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.
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.
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.
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.
Collisions between atoms and molecules govern how energy is exchanged and redistributed in the gas phase, underpinning phenomena from spectral line broadening and interstellar molecular excitation to the cooling and trapping of molecules. A quantitative account of these collisions requires connecting the theoretical description of the interaction between the collision partners to measurable quantities such as cross sections. However, scattering theory as presented in quantum mechanics textbooks does not readily translate into numerical methods. Here, we formulate the scattering problem in Dirac notation, as customary in present-day quantum mechanics courses, and show how to build a practical numerical framework from it. We derive the "coupled channel" representation of the scattering problem with correct asymptotic behaviour and introduce Numerov's method to obtain time-independent wavefunctions, from which we calculate cross sections, the observables measured in scattering experiments. Starting from single channel scattering, we build up to the realistic collision of a ground-state atom with a diatomic molecule.
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 ].
Results for elastic atom-atom scattering are obtained as a first practical application of RmatReact, a new code for generating high-accuracy scattering observables from potential energy curves. RmatReact has been created in response to new experimental methods which have paved the way for the routine production of ultracold (μK) atoms and molecules, and hence the experimental study of chemical reactions involving only a small number of partial waves. Elastic scattering between argon atoms is studied here. There is an unresolved discrepancy between different Ar2 potential energy curves which give different numbers of vibrational bound states and different scattering lengths for the Ar2 dimer. Depending on the number of bound states, the scattering length is either large and positive or large and negative. Scattering observables, specifically the scattering length, effective range, and partial and total cross-sections, are computed at low collision energies and compared to previous results. In general, good agreement is obtained, although our full scattering treatment yields resonances which are slightly lower in energy and narrower than previous determinations using the same potential energy curve.
Results for elastic atom-atom scattering are obtained as a first practical application of RmatReact, a new code for generating high-accuracy scattering observables from potential energy curves. RmatReact has been created in response to new experimental methods which have paved the way for the routine production of ultracold (μK) atoms and molecules, and hence the experimental study of chemical reactions involving only a small number of partial waves. Elastic scattering between argon atoms is studied here. There is an unresolved discrepancy between different Ar2 potential energy curves which give different numbers of vibrational bound states and different scattering lengths for the Ar2 dimer. Depending on the number of bound states, the scattering length is either large and positive or large and negative. Scattering observables, specifically the scattering length, effective range, and partial and total cross-sections, are computed at low collision energies and compared to previous results. In general, good agreement is obtained, although our full scattering treatment yields resonances which are slightly lower in energy and narrower than previous determinations using the same potential energy curve.