Non-traditional applications of variational methods were proposed both for critical parameters (when the energy crosses the ionization threshold) and for the energy below and above ionization threshold.

When the critical parameter enters the Schrodinger equation linearly, then the equation for critical parameters can be considered as a generalized eigenvalue equation with a non-trivial weight operator. The expectation value of the "generalized energy" produces an upper bound for the critical parameter. This variational principle is optimized in order to give accurate estimation of the critical parameter itself rather than the energy. As an example, we consider a two-electron atom with a charge of the nucleus treated as a continuous parameter. Numerical tests confirm fast convergence of our results with increase of the size of Hylleraas basis set. Similar results were obtained for the 5-dimensional two-electron atom, which is equivalent to a doubly excited state of the 3-dimensional atom. For the 7-dimensional atom, we have found that the critical charge is exactly one. The critical charges were found for two-electron atoms subject to external magnetic field as a function of magnetic field strength.

For ionization energies, an ordinary variational principle for the energy functional was used, but with allowance of complex variational parameters. Above the ionization threshold, a minimum of the energy functional turnes into a complex stationary point. It means that the variational principle produces complex energy that approximates position (real part) and half width (imaginary part) of the corresponding quasi-stationary state. We calculated variational energies of few-electron atoms as a function of charge of the nucleous using simple trial functions in the form of a product of exponents (including permutations). Analytical properties of the energy as a function of the nuclear charge were studied in detail for singlet and triplet states of helium and for ground states of three to four electron atoms. It was found that the behavior near the critical charge falls into one of two categories resembling the first and the second order phase transitions in statistical physics. Above the ionization threshold, we can estimate, for example, the energy of the unstable ion He^--.

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Designed by A. Sergeev.