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The question of stability of a given quantum system made up of charged particles is of fundamental interest in atomic, molecular, and nuclear physics. In this work, the stability for the negatively charged positronium (Ps)-like ions or the three-body system (    Z  e +  ,  e −  ,  e −) with Yukawa potentials is studied using correlated exponential wavefunctions based on the Ritz variational method. We obtained the critical screening parameterμ Cas a function of the continuously varied nuclear charge    Z   , the critical nuclear chargeZ Cas a function of the screening parameter    μ   , and the ionization energies in terms of the screening parameter    μ    and    Z   . The critical nuclear charge for the bare Coulomb system (    Z  e +  ,  e −  ,  e −) obtained using 700-term correlated exponential wavefunctions is in accord with the reported results. The ionization energy,μ C, andZ Cfor the Yukawa system (    Z  e +  ,  e −  ,  e −) exhibit interesting behaviors. The present study describes the possible nonexistence of Borromean binding as well as Efimov states. The possible existence of quasi-bound resonances states for the negatively charged screened Ps-like ions is briefly discussed.

Critical Stability of the Negatively Charged Positronium-Like Ions with Yukawa Potentials and Varying Z