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

Abstract

Articles you may be interested inVariable 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.

Selective adsorption resonances: Quantum and stochastic approaches

“…These approximations are relatively accurate, though only the first or second neighboring points, respectively, are taken into account. Of course, they can be improved by considering higher order approximations [242,243], however the convergence still follows a power series law precisely because of the inherent locality of this method. This slow convergence is related to the nonlocal character of the kinetic operator; the accurate evaluation of its action on the wave function at a given point of the grid requires to consider the value of the wave function at any other site across the grid.…”

Dispersion fitted finite difference method with applications to molecular quantum mechanics

“…[10][11][12][13][14][15] We can also view them as arising from fits of the FD dispersion relation, Eq. [10][11][12][13][14][15] We can also view them as arising from fits of the FD dispersion relation, Eq.…”

“…6,7 In molecular quantum mechanics, our main interest, grid representations are well suited to iterative methods, e.g., the Lanczos method 8 for bound states and wave packet methods for dynamics, 9 because they lead to favorable computational scalings. [10][11][12][13] In current molecular quantum mechanics, FD methods are less commonly used than pseudospectral methods. These repeated Hy products are the computational bottleneck.…”

An improved Lobatto discrete variable representation by a phase optimisation and variable mapping method

“…Finite difference (FD) methods is another class of popular numerical methods for solving the Schrödinger equation [36][37][38], which is convenient for paralleling and more suitable for potentials involving singularities. Usually low order FD methods require rather dense grids to achieve numerical results with high accuracy or one turns with higher order FD methods for accurate numerical results.…”