A reformulation of Hartree-Fock theory for time-independent molecular properties with perturbation-dependent basis sets and which refers strictly to the atomic-orbital basis is presented. The formalism is based on a recently proposed exponential parametrization of the one-electron atomic-orbital density matrix. In the presented formulation, only multiplications and additions of sparse matrices are needed. Linear scaling with system size is therefore obtainable, making this formulation ideally suited to large molecular systems. The paper contains general formulas for molecular energy derivatives up to fourth order, with special attention given to molecular gradients, molecular Hessians, magnetizabilities, and nuclear magnetic shieldings.