Huizhong Lu

EXPONENTIAL PROPAGATORS (INTEGRATORS) FOR THE TIME-DEPENDENT SCHRÖDINGER EQUATION

The time-dependent Schrodinger Equation (TDSE) is a parabolic partial differential equation (PDE) comparable to a diffusion equation but with imaginary time. Due to its first order time derivative, exponential integrators or propagators are natural methods to describe evolution in time of the TDSE, both for time-independent and time-dependent potentials. Two splitting methods based on Fer and/or Magnus expansions allow for developing unitary factorizations of exponentials with different accuracies in the time step △t. The unitary factorization of exponentials to high order accuracy depends on commutators of kinetic energy operators with potentials. Fourth-order accuracy propagators can involve negative or complex time steps, or real time steps only but with gradients of potentials, i.e. forces. Extending the propagators of TDSEs to imaginary time allows to also apply these methods to classical many-body dynamics, and quantum statistical mechanics of molecular systems.

Moving adaptive grid methods for numerical solution of the time-dependent molecular Schrödinger equation in laser fields

We present a moving adaptive grid method for solving the time-dependent Schrodinger equation, TDSE, for molecules in intense laser fields, applicable in the nonperturbative nonlinear regime where dissociation ionization occurs. The method is based on a Lagrangian, moving coordinate system. In this representation, the reference system is moving with the laser pulse so that the classical movement of free particles in the field, i.e., in the asymptotic region where electron–molecule potentials are negligible but the laser field is still present, is exactly described. As a consequence, the asymptotic quantum wave functions are exact in presence of a laser pulse. We have tested several discrete propagator methods for the TDSE in different gauges in a Born–Oppenheimer simulation of H2+ in a short, intense laser pulse. Our comparison of convergence between the same discretization methods for different gauges have demonstrated the superiority of the present Lagrangian adaptive grid method to treat the response of...

Electron interference in molecular photoionization by attosecond laser pulses.

Molecular photoionization by intense attosecond linearly and circularly polarized X-ray laser pulses is investigated from numerical solutions of time-dependent Schrödinger equations for the one-electron systems H2(+) and H3(++). Both momentum stripes and rings in photoelectron angular distributions are observed. The first with momentum intervals Δp(s)=2 π/R, where R is the molecular internuclear distance, results from interference of the coherent continuum scattering electron wave packets, which is shown to be insensitive to the laser polarization and wavelength. Diffraction of the directly ionized electrons leads to the momentum rings defined by the angle theta(p(r)R=cos(-1)(2nπ)/p(r)R between the electron momentum p(r) and the molecular internuclear R axis. These patterns are well described by multi-center interference models. Such complex patterns allow us to probe intermolecular structures.

Harmonic generation in a 1D model of H2 with single and double ionization

Single and double electron ionization probabilities are calculated numerically for the first three electronic states of a 1D model of H2 in an ultrashort (t < 10 fs) intense (I ≥ 1014 W cm−2) longwavelength (λ = 800 nm) laser pulse. High order harmonic generation spectra are also presented in intensity regimes where single and double ionizations dominate. Numerical results are presented at equilibrium and at the critical internuclear distance Rc where charge resonance enhanced ionization, CREI, occurs. Comparison of the harmonic generation plateaus at different intensities and the two different internuclear distances allows for an evaluation of the influence of electron correlation and symmetry of the two-electron wavefunction.

Generalized space translation and new numerical methods for time-dependent Schroedinger equations of molecules in intense laser fields

Abstract We present new numerical methods, based on a Generalized Space Translation representation (GST), for solving the Time-Dependent Schroedinger equation (TDSE), to treat the nonlinear, nonperturbative interaction of molecules with intense laser pulses. Adopting a Lagrangian, moving coordinate system, we generalize the previous Space Translation method (ST), used in atom–laser interaction problem. In this representation (gauge), the reference system is moving with the laser pulse so the classical movement of free particles in the field, i.e. in the asymptotic region where electron-molecule potentials are negligible but the laser field is still present, is exactly described. As a consequence, the asymptotic quantum wave functions are exact in presence of the laser pulse. To solve numerically the GST–TDSE, all standard discrete propagators for the time discretization and all efficient space discretization methods can be applied. In order to illustrate different possibilities for the choice of discretization methods, we have tested several types of split-operator (SO) and alternating direction implicit (ADI) methods combined with adaptive finite difference methods for the 3D Born–Oppenheimer simulation of H 2 + in a short intense laser pulse field. Our comparison of convergence between the same discretization methods for different gauges have demonstrated the superiority of the GST method. As examples, we present for the first time the ionization of H 2 + exposed to a circularly or linearly polarized intense laser pulse perpendicular to the internuclear axis by an exact 3D simulation.

Electron correlation and double ionization of a 1D H2 in an intense laser field

Single and double electron ionization probabilities are calculated for a one-dimensional (1D) H2 molecule from the exact numerical solution of the systems time-dependent Schrödinger equation (TDSE) at 800 nm and intensities 1013W cm-2<I<1015 W cm−2. Comparison is made for three electronic states, and of the molecule. Using the exact time-dependent densities of the three states allows for comparison of density functional probabilities obtained from exact one-electron densities with the exact probabilities.

Numerical methods for molecular time-dependent schrödinger equations — bridging the perturbative to nonperturbative regime

Publisher Summary This chapter explains numerical methods for molecular time-dependent Schrodinger equations. This chapter focuses on methods of solving accurately TDSEs to describe laser-molecule interactions from the perturbative to nonperturbative regime taking into consideration that modern laser technology is ever evolving towards controllable, more intense and shorter laser pulses. It emphasized the simplest one-electron H 2 + molecule with a single nuclear degree of freedom. This simple system is the benchmark as it allows for exact numerical treatment of competing electron-nuclear dynamics that is, including all non-Born-Oppenheimer corrections. The advantage of the space translation (ST) method is that asymptotically, at the edge of the usual large grids necessary for high intensity calculations, radiative (laser-molecule) interactions vanish so that one can project the exact numerical solutions at grid boundaries onto well known analytic free particle states, either electronic or nuclear.

Molecular photoelectron interference effects by intense circularly polarized attosecond x-ray pulses

We present photoelectron interference effects in molecular photoionization by intense circularly polarized attosecond X-ray laser pulses. Simulations are performed on single electron molecular systems, H2+

Numerical method for dynamic imaging of nonlinear polyatomic molecules

_{2}^{+}

Correlated Electron-Nuclear Motion Visualized Using a Wavelet Time-Frequency Analysis

, HeH2+, and H32+