In this paper we systematically investigate the stability of the restricted Hartree-Fock (RHF) solutions for all closed-shell atoms and ions up to xenon-like systems by means of a symmetry-adapted Thouless stability matrix. We express the RHF solution and the lowest eigenvalue of the stability matrix in the form of a series in $1/Z$; $Z$ is the nuclear charge. Using Padé and Weniger sequence transformations, we first determine the onset of the pure singlet instability, i.e., the instability preserving all the symmetries of the underlying Hamiltonian, and identify it with the critical charge $Z_c$, i.e., the smallest charge supporting a bound-state RHF solution. This thus finally gives a physical meaning to the pure singlet instability. Consequently, we find that no basis-set-independent RHF solution for any doubly charged anion exists. Second, we determine the onset of instabilities associated with the breaking one of the symmetries of the Hamiltonian and give a simple qualitative criterion for their appearance. In particular, we find that once the shells are no longer filled according to the Aufbau principle for hydrogenic energy levels, cations are generally unstable with respect to monoexcitations violating the spherical symmetry.

• Received 28 November 2019
• Revised 28 April 2020
• Accepted 12 May 2020

DOI:https://doi.org/10.1103/PhysRevA.101.062504