Solid-state qubits can be implemented with electrostatically confined quantum dots in semiconductors, allowing gate voltages to independently control the electrochemical potentials of each quantum dot. Quantum dots offer high levels of reliability and scalability. In this paper, along with our proposed approach based on the Generalized Hubbard model followed by Fermi's Golden rule, the charge stability diagram of a double quantum dots system with two electrons has been studied extensively. The validity of the presented approach is confirmed by experimental data. Using Fermi's Golden rule for mapping the charge stability diagram, we have deeply studied the temperature effects arising from both the Hamiltonian and transport. In addition, spin-exchange, pair-hopping, and the occupation-modulated hopping parameters on the states of the charge stability diagram are deeply discussed. Furthermore, we incorporate the Zeeman energies in the Hubbard model in order to theoretically study the spin splitting caused by an external magnetic field applied to the quantum dots. In particular, the aim of this paper is to rely on fundamental physical concepts in order to model and optimize the singlet-triplet qubit in quantum dots. In this study, the probabilities associated with singlet and triplet states have been modeled and analyzed under the impacts of intrinsic and extrinsic parameters. This will help us to find the optimal condition for coupling between double dots and provides us the design rules in terms of physical parameters to efficiently design, measure and sense, initialize, manipulate, and readout of the qubit state.