Note on critical cage size for ionization of confined two-electron systems

The use of independent confinement models for the neutral and ionized two-electron atom or molecule to obtain the critical cage size for ionization is shown to be formally inconsistent. Instead, a treatment using the same Hamiltonian for the evolution of the ground state energy of the system is proposed, which allows for a consistent estimate of the first and second ionization energies as a function of cage size. Our calculations are based on the variational method applied to the helium atom confined by a spherical cage with impenetrable and penetrable confining walls.