## Variational principles for critical parameters and ionization energies of
quantum systems

Alexei Sergeev and Sabre Kais

*Purdue University, Department of Chemistry*

*West Lafayette, IN 47906*

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--.