Colloquium: Multiconfigurational time-dependent Hartree approaches for indistinguishable particles

In this Colloquium, the wave-function-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. The historical developments that have led to the formulation of the MCTDH-X method and motivate the necessity for wave-function-based approaches are briefly described. The derivation of the unified MCTDH-F and MCTDH-B equations of motion for complete and also specific restricted configuration spaces are sketched. The strengths and limitations of the MCTDH-X approach are assessed via benchmarks against an exactly solvable model and via convergence checks. Applications to some instructive and experimentally realized quantum many-body systems are highlighted: The dynamics of atoms in Bose-Einstein condensates in magneto-optical and optical traps and of electrons in atoms and molecules. The current development and frontiers in the field of MCTDH-X are discussed: Theories and numerical methods for indistinguishable particles, for mixtures of multiple species of indistinguishable particles, the inclusion of nuclear motion for the nonadiabatic dynamics of atomic and molecular systems, as well as the multilayer and second-quantized-representation approaches, and the orbital-adaptive time-dependent coupled-cluster theory.