A discrete time-dependent method for metastable atoms and molecules in intense fields

The full-dimensional time-dependent Schrodinger equation for the electronic dynamics of single-electron systems in intense external fields is solved directly using a discrete method. Our approach combines the finite-difference and Lagrange mesh methods. The method is applied to calculate the quasienergies and ionization probabilities of atomic and molecular systems in intense static and dynamic electric fields. The gauge invariance and accuracy of the method is established. Applications to multiphoton ionization of positronium, the hydrogen atom and the hydrogen molecular ion are presented. At very high laser intensity, above the saturation threshold, we extend the method using a scaling technique to estimate the quasienergies of metastable states of the hydrogen molecular ion. The results are in good agreement with recent experiments. (C) 2004 American Institute of Physics.

Combined R-matrix eigenstate basis set and finite-difference propagation method for the time-dependent Schrodinger equation: The one-electron case

In this work we present the theoretical framework for the solution of the time-dependent Schrödinger equation (TDSE) of atomic and molecular systems under strong electromagnetic fields with the configuration space of the electron’s coordinates separated over two regions; that is, regions I and II. In region I the solution of the TDSE is obtained by an R-matrix basis set representation of the time-dependent wave function. In region II a grid representation of the wave function is considered and propagation in space and time is obtained through the finite-difference method. With this, a combination of basis set and grid methods is put forward for tackling multiregion time-dependent problems. In both regions, a high-order explicit scheme is employed for the time propagation. While, in a purely hydrogenic system no approximation is involved due to this separation, in multielectron systems the validity and the usefulness of the present method relies on the basic assumption of R-matrix theory, namely, that beyond a certain distance (encompassing region I) a single ejected electron is distinguishable from the other electrons of the multielectron system and evolves there (region II) effectively as a one-electron system. The method is developed in detail for single active electron systems and applied to the exemplar case of the hydrogen atom in an intense laser field.

SOLVING THE INHOMOGENEOUS SCHRODINGER-EQUATION ON A BIDIRECTIONAL PIPELINE OF TRANSPUTERS

In this paper we describe the design of a parallel solution of the inhomogeneous Schrodinger equation, which arises in the construction of continuum orbitals in the R-matrix theory of atomic continuum processes. A prototype system is described which has been programmed in occam2 and implemented on a bi-directional pipeline of transputers. Some timing results for the prototype system are presented, and the development of a full production system is discussed.

Measuring math anxiety in Italian college and high school students: Validity, reliability and gender invariance of the Abbreviated Math Anxiety Scale (AMAS)

Given that the ability to manage numbers is essential in a modern society, mathematics anxiety – which has been demonstrated to have unfortunate consequences in terms of mastery of math – has become a subject of increasing interest, and the need to accurately measure it has arisen. One of the widely employed scales to measure math anxiety is the Abbreviated Math Anxiety Scale (AMAS) (Hopko, Mahadevan, Bare & Hunt, 2003). The first aim of the present paper was to confirm the factor structure of the AMAS when administered to Italian high school and college students, and to test the invariance of the scale across educational levels. Additionally, we assessed the reliability and validity of the Italian version of the scale. Finally, we tested the invariance of the AMAS across genders. The overall findings provide evidence for the validity and reliability of the AMAS when administered to Italian students.

Modeling relativistic soliton interactions in overdense plasmas: A perturbed nonlinear Schrodinger equation framework

We investigate the dynamics of localized solutions of the relativistic cold-fluid plasma model in the small but finite amplitude limit, for slightly overcritical plasma density. Adopting a multiple scale analysis, we derive a perturbed nonlinear Schrodinger equation that describes the evolution of the envelope of circularly polarized electromagnetic field. Retaining terms up to fifth order in the small perturbation parameter, we derive a self-consistent framework for the description of the plasma response in the presence of localized electromagnetic field. The formalism is applied to standing electromagnetic soliton interactions and the results are validated by simulations of the full cold-fluid model. To lowest order, a cubic nonlinear Schrodinger equation with a focusing nonlinearity is recovered. Classical quasiparticle theory is used to obtain analytical estimates for the collision time and minimum distance of approach between solitons. For larger soliton amplitudes the inclusion of the fifth-order terms is essential for a qualitatively correct description of soliton interactions. The defocusing quintic nonlinearity leads to inelastic soliton collisions, while bound states of solitons do not persist under perturbations in the initial phase or amplitude

Generation of macroscopic superposition states with small nonlinearity

We suggest a scheme to generate a macroscopic superposition state ("Schrodinger cat state") of a free-propagating optical field using a beam splitter, homodyne measurement, and a very small Kerr nonlinear effect. Our scheme makes it possible to reduce considerably the required nonlinear effect to generate an optical cat state using simple and efficient optical elements.

Dissociative ionization of molecules in intense laser fields

Accurate and efficient grid based techniques for the solution of the time-dependent Schrodinger equation for few-electron diatomic molecules irradiated by intense, ultrashort laser pulses are described. These are based on hybrid finite-difference, Lagrange mesh techniques. The methods are applied in three scenarios, namely H-2(+) with fixed internuclear separation, H-2(+) with vibrating nuclei and H-2 with fixed internuclear separation and illustrative results presented.

Dynamic tunnelling ionization of H-+(2) in intense fields

Intense-field ionization of the hydrogen molecular ion by linearly polarized light is modelled by direct solution of the fixed-nuclei time-dependent Schrodinger equation and compared with recent experiments. Parallel transitions are calculated using algorithms which exploit massively parallel computers. We identify and calculate dynamic tunnelling ionization resonances that depend on laser wavelength and intensity, and molecular bond length. Results for lambda similar to 1064 nm are consistent with static tunnelling ionization. At shorter wavelengths lambda similar to 790 nm large dynamic corrections are observed. The results agree very well with recent experimental measurements of the ion spectra. Our results reproduce the single peak resonance and provide accurate ionization rate estimates at high intensities. At lower intensities our results confirm a double peak in the ionization rate as the bond length varies.

High-energy cutoff in the spectrum of strong-field nonsequential double ionization

Electron energy distributions of singly and doubly ionized helium in an intense 390 nm laser field have been measured at two intensities (0.8 PW/cm(2) and 1.1 PW/cm(2), where PW equivalent to 10(15) W/cm(2)). Numerical solutions of the full-dimensional time-dependent helium Schrodinger equation show excellent agreement with the experimental measurements. The high-energy portion of the two-electron energy distributions reveals an unexpected 5U(p) cutoff for the double ionization (DI) process and leads to a proposed model for DI below the quasiclassical threshold.

Plane wave and Coulomb asymptotics

A simple plane wave solution of the Schrodinger-Helmholtz equation is a quantum eigenfunction obeying both energy and linear momentum correspondence principles. Inclusion of the outgoing wave with scattering amplitude f asymptotic development of the plane wave, we show that there is a problem with angular momentum when we consider forward scattering at the point of closest approach and at large impact parameter given semiclassically by (l + 1/2)/k where l is the azimuthal quantum number and may be large (J. Leech et al., Phys. Rev. Lett. 88. 257901 (2002)). The problem is resolved via non- uniform, non-standard analysis involving the Heaviside step function, unifying classical, semiclassical and quantum mechanics, and the treatment is extended to the case of pure Coulomb scattering.

Boundary conditions in the simplest model of linear and second harmonic magneto-optical effects

This paper is concerned with linear and nonlinear magneto- optical effects in multilayered magnetic systems when treated by the simplest phenomenological model that allows their response to be represented in terms of electric polarization, The problem is addressed by formulating a set of boundary conditions at infinitely thin interfaces, taking into account the existence of surface polarizations. Essential details are given that describe how the formalism of distributions (generalized functions) allows these conditions to be derived directly from the differential form of Maxwell's equations. Using the same formalism we show the origin of alternative boundary conditions that exist in the literature. The boundary value problem for the wave equation is formulated, with an emphasis on the analysis of second harmonic magneto-optical effects in ferromagnetically ordered multilayers. An associated problem of conventions in setting up relationships between the nonlinear surface polarization and the fundamental electric field at the interfaces separating anisotropic layers through surface susceptibility tensors is discussed. A problem of self- consistency of the model is highlighted, relating to the existence of resealing procedures connecting the different conventions. The linear approximation with respect to magnetization is pursued, allowing rotational anisotropy of magneto-optical effects to be easily analyzed owing to the invariance of the corresponding polar and axial tensors under ordinary point groups. Required representations of the tensors are given for the groups infinitym, 4mm, mm2, and 3m, With regard to centrosymmetric multilayers, nonlinear volume polarization is also considered. A concise expression is given for its magnetic part, governed by an axial fifth-rank susceptibility tensor being invariant under the Curie group infinityinfinitym.

Tools for analysing configuration interaction wavefunctions

The configuration interaction (CI) approach to quantum chemical calculations is a well-established means of calculating accurately the solution to the Schrodinger equation for many-electron systems. It represents the many-body electron wavefunction as a sum of spin-projected Slater determinants of orthogonal one-body spin-orbitals. The CI wavefunction becomes the exact solution of the Schrodinger equation as the length of the expansion becomes infinite, however, it is a difficult quantity to visualise and analyse for many-electron problems. We describe a method for efficiently calculating the spin-averaged one- and two-body reduced density matrices rho(psi)((r) over bar; (r) over bar' ) and Gamma(psi)((r) over bar (1), (r) over bar (2); (r) over bar'(1), (r) over bar'(2)) of an arbitrary CI wavefunction Psi. These low-dimensional functions are helpful tools for analysing many-body wavefunctions; we illustrate this for the case of the electron-electron cusp. From rho and Gamma one can calculate the matrix elements of any one- or two-body spin-free operator (O) over cap. For example, if (O) over cap is an applied electric field, this field can be included into the CI Hamiltonian and polarisation or gating effects may be studied for finite electron systems. (C) 2003 Elsevier B.V. All rights reserved.

Ionization dynamics of laser-driven H-2(+)

We set out aspects of a numerical algorithm used in solving the full-dimensionality time-dependent Schrodinger equation describing the electronic motion of the hydrogen molecular ion driven by an intense, linearly polarized laser pulse aligned along the molecular axis. This algorithm has been implemented within the fixed inter-nuclear separation approximation in a parallel computer code, a brief summary of which is given. Ionization rates are calculated and compared with results from other methods, notably the time-independent Floquet method. Our results compare very favourably with the precise predictions of the Floquet method, although there is some disagreement with other wavepacket calculations. Visualizations of the electron dynamics are also presented in which electron rescattering is observed.

Raman and ab initio studies of simple and binary 1-Alkyl-3-methylimidazolium ionic liquids

Raman spectra of the ionic liquids, 1-butyl-3-methylimidazolium hexafluorophosphate ([C(4)mim][PF(6)]), 1-hexyl-3-methylimidazolium chloride ([C(6)mim]Cl), and 1-hexyl-3-methylimidazolium hexafluorophosphate ([C(6)mim][PF(6)]), and binary mixtures thereof, have been assigned using ab initio MP2 calculations. The previously reported anti and gauche forms of the [C(4)mim](+) cation have been observed, and this study reveals this to be a general feature of the long-chain I-alkyl derivatives. Analysis of mixtures Of [C(6)mim]Cl and [C(6)mim][PF(6)] has provided information on the nature of the hydrogen bonding between the imidazolium headgroup and the anions, and the invariance of the essentially 50:50 mixture of the predominant conformers informs on the nature of glass formation in these systems.