  A new approach is described to the evaluation of the S-matrix in
three-dimensional atom-diatom reactive quantum scattering theory. The theory is
developed based on natural collision coordinates where progress along the
reaction coordinate can be viewed as fulfilling the same role as time in a
time-dependent formulation. By writing the full wavefunction in coupled-channel
form it is proved that the 3D quantum reactive scattering problem can be
treated in the same way as an inelastic single-arrangement problem. In
particularly, two types of coupled-channel representations, which are reduced
to two different systems of coupled first order ordinary differential equations
describing the inelastic scattering, are used. The first system of coupled
differential equations is constructed on a set of points (grid) of the
coordinate reaction curve after solution of many 1D Schroedinger problems in
the directions normal to the reaction coordinate. The second expression for
inelastic scattering is found using exactly solvable nonstationary 1D
Schroedinger equation (etalon equation method), which is introduced for
describing the localization properties of the full wavefunction along the curve
of coordinate reaction. In this case we avoid a large amount of computation
involved in solving the 1D Schroedinger problem along the reaction coordinate
by using a slightly difficult initial conditions for the inelastic scattering
equations. In both cases by solving the system of coupled first order ordinary
differential equations, the full wavefunction and all S-matrix elements are
obtained simultaneously without further calculations. Our analysis shows that
the methods we have developed constitute the simplest algorithms for computing
the reactive scattering S-matrices.
