Periodic Orbit Assisted Grid Solutions of Time Dependent Schroedinger Equation

High order finite difference algorithms for solving the Schroedinger equation in molecular dynamics. I.

        Raul Guantes and Stavros C. Farantos

       Institute of Electronic Structure and Laser
           Foundation for Research and Technology - Hellas,


      Department of Chemistry
        University of Crete
         Iraklion, Crete 711 10, Greece


The view of considering global Pseudospectral methods (Sinc and Fourier) as the infinite order limit of local Finite Difference methods, and vice versa, Finite Difference as a certain sum acceleration of the Pseudospectral methods is exploited to investigate high order Finite Difference algorithms for solving the Schroedinger equation in Molecular Dynamics. A Morse type potential for Iodine molecule is used to compare the eigenenergies obtained by a Sinc Pseudospectral method and a high order Finite Difference approximation of the action of the kinetic energy operator on the wave function. 2-D and 3-D model potentials are employed to compare spectra obtained by Fast Fourier Transform techniques and variable order Finite Difference. It is shown, that it is not needed to employ very high order approximations of Finite Differences to reach the numerical accuracy of Pseudospectral techniques. This, in addition to the fact that for complex configuration geometries and high dimensionality, local methods require less memory and are faster than Pseudospectral methods, put Finite Difference among the effective algorithms for solving the Schroedinger equation in realistic molecular systems.

Published in The Journal of Chemical Physics, Vol. 111, 10827, 1999

High order finite difference algorithms for solving the Schroedinger equation in molecular dynamics. II. Periodic variables.

Raul Guantes and Stavros C. Farantos


Variable high order Finite Difference methods are applied to calculate the action of molecular Hamiltonians on the wave function using centered equi-spaced stencils, mixed centered and one-sided stencils and periodic Chebyshev and Legendre grids for the angular variables. Results from one dimensional model Hamiltonians and the three dimensional spectroscopic potential of SO2 demonstrate that as the order of Finite Difference approximations of the derivatives increases the accuracy of Pseudospectral methods is approached in a regular manner. The high order limit of Finite Differences to Fourier and general orthogonal polynomial Discrete Variable Representation methods is analytically and numerically investigated.

Published in The Journal of Chemical Physics, Vol. 113,  10429, 2000

A method for solving the molecular Schrödinger equation in Cartesian coordinates via angular momentum projection operators

J. Suarez, S.C. Farantos, S. Stamatiadis, L. Lathouwers


A method for solving the Schrödinger equation of N-atom molecules in 3N − 3 Cartesian coordinates usually defined by Jacobi vectors is presented. The separation and conservation of the total angular momentum are obtained not by transforming the Hamiltonian in internal curvilinear coordinates but instead, by keeping the Cartesian formulation of the Hamiltonian operator and projecting the initial wavefunction onto the proper irreducible representation angular momentum subspace. The increased number of degrees of freedom from 3N − 6 to 3N − 3, compared to previous methods for solving the Schrödinger equation, is compensated by the simplicity of the kinetic energy operator and its finite difference representations which result in sparse Hamiltonian matrices. A parallel code in Fortran 95 has been developed and tested for model potentials of harmonic oscillators. Moreover, we compare data obtained for the three-dimensional hydrogen molecule and the six-dimensional water molecule with results from the literature. The availability of large clusters of computers with hundreds of CPUs and GBytes of memory, as well as the rapid development of distributed (Grid) computing, make the proposed method, which is unequivocally highly demanding in memory and computer time, attractive for studying Quantum Molecular Dynamics.

Published in Computer Physics Communication, Vol. 180,  2025, 2009


Tue Nov 30 15:36:34 EET 2010