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Cusp conditions for two electrons atoms in the Hylleraas aproximation

In this work it was shown how cusp conditions in radial coordinates are implied in the fundamental idea of the R12 method used in electronic structure calculations. We also derived the cusp conditions in Hylleraas coordinates through Hamiltonian singularities analysis. The functions described in literature which depend on these variables were investigated and it was found that these functions do not necessarily satisfy the cusp conditions, although they return good energy results with few parameters. Starting from the cusp conditions derived in the Hylleraas coordinates, three restrictions were established for the wavefunctions. An example of this correct cusp behavior for a particular wavefunctions is given. This issue is of great importance since the cusp conditions guarantee a better convergence for numerical results.