Quantum chemical calculations are based on optimizing parameters including the basis set in order to obtain the best possible results. Applying the subroutines with analytical derivatives instead of numerical derivatives leads to decrease the number of iteration steps through the direct optimizing process. All the components of the first and second derivatives with respect to all parameters can be derived analytically and programmed. Evaluation of the correct structure for molecules is the main goals of the computational chemistry coding to introduce new methods. The correct description of the lone pair electrons can predict the electronic properties which can be compared to experimental values by using many models based on FSGOs. Reading the book, you will learn: a- Detailed mathematical techniques to evaluate and computerize the exact expression for analytical derivatives of a simple kind of the molecular orbital descriptions like as the explicit expression of total energy in FSGO method, b- Applying simplified PV-RR method for Hydrogen and LiH molecules using the basis set obtained by DFSGO method, and c- The correct description for lone pair orbitals in some cyclic compounds.