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The Journal of Chemical Physics

A multiconfiguration time-dependent Hartree approximation based on natural single-particle states

Antonius P. J. Jansen

Published Sep 1, 1993• 48 citations
Publisher

Abstract

The time-dependent Schrödinger equation is solved with a multiconfiguration form for the state. It is shown that the use of natural single-particle states results in a minimal number of configurations. The equations of motion of the natural single-particle states are derived by differentation with respect to time of the defining equations, which are eigenvalue equations of reduced density matrices. The method is tested on a two-dimensional model of H2 dissociation on a transition metal surface. It is shown that numerically exact solutions can be obtained. Computation times and memory requirements are less than for methods that solve the time-dependent Schrödinger equation directly.

DOI
10.1063/1.466101

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The time-dependent Schrödinger equation is solved with a multiconfiguration form for the state. It is shown that the use of natural single-particle states results in a minimal number of configurations. The equations of motion of the natural single-particle states are derived by differentation with respect to time of the defining equations, which are eigenvalue equations of reduced density matrices. The method is tested on a two-dimensional model of H2 dissociation on a transition metal surface. It is shown that numerically exact solutions can be obtained. Computation times and memory requirements are less than for methods that solve the time-dependent Schrödinger equation directly.
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.
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.
A numerical integrator is proposed for solving the multiconfiguration time-dependent Hartree (MCTDH) equations of motion, which are widely used in computations of molecular quantum dynamics. In contrast to existing integrators, the proposed algorithm does not require inverses of ill-conditioned density matrices and obviates the need for their regularization, allowing for large stepsizes also in the case of near-singular density matrices. The nonlinear MCTDH equations are split into a chain of linear differential equations that can all be efficiently solved by Lanczos approximations to the action of Hermitian-matrix exponentials, alternating with orthogonal matrix decompositions. The integrator is an extension to the Tucker tensor format of recently proposed projector-splitting integrators for the dynamical low-rank approximation by matrices and tensor trains (or matrix product states). The integrator is time-reversible and preserves both the norm and the total energy.
The feasibility of using a basis set approach to the study of quantum dissipative dynamics is investigated for the spin-boson model, a system of two discrete states linearly coupled to a harmonic bath. The infinite Hamiltonian is discretized to a finite number of degrees of freedom. Traditional basis set approach, in a multiconfiguration time-dependent Hartree context, is used to solve the time-dependent Schrödinger equations by explicitly including all the degrees of freedom (“system”+“bath”). Quantities such as the reduced density matrix are then evaluated via a quadrature summation/Monte Carlo procedure over a certain number of time-dependent wave functions. Numerically exact results are obtained by systematically increasing the number of bath modes used to represent the condensed phase environment, as well as other variational parameters (number of basis functions, configurations, etc.). The potential of the current method is briefly discussed.
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 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.