Non-orthogonal basis functions in the solution of diffusional problems

The diffusion equation in presence of strong potential gradients can be conveniently solved by means of a suitable set of non-orthogonal basis functions that mimic the asymptotic behaviour of the eigensolutions of the time evolution operator. This method is discussed in the context of the calculation of spectroscopic observables by means of the Lanczos algorithm, that is by representing the spectral density as a continued fraction. The modified Lanczos algorithm, which takes into account the non-orthogonality of the basis functions, is analysed in detail. In order to test the performance of the new procedure, the planar rotor subject to a cosine potential is considered, and the set of non-orthogonal functions having the correct asymptotic behaviour is derived in both cases of single and double minimum. The results of the numerical calculations clearly show the advantages of the proposed method when dealing with systems characterized by hindered motions.