  A novel minimal-size pairing operator $\Delta^{\dagger}_{D0}$ with $d$-wave
symmetry in CuO$_2$ planes is introduced. This pairing operator creates on-site
Cooper pairs at the four oxygens that surround a copper atom. Via the time
evolution of $\Delta^{\dagger}_{D0}$, an additional inter-orbital pairing
operator $\Delta^{\dagger}_{Dpd}$ with $d$-wave symmetry is generated that
pairs fermions located in a Cu and its four surrounding O's. The subsequent
time evolution of $\Delta^{\dagger}_{Dpd}$ generates an intra-orbital $d$-wave
pairing operator $\Delta^{\dagger}_{Dpp}$ involving the four O atoms that
surround a Cu, as well as the $d$-wave operator $\Delta^{\dagger}_D$
traditionally used in single-band models for cuprates. Because we recover the
larger size operators extensively used in the three-orbital Hubbard model, we
suggest that long-range order using the canonical extended operators occurs
together with long-range order in the new minimal operators. However, our
minimal $d$-wave operators could be more practical to study $d$-wave
superconductivity because in the finite-size relatively small systems
accessible to computational techniques it is easier to observe long-range order
using local operators. Moreover, an effective model with the usual
tight-binding hopping of the CuO$_2$ planes supplemented by an attractive
potential $V$ in the $d$-wave channel is introduced. Using mean-field
techniques we show that a paired ground state is stabilized for any finite
value of $V$.
