The quantum mechanics of two-electron systems is reviewed, starting with the ground state of the helium atom and heliumlike ions with central charge . The case of is rather straightforward. In contrast, for negative hydrogen ion with , the stability of cannot be achieved using a product of individual electron wavefunctions and requires explicit account of the anticorrelation among the two electrons. The wavefunction proposed by Chandrasekhar is revisited, where the permutation symmetry is first broken and then restored by a counterterm. More difficult problems can be studied using the same strategy such as the stability of hydrogenlike ions for any value of the proton-to-electron mass ratio , the energy of the lowest spin-triplet state of helium and heliumlike ions, and the stability of the doubly excited hydrogen ion with unnatural parity. The positronium molecule, which was predicted years ago and discovered recently, can also be shown to be stable against spontaneous dissociation. Emphasis is placed on symmetry breaking, which can either spoil or improve the stability.
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