Amplitude scattering matrix symmetry

The formalism of electron-atom scattering has been extensively dealt with elsewhere./27,28,29/ We shall only recall its main features here. Because of the assumed spherical symmetry, the partial-wave scattering approach is convenient. Namely, an incoming spherical wave h[2 kr) y/ (r), (—can scatter only into the outgoing spherical wave hf kr) Y/"(r) (here hf and hf2 are Hankel functions of the first and second kinds, k = 2n/h(2mE) i, E is the kinetic energy and r — r ). This occurs with amplitude t( (f is an element of the diagonal atomic <-matrix), which is related to the phase shifts 6l through... 

Due to the muffin-tin approximation of the local multi-centre potential, the applicability of the scattered-wave (SW) method is limited to relatively small and high-symmetry clusters and molecules. On the other hand, it is the prerequisite for generalizing the in-out integration scheme of atomic one-electron wavefiinctions to the multi-centre case. This approach furnishes the multi-centre electronic structure in a particular atom sphere in its single-centre representation. Thus atomic transition-matrix elements can be readily rescaled in terms of near-nucleus electron amplitudes. 

The scattering amplitudes or density matrix elements required for the description of the excited states are given with respect to a suitable reference frame. The optimum solution is to use a frame-independent parametrisation of the dynamical information, closely related to the symmetry process. 

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