Mapped grid methods for long-range molecules and cold collisions

Abstract

The paper discusses ways of improving the accuracy of numerical calculations for vibrational levels of diatomic molecules close to the dissociation limit or for ultracold collisions, in the framework of a grid representation. In order to avoid the implementation of very large grids, Kokoouline et al. [J. Chem. Phys. 110, 9865 (1999)] have proposed a mapping procedure through introduction of an adaptive coordinate x subjected to the variation of the local de Broglie wavelength as a function of the internuclear distance R. Some unphysical levels ("ghosts") then appear in the vibrational series computed via a mapped Fourier grid representation. In the present work the choice of the basis set is reexamined, and two alternative expansions are discussed: Sine functions and Hardy functions. It is shown that use of a basis set with fixed nodes at both grid ends is efficient to eliminate "ghost" solutions. It is further shown that the Hamiltonian matrix in the sine basis can be calculated very accurately by using an auxiliary basis of cosine functions, overcoming the problems arising from numerical calculation of the Jacobian J(x) of the R-->x coordinate transformation.

Short-pulse photoassociation in rubidium below theD1line

“…(1)(2)(3), are represented on a grid employing a mapped grid method [30,36]. This allows to employ a fairly extended grid (R max ≈ 18500 a 0 ) with a comparatively small number of grid points (N grid = 1027).…”

Optimizing the photoassociation of cold atoms by use of chirped laser pulses

“…The time-dependent Schrödinger equation corresponding to the two-electronic coupled channels (a 3 Σ + u (6s+6s) and 0 − g (6s + 6p 3/2 )) was solved using a propagation method [27,28], and, due to a Mapped Sine Grid representation [29], it was possible to take as the initial state in the photoassociation a stationary collision state corresponding to a temperature of about 50 µK. The chirped pulse has been designed in view of creating a vibrational wavepacket in the excited state, which, after the pulse, moves to shorter internuclear distances and is focussing at the barrier of the double well potential, thus preparing a good initial state for the stabilization process into low vibrational levels of the a 3 Σ + u (6s + 6s) potential.…”

Optimization of generalized multichannel quantum defect reference functions for Feshbach resonance characterization

“…We briefly recall below the main features of the MFGH approach combined with an optical potential (MFGH+OP), which has been extensively described in several previous papers of the group [32,33,34,38,39]. We want to solve the system of two coupled Schrödinger equations for the (A, b) system expressed in the diabatic representation, which is conveniently written in a matrix form according to:…”