We perform the energy minimization of the paired composite fermion (CF) wave functions, proposed by M¨oller and Simon (MS) [PRB 77, 075319 (2008)] and extended by Yutushui and Mross (YM) [PRB 102, 195153 (2020)], where the energy is minimized by varying the CF pairing function, in the case of an approximate model of the Coulomb interaction in the second Landau level for pairing channels ℓ = −1, 3, 1 which are expected to be in the Pfaffian, anti-Pfaffian and particle-hole symmetric (PH) Pfaffian phases respectively. It is found that the energy of the ℓ = −1 MS wave function can be reduced substantially below that of the Moore-Read wave function at small system sizes, however, in the ℓ = 3 case the energy cannot be reduced much below that of the YM trial wave function. Nonetheless, both our optimized and unoptimized wave functions with ℓ = −1, 3 extrapolate to roughly the same energy per particle in the thermodynamic limit. For the ℓ = 1 case, the optimization makes no qualitative difference and these PH-Pfaffian wave functions are still energetically unfavourable. The effective CF pairing is analyzed in the resulting wave functions, where the effective pairing for the ℓ = −1, 3 channels is found to be well approximated by a weak-pairing BCS ansatz and the ℓ = 1 wave functions show no sign of emergent CF pairing
