For calculating the MP2 energy from the Hartree-Fock, as mentioned in the comments, single excitations don't contribute due to Brillouin's theorem, which states that the Hamiltonian matrix elements between the optimized HF ground state and any singly excited determinant are explicitly zero.
There is no such restriction if we were to compute a perturbed energy for a DFT wavefunction, so in principle single excitations should contribute. However, when Perturbation Theory-Kohn Sham (PT-KS) was initially attempted¹, it produced spuriously low energies, with the error being traced to the single excitation energy. They attribute this to the single excitation energy being negative definite and having occupied-virtual orbital energy differences in the denominator, which tend to be underestimated by DFT.
Since double hybrid DFT is already semiempirical due to the parameter fitting for the HF exchange, DFT xc, and MP2 correlation, it made sense to neglect these single excitations that had already proven problematic rather than proceed with the nonempirical form of the PT contribution.
- P. Mori-Sanchez, Q. Wu, and W. Yang, J. Chem. Phys. 123, 062204