Boundary Solutions of the Two-electron Schr\"{o}dinger Equation at Two-particle Coalescences of the Atomic Systems

The limit relations for the partial derivatives of the two-electron atomic wave functions at the two-particle coalescence lines have been obtained numerically using accurate CFHHM wave functions. The asymptotic solutions of the proper two-electron Schr\"{o}dinger equation have been derived for both electron-nucleus and electron-electron coalescence. It is shown that the solutions for the electron-nucleus coalescence correspond to the ground and singly excited bound states, including triplet ones. The proper solutions at small distances $R$ from the triple coalescence point were presented as the second order expansion on $R$ and $\ln R$. The vanishing of the Fock's logarithmic terms at the electron-nucleus coalescence line was revealed in the frame of this expansion, unlike the case of electron-electron coalescence. On the basis of the obtained boundary solutions the approximate wave function corresponding to both coalescence lines have been proposed in the two-exponential form with no variational parameters.