The potential energy surfaces, dipole moments, and spectroscopic constants of the ground states of CuH and CuF are calculated by using several single reference (SR) many electron theories. The methods used in this particular study are the coupled-cluster doubles (CCD), singles and doubles (CCSD), Brueckner doubles (BD), and quadratic configuration interaction singles and doubles (QCISD). All these methods are supposed to give the same results when the Hartree-Fock (HF) orbitals coincide with the approximate Brueckner orbitals. Furthermore, we examine the effect of the inclusion of perturbational triples to these methods, i.e., CCD(T), CCSD(T), BD(T), and QCISD(T). Since the chosen molecules, CuH and CuF, have large T-1 amplitudes, the comparisons of the different methods offer critical examinations of the SR theories. For CuH, all the strict SR theories, i.e., the CCD, CCSD, ED, CCD(T), CCSD(T), and BD(T), result in parallel potential energy curves. The QCISD energy is, however, too low in comparison with the results of the other methods. Furthermore, the inclusion of triples, QCISD(T), gives an obviously wrong potential energy curve. Analyses of the calculated dipole moment based on the finite field method and the diagnostics of T-1 amplitudes clearly demonstrate that in the QCISD method overestimates the single electron excitations around the equilibrium distance. The absence of disconnected T-1 products in the QCISD is responsible for this failure. These effects are even more pronounced in the CuF case. The QCISD equilibrium bond distance, r(e)=1.767 Angstrom, matches the results of the other methods. However, the calculated dipole moment does not compare with the experimental nor with the other CC results, and D-e is overestimated. Severe failure was found for the QCISD(T) energy : the calculated curve possesses an unphysical double well profile. The dipole moment is overestimated by a factor of 3 while the calculated D-e is too low. The results indicate that the HF orbitals become less suitable at the bonding region and the QCl results consequently become less reliable within the SR many electron theories.