Instability conditions of the restricted Hartree-Fock (RHF) solutions for the triplet state are investigated in connection with the orbital degeneracy of the H4system of theD4h,TdandD3hsymmetries. The relative stabilities of the RHF solutions and of the solutions obtained from the instability conditions are discussed. The correlation effects involved in the ground Hartree-Fock solutions are also discussed systematically. It is shown that the instability conditions of the RHF solutions are useful in investigating the correlation effects for the triplet species as well as singlet and doublet species.