Hartree-Fock (HF) is known to suffer from drawbacks in the description of the relative stabilities between the hemi-bonded (HB) and proton-transferred (PT) isomers of the water dimer cation, (H2O)(2)(+). The energy difference predicted by HF is too large, approximately 27 kcal/mol, which is lowered to 7 kcal/mol when correlation effects are added. The error in HF has been previously attributed to the large dynamic correlation effects in the HB structure as well to the large symmetry breaking this structure exhibits. In this study we use the recently developed projected Hartree-Fock (PHF) methods to study the relative stability of the two isomers of (H2O)(2)(+) as well as its second and third row analogs, namely, (H2S)(2)(+) and (H2Se)(2)(+). In PHF, symmetries are broken and restored in a variation-after-projection approach and thus can deal easily with systems for which HF itself spontaneously breaks symmetry. We use different flavors of PHF (SURF, KSUHF, SGHF, and KSGHF) to explore their ability in capturing dynamic correlation effects and to compare their performance to different wave function based methods. We study the role of the symmetry-breaking in the above systems, using wave function based methods with unrestricted and restricted wave functions as well as performing a single-shot symmetry restoration (a projection-after-variation scheme).