The QQT is a quasi-quantum mechanical treatment of the collision between molecules. Instead of a partial wave expansion approach, it uses a kind of Feynman path integral method that exploits the path length differences originating from the different orientations of an anisotropic molecule. As a result, the QQT provides valuable physical insight while requiring very little computational effort. The current paper gives a systematic derivation of the QQT and explains its underlying principles. The expression for the scattering amplitude is shown to be self-consistent, without any normalisation factors, when the rotational energy level spacing is negligible. The constant curvature approximation that is presented makes the QQT conceptually even more simple, and its effect on the calculated differential cross-sections (DCSs) turns out to be small. As examples we present QQT calculations of the DCSs for Ne-CO(1) and He-NO(2), at collision energies of, respectively, 511 cm-1 and 514 cm-1. The anisotropy of the hard shell potential energy surface for Ne-CO in terms of the incoming de Broglie wavelength is about twice as large as for He-NO. This leads to state-to-state DCSs that have up to three maxima of comparable amplitude, instead of only one large maximum as is found for He-NO. The QQT results for these two applications are compared with results from close coupling calculations.