We present a time-dependent hyperspherical, wave packet method for calculating three atom state-to-state S-matrix elements.

The wave packet is propagated in time using adiabatically adjusting, principal axes hyperspherical (APH) coordinates that treat all arrangement channels equivalently, allowing the simultaneous analysis of the products in all three arrangement channels.

We take advantage of the symmetry of the potential energy surface and decompose the initial wave packet into its component irreducible representations, propagating each component separately.

Each packet is analyzed by projecting it onto the hyperspherical basis at a fixed, asymptotic hyperradius, and

irreducible representation dependent S-matrix elements are obtained by matching the hyperspherical projections to symmetry-adapted Jacobi coordinate boundary conditions.

We obtain arrangement channel-dependent S-matrix elements as linear combinations of the irreducible representation dependent elements.

We derive and implement a new three-dimensional Sylvester-like algorithm that reduces the number of multiplications required to apply the Hamiltonian to the wave packet, dramatically increasing the computational efficiency.

A convergence study is presented to show the behavior of the results as the initial parameters are varied and to determine the values of those parameters that give accurate results.

State-to-state H+H2 and F+H2 results for zero total angular momentum are presented and show excellent agreement with time-independent benchmark results.