We complete our recently introduced theoretical framework treating the double-quantum-dot system with a generalized form of Hubbard model. The effects of all quantum parameters involved in our model on the charge-stability diagram are discussed in detail. A general formulation of the microscopic theory is presented and, truncating at one orbital per site, we study the implication of different choices of the model confinement potential on the Hubbard parameters as well as the charge-stability diagram. We calculate the charge-stability diagram keeping three orbitals per site and find that the effect of additional higher-lying orbitals on the subspace with lowest-energy orbitals only can be regarded as a small renormalization of Hubbard parameters, thereby justifying our practice of keeping only the lowest orbital in all other calculations. The role of the harmonic-oscillator frequency in the implementation of the Gaussian model potential is discussed, and the effect of an external magnetic field is identified to be similar to choosing a more localized electron wave function in microscopic calculations. The full matrix form of the Hamiltonian, including all possible exchange terms and several peculiar charge-stability diagrams due to unphysical parameters, is presented in the Appendices, thus emphasizing the critical importance of a reliable microscopic model in obtaining the system parameters defining the Hamiltonian. © 2011 American Physical Society.