We present a new coupled Hartree-Fock(HF)/Kohn-Sham DFT perturbation method that accounts for the effect of enlarging the basis set in electronic structure calculations.In contrast with previous approaches, our dual basis set treatment yields not only a correction for the total energy, but also for the orbital eigenvalues and density. The zero-th order solution is obtained from the projection of the small basis set coefficients.Diagonalization of the full Fock matrix in the large basis set is avoided. In this first paper of a series, we develop the theoretical foundations of our approach for molecules, including the coupled-perturbed equations through second order and the energy expressions through fourth order -as our method complies with Wigner's 2n + 1 rule. The first-order perturbation equation turns out to be uncoupled and odd-order terms in the energy expansion vanish.In calculations on simple molecules, our method recovers over 93% (84%) of the missing DFT(HF) energy when going from the cc-pVDZ to the aug-cc-pVDZ basis,