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

* and*

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

**Abstract**

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

**Abstract**

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 SO_{2} 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

**Abstract**

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