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Using an impurity atom in crystal silicon as a spin-1/2 qubit has been made experimentally possible recently where the impurity atom acts as a quantum dot (QD). Quantum transport in and out of such a donor QD occurs in the sequential tunneling regime where a physical quantity of importance is the charging (addition) energy, which measures the energy necessary for adding an electron into the donor QD. In this work, we present a first-principles method to quantitatively predict the addition energy of the donor QD. Using density functional theory (DFT), we determine the impurity states that serve as the basis set for subsequent exact diagonalization calculation of the many-body states and energies of the donor QD. Due to the large effective Bohr radius of the conduction electrons in Si, very large supercells containing more than 10 000 atoms must be used to obtain accurate results. For the donor QD of a phosphorus impurity in bulk Si, the combined DFT and exact diagonalization predicts the first addition energy to be 53 meV, in good agreement with the corresponding experimental value. For the donor QD of an arsenic impurity in Si, the first addition energy is predicted to be 44.2 meV. The calculated many-body wave functions provide a vivid electronic picture of the donor QD.