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The Journal of Physical Chemistry A

Mapped Grid Methods Applied to the Slow Variable Discretization–Enhanced Renormalized Numerov Approach

Juan Blandón, Gregory A. Parker, C. Madrid

Published Jan 21, 2016• 3 citations
Publisher

Abstract

We introduce a hyperspherical coordinate mapping procedure to the slow variable discretization-enhanced renormalized Numerov method that allows for more accurate and cost-effective calculations of cold and ultracold atom-dimer scattering. The mapping procedure allows optimization of the numerical grid point spacing by adjusting to the shape of the interaction potential. We show results for elastic scattering in HeH2 and compare the results to previous MOLSCAT calculations by Forrey et al. [ Phys. Rev. A 1999, 59, 2146 ].

DOI
10.1021/acs.jpca.5b10997

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We introduce a hyperspherical coordinate mapping procedure to the slow variable discretization-enhanced renormalized Numerov method that allows for more accurate and cost-effective calculations of cold and ultracold atom-dimer scattering. The mapping procedure allows optimization of the numerical grid point spacing by adjusting to the shape of the interaction potential. We show results for elastic scattering in HeH2 and compare the results to previous MOLSCAT calculations by Forrey et al. [ Phys. Rev. A 1999, 59, 2146 ].
In view of modelization of experiments involving cold atoms and molecules, we develop a method that allows us to calculate weakly bound levels of triatomic molecules. The method combines (1) the hyperspherical coordinates to describe interparticle motion in the three-body system, (2) the solution of the Schr\"odinger equation in two steps: determination of adiabatic states for a fixed hyper-radius and then solution of a set of coupled hyper-radial equations using the slow variable representation of Tolstikhin et al. [J. Phys. B: At. Mol. Opt. Phys. 29, L389 (1996)], (3) and a mapping procedure that reduces considerably the number of basis functions needed to represent wave functions of weakly bound levels. We apply the method to the three different systems: the helium trimer 4He3^{4}\mathrm{He}_{3}4He3​, isotopomers of the H3+\mathrm{H}_{3}{}^{+}H3​+ ion, and finally a model three-body problem involving three nucleons. For all these systems, we show that the suggested method provides accurate results.
Abstract The log-derivative (LOGD) and renormalized Numerov (RN) methods are popularly used in inelastic and reactive scattering calculations. The computational precision of two methods and their improved versions are compared in two-body diatomic and bimolecular collisions at low temperatures. In diatomic 40K-133Cs collision example the single-channel calculations show the solution-improved RN method has the advantage over others when large fixed propagating interval used. The relative error of local scattering length in different propagating sectors are explored. In multi-channel 40K-133Cs collisions the predicted positions of Feshbach resonances by using mapping grid points indicate strong potential-following character in LOGD methods, which is consistent with the analysis on local scattering length in single-channel calculations. In multiple open-channel calculations the elastic and inelastic scatterings and their dependence on collision energy are compared as well. In bimolecular 23Na87Rb-23Na87Rb long-range reactive scattering example the elastic and reactive rate coefficients are investigated. The original LOGD method displays surprisingly relatively high accuracy with less mapping grid points and a wide range of mapping parameters. By increasing the number of mapping grid points the improved LOGD exhibits definitely superiority on precision with proper mapping parameters. We demonstrate our consequent is applicable in other colliding systems.
Quantum mechanical calculations of ultralow-energy collision of Coulomb three-body systems in the hyperspherical elliptic coordinates are presented. The nonadiabatic coupling between the hyperradius and hyperangular variables is treated with the slow-variable discretization method in combination with the RRR-matrix propagation technique. For scattering state calculations, the two-dimensional matching procedure using Gailitis's method [M. Gailitis, J. Phys. B 9, 843 (1976); C. Noble and R. Nesbet, Comput. Phys. Commun. 33, 399 (1984)] is implemented to determine the boundary conditions between the internal and the asymptotic wave functions. This method is proved to be very efficient and gives very accurate results. Taking advantage of this method, we accurately calculate the scattering phase shifts and the scattering lengths of Coulomb three-body systems with mass ratio varying over several orders of magnitudes. We observed jumps of the scattering length from $\ensuremath{-}\ensuremath{\infty}$ to $\ensuremath{\infty}$ at certain mass ratios and monotonic decreases between two jumps. These are closely related to the binding energy of the highest bound state through Levinson's theorem [N. Levinson, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 25, 9 (1949)]. Our calculations provide a comprehensive perspective to the scattering length from the variation of mass ratio of Coulomb three-body systems.
The renormalized Numerov algorithm is applied to solving time-independent Schrödinger equation relating to atom-atom collisions at ultralow temperature. The proprieties of Feshbach resonance in 39K-133Cs collisions are investigated as an example. The results show that the renormalized Numerov method can give excellent results for ultracold colliding process. In contrast to improved log derivative method, the renormalized Numerov method displays a certain weakness in computational efficiency under the same condition. However, it is much stable in a wide range of grid step size. Hence a new propagating method is proposed by combining renormalized Numerov and logarithmic derivative method which can save computational time with a better accuracy. This algorithm can be used to solve close-coupling Schrödinger equation at arbitrary temperature for two-body collisions.
We investigate the prospects of combining a standard momentum space approach for ultracold three-body scattering with efficient coordinate space schemes to solve the underlying two-body problem. In many of those schemes, the two-body problem is numerically restricted up to a finite interparticle distance rb{r}_{\mathrm{b}}rb​. We analyze the effects of this two-body restriction on the two- and three-body level using pairwise square-well potentials that allow for analytic two-body solutions and more realistic Lennard-Jones van der Waals potentials to model atomic interactions. We find that the two-body ttt operator converges exponentially in rb{r}_{\mathrm{b}}rb​ for the square-well interaction. Setting rb{r}_{\mathrm{b}}rb​ to 2000 times the range of the interaction, the three-body recombination rate can be determined accurately up to a few percent when the magnitude of the scattering length is small compared to rb{r}_{\mathrm{b}}rb​, while the position of the lowest Efimov features is accurate up to the percent level. In addition, we find that with the introduction of a momentum cutoff, it is possible to determine the three-body parameter in good approximation even for deep van der Waals potentials.
Comparison between three different numerical techniques for solving a coupled channel Schrödinger equation is presented. The benchmark equation, which describes the collision between two ultracold atoms, consists of two channels, each containing the same diagonal Lennard-Jones potential, one of positive and the other of negative energy. The coupling potential is of an exponential form. The methods are (i) a recently developed spectral type integral equation method based on Chebyshev expansions, (ii) a finite element expansion, and (iii) a combination of an improved Numerov finite difference method and a Gordon method. The computing time and the accuracy of the resulting phase shift is found to be comparable for methods (i) and (ii), achieving an accuracy of ten significant figures with a double precision calculation. Method (iii) achieves seven significant figures. The scattering length and effective range are also obtained.