An efficient and robust integration scheme for the equations of motion of the multiconfiguration time-dependent Hartree (MCTDH) method

Abstract

An efficient and robust integration scheme tailored to the equations of motion of the multiconfiguration time-dependent Hartree (MCTDH) method is presented. An error estimation allows the automatical adjustment of the step size and hence controls the integration error. The integration scheme decouples the MCTDH equations of motion into several disjoined subsystems, of which one determines the time evolution of the MCTDH-coefficients. While the conventional MCTDH equations are non-linear, the working equation for the MCTDH-coefficients becomes linear in the present integration scheme. To investigate the integrator's performance it is applied to the photodissociation process of methyl iodide. The results of the novel integration scheme are in perfect agreement to those obtained by solving the MCTDH working equations conventionally. The computation time, however, is reduced by a factor of about ten when the new integration scheme is used to propagate large systems. PACS: 02.70.Ns, 31.70.Hq I IntroductionThe time-dependent quantum mechanical treatment of molecular dynamics, which numerically solves the time-dependent Schrödinger equation, has gained rapidly increasing interest and importance during the last fifteen years. Although for time-independent Hamiltonians this approach is formally equivalent to solving the time-independent Schrödinger equation, it offers a number of advantages. While in the timeindependent framework one has to solve an eigenvalue problem, one is faced in the time-dependent picture with an initial value problem, which is mathematically simpler. Moreover, time-dependent methods can handle systems that involve continuum states (e. g. chemical reactions or photodissociation processes) very easily. Employing a time-independent procedure in this case, the wave function becomes a continuum function subject to complicated scattering boundary conditions. In contrast, turning to a time-dependent treatment the wave function remains square integrable, and there is virtually no difference between propagating a wave-packet that is a superposition of bound or of continuum states. Besides these rather technical advantages, time-dependent schemes offer a deeper insight into the dynamics, and hence lead to a better understanding of the physical process under investigation. Finally, if the Hamiltonian is itself time-dependent, one must of course adopt the time-dependent framework.Apart from the development of the computational facilities there has been a considerable progress over the last fifteen years in improving the algorithms for solving the time-dependent Schrödinger equation numerically exactly. Powerful integrators [1,2] and representation schemes [3][4][5][6] are now available. Nevertheless, a numerically exact quantum mechanical treatment of the wave-packet dynamics is in general restricted to three-or four-dimensional systems due to the exponential increase of numerical effort and main memory with the number of degrees of freedom.To overcome these limitations it is essential to develop approximate s...

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