for “ Quantum reactive scattering of ultracold NH ( X 3 Σ − ) radicals in a magnetic trap ”

QUANTUM REACTIVE SCATTERING METHOD In this section, we outline the quantum scattering algorithm used to calculate the total NH + NH reactive cross sections. Before discussing our approach in detail, let us recall the basics of already established single-arrangement reactive scattering methods. Conventional single-arrangement algorithms combine a standard inelastic scattering calculation, e.g., renormalized Numerov or log-derivative propagation [1], with complex " capture " reactive boundary conditions. These boundary conditions allow collision flux to disappear into reactive channels at a sufficiently small value of the radial coordinate R. The corresponding solutions to the coupled-channels equations become complex-valued and, as a consequence, the scattering S-matrix is no longer unitary. The deviation from unitarity is then used to extract the reactive collision cross sections [2, 3]. Here we employ a different strategy and first calculate, for a given interval [R 0 , R n ], two sets of real-valued solutions to the coupled-channels equations. This is most conveniently done using the renormalized Numerov propagation algorithm [1]. The " regular " solutions F satisfy F (R 0) = 0 and F (R n) = 1, and the " irregular " solutions G are defined by G(R 0) = 1 and G(R n) = 0. A suitable linear combination of these functions, determined by the scattering boundary conditions, provides the final scattering wave function. We note that the irregular functions are equivalent to the regular solutions on the reversed radial grid, i.e. the regular solutions obtained by propagating from R n to R 0. It is, however, also possible to obtain both sets of solutions in a single propagation run, as detailed in Ref. [4]. The regular and irregular functions form a complete set of linearly independent solutions for a given energy E. Hence, once the propagation is completed, we may enforce all possible boundary conditions to construct all possible scattering wave functions for a given Hamiltonian and energy. This represents a major advantage over conventional reactive scattering (log-derivative propagation) methods, for which the boundary conditions must be imposed at the start of the propagation [2, 3]. The fact that the boundary conditions can be applied after our propagation routine also implies that the scattering code can be parallelized to speed up the calculation. Such a parallelization may be realized by solving the coupled-channels equations