Method of Calculating Cross Sections for Molecular Collisions

The calculation of cross sections for slow inelastic collisions of molecules, such as occur in chemical reactions, shows an interesting combination of classical and quantum-mechanical features. The de Broglie wavelength is small compared with the range of the intermolecular potential so that there exists a relatively well-defined quasi-classical orbit; but in some cases the coupling constant is small compared with unity, and this gives the problem certain quantum-mechanical characteristics. On account of the condition relating to the de Broglie wavelength, a calculation of the total collision cross section requires the knowledge of the partial waves for a large number of values of the angular momentum lh/. A method is devised to suit these different features. The WKB method lends itself well to this application, with a quasi-classical amplitude and a quantum-mechanical phase for the wave function. We also use the concept of an impact parameter b rather than the orbital quantum number l: this is well suited to describing a quasi-classical orbit, and hence for calculating the total cross section in a case like the present one. As an example of the method, the cross section for excitation of the first vibrational state of a hydrogen molecule resulting from collision with another hydrogen molecule is calculated.