Multiconfigurational Time-Dependent Hartree approaches to the dynamics of indistinguishable particles (MCTDH-F for Fermions and MCTDH-B for Bosons) are reviewed. MCTDH-B and MCTDH-F or, together, MCTDH-X are methods for describing correlated quantum systems of identical particles by solving the time-dependent Schrödinger equation from first principles. MCTDH-X is used to accurately model the dynamics of real-world quantum many-body systems in atomic, molecular, and optical physics. The key feature of these approaches is the time-dependence and optimization of the single-particle states employed for the construction of a many-body basis set, which yields nonlinear working equations. We briefly describe the historical developments that have lead to the formulation of the MCTDH-X methods and motivate the necessity for wavefunction-based approaches. We sketch the derivation of the unified MCTDH-F and MCTDH-B equations of motion for complete and also specific restricted configuration spaces. The strengths and limitations of the MCTDH-X approach are assessed via benchmarks against an exactly solvable model and via convergence checks. We highlight some applications to instructive and experimentally-realized quantum many-body systems: the dynamics of atoms in Bose-Einstein condensates in magneto-optical and optical traps and of electrons in atoms and molecules. We discuss the current development and frontiers in the field of MCTDH-X: theories and numerical methods for indistinguishable particles, for mixtures of multiple species of indistinguishable particles, the inclusion of nuclear motion for the nonadia-batic dynamics of atomic and molecular systems, the so-called multilayer generalizations to the MCTDH-F and MCTDH-B methods, and the time-dependent orbital-adaptive coupled cluster theory are discussed.
We derive a multiconfigurational time-dependent Hartree theory for systems with particle conversion. In such systems particles of one kind can convert to another kind and the total number of particles varies in time. The theory thus extends the scope of the available and successful multiconfigurational time-dependent Hartree methods---which were solely formulated for and applied to systems with a fixed number of particles---to a broader class of physical systems and problems. As a guiding example we treat explicitly a system where bosonic atoms can combine to form bosonic molecules and vice versa. In the theory for particle conversion, the time-dependent many-particle wave function is written as a sum of configurations made of a different number of particles, and assembled from sets of atomic and molecular orbitals. Both the expansion coefficients and the orbitals forming the configurations are time-dependent quantities that are fully determined according to the Dirac-Frenkel time-dependent variational principle. By employing the Lagrangian formulation of the Dirac-Frenkel variational principle we arrive at two sets of coupled equations of motion, one for the atomic and molecular orbitals and one for the expansion coefficients. The first set is comprised of first-order differential equations in time and nonlinear integrodifferential equations in position space, whereas the second set consists of first-order differential equations with coefficients forming a time-dependent Hermitian matrix. Particular attention is paid to the reduced density matrices of the many-particle wave function that appear in the theory and enter the equations of motion. There are two kinds of reduced density matrices: particle-conserving reduced density matrices which directly only couple configurations with the same number of atoms and molecules, and particle nonconserving reduced density matrices which couple configurations with a different number of atoms and molecules. Closed-form and compact equations of motion are derived for contact as well as general two-body interactions, and their properties are analyzed and discussed.
A unified view on linear response of interacting systems utilizing multiconfigurational time-dependent Hartree methods is presented. The cases of one-particle and two-particle response operators for identical particles and up to all-system response operators for distinguishable degrees-of-freedom are considered. The working equations for systems of identical bosons and identical fermions, as well for systems of distinguishable particles are explicitly derived. These linear-response theories - applicable for discrete excitation spectra - provide numerically exact excitation energies and system's properties, when numerical convergence is achieved in the calculations.
We report on an implementation of the multiconfigurational time-dependent Hartree method (MCTDH) for spin-polarized fermions (MCTDHF). Our approach is based on a mapping for operators in Fock space that allows a compact and efficient application of the Hamiltonian and solution of the MCTDHF equations of motion. Our implementation extends, builds on, and exploits the recursive implementation of MCTDH for bosons (r-mctdhb) package. Together with r-mctdhb, the present implementation of MCTDHF forms the mctdh-x package. We benchmark the accuracy of the algorithm with the harmonic interaction model and a time-dependent generalization thereof. These models consider parabolically trapped particles that interact through a harmonic interaction potential. We demonstrate that MCTDHF is capable of solving the time-dependent many-fermion Schr\"odinger equation to an arbitrary degree of precision and can hence yield numerically exact results even in the case of Hamiltonians with time-dependent one-body and two-body potentials. We study the problem of two initially parabolically confined and charged fermions tunneling through a barrier to open space. We demonstrate the validity of a model proposed previously for the many-body tunneling to open space of bosonic particles with contact interactions [Proc. Natl. Acad. Sci. USA 109, 13521 (2012)]. The many-fermion tunneling can be built up from sequentially happening single-fermion tunneling processes. The characteristic momenta of these processes are determined by the chemical potentials of trapped subsystems of smaller particle numbers: The escaped fermions convert the different chemical potentials into kinetic energy. Using the two-body correlation function, we present a detailed picture of the sequentiality of the process and are able to tell tunneling from over-the-barrier escape.