Attila Askar

Explicit integration method for the time‐dependent Schrodinger equation for collision problems

To date, only the implicit (Crank–Nicholson) integration method has ben used for numerical integration of the Schrodinger equation for collision processes. The standard explicit methods are known to be unstable and a high price is paid for the implicit method due to the inversion of the large matrices involved. Furthermore, the method is prohibitive in more than two dimensions due to restrictions on memory and large computation times. An explicit method (i.e., a method which doesn’t require the solution of simultaneous equations) is presented, and is shown to be stable in n dimensions to the same order of accuracy as the implicit method with the unitarity being secured to two orders higher accuracy than that for the wave function.

Quantum fluid dynamics in the Lagrangian representation and applications to photodissociation problems

This paper considers the practical utility of quantum fluid dynamics (QFD) whereby the time-dependent Schrodinger’s equation is transformed to observing the dynamics of an equivalent “gas continuum.” The density and velocity of this equivalent gas continuum are respectively the probability density and the gradient of the phase of the wave function. The numerical implementation of the QFD equations is carried out within the Lagrangian approach, which transforms the solution of Schrodinger’s equation into following the trajectories of a set of mass points, i.e., subparticles, obtained by discretization of the continuum equations. The quantum dynamics of the subparticles which arise in the present formalism through numerical discretization are coupled by the density and the quantum potential. Numerical illustrations are performed for photodissociation of NOCl and NO2 treated as two-dimensional models. The dissociation cross sections σ(ω) are evaluated in the dramatically short CPU times of 33 s for NOCl and ...

A structural model of a micropolar continuum

Abstract In their crystal lattice theory, Born-Von Karman have shown that the long wave approximation to their model yields the equations of the classical theory of elasticity. The Born-Von Karman model assumed that the interaction between the particles can be characterized by a simple spring which allows extensions only. Also, in their model, the dimensions of the particles are shrunk to zero thus resulting in non-orientable mathematical points. In this paper, a two dimensional model composed of orientable points, joined by extensible and flexible rods is presented in order to explain the foundations of the Cosserat [1] or the micropolar continuum [2]. As the long wave approximation from this model, one gets the coupled displacement and microration equations of the micropolar medium, similar to those given by Eringen and Suhubi[2, 3] and by Mindlin[4, 5].

Multidimensional wave packet dynamics within the fluid dynamical formulation of the Schrodinger equation

This paper explores the quantum fluid dynamical (QFD) representation of the time-dependent Schrodinger equation for the motion of a wave packet in a high dimensional space. A novel alternating direction technique is utilized to single out each of the many dimensions in the QFD equations. This technique is used to solve the continuity equation for the density and the equation for the convection of the flux for the quantum particle. The ability of the present scheme to efficiently and accurately describe the dynamics of a quantum particle is demonstrated in four dimensions where analytical results are known. We also apply the technique to the photodissociation of NOCl and NO2 where the systems are reduced to two coordinates by freezing the angular variable at its equilibrium value.

Finite element methods for reactive scattering

Abstract The finite element method is applied to collinear reactive scattering problems. In this way no basis set expansion of the wave function is required and a direct solution of the two-dimensional partial differential equation is achieved. It is shown how to generally formulate this approach and achieve fast and accurate results. As a test calculation the method was applied to H + H 2 , yielding excellent agreement with close coupling results. Since no basis sets are used in the finite element calculation, no question of basis set convergence or closed channel behavior arises. Some discussion on applications to higher dimensions is also included.

Finite element method for bound state calculations in quantum mechanics

The finite element method, which in other fields has replaced finite difference and variational methods, is applied to the radially symmetric case of the hydrogen atom. The method is shown to have computational advantages over the finite difference and Rayleigh−Ritz methods. (AIP)

Long time scale molecular dynamics subspace integration method applied to anharmonic crystals and glasses

A subspace dynamics method is presented to model long time dynamical events. The method involves determining a set of vectors that span the subspace of the long time dynamics. Specifically, the vectors correspond to real and imaginary low frequency normal modes of the condensed phase system. Most importantly, the normal mode derived vectors are only used to define the subspace of low frequency motions, and the actual time dependent dynamics is fully anharmonic. The resultant projected set of Newton’s equations is numerically solved for the subspace motions. Displacements along the coordinates outside the subspace are then constrained during the integration of the equations of motion in the reduced dimensional space. The method is different from traditional constraint methods in that it can systematically deduce and remove both local and collective high frequency motions of the condensed phase system with no a priori assumptions. The technique is well suited to removing large numbers of degrees of freedom,...

A comparison between finite element methods and spectral methods as applied to bound state problems

The finite element and spectral methods are applied to two‐dimensional bound state problems. A comparison of the spectral method, which requires a global basis set expansion of the wave functions, and the finite element method, which requires no such such expansion, is presented. A procedure is given for formulating the finite element approach and for achieving fast and accurate results. The convergence of the finite element calculations is considered and shown to be well behaved. A discussion of the extension of the finite element method to higher dimensions is also included.

Variational time‐dependent perturbation scheme based on the hydrodynamic analogy to Schrödinger's equation

A variational principle correct through the second order is presented for time‐dependent perturbations about a bound state. The formulation is based on the hydrodynamic analogy to quantum mechanics and is obtained by linearization of a general principle within the spirit of the acoustic approximation in classical fluid mechanics. The method requires the knowledge of the wavefunction for the unperturbed state only, unlike in the standard procedure of expansion in terms of a complete set of unperturbed wavefunctions. In dealing with the changes in the amplitude and phase of the wavefunction which are nonoscillatory, the method is expected to offer advantages over the usual variational perturbation methods which utilize the oscillatory wavefunctions directly. The method is applied to the calculation of the dynamic polarizability of the hydrogen atom and reasonable accuracy is obtained for an overly simple choice of basis functions.

The use of global wavefunctions in scattering theory

Abstract This paper considers the potential savings associated with the use of global wavefunctions in scattering theory. The normal close coupling approach exp