A new method of simultaneously scaling and shifting the Lennard-Jones (LJ) potential in molecular dynamics (MD) and thermodynamic integration (TI) simulations is presented. The approach allows the smooth creation or annihilation of atoms or molecules in an ensemble of solvent molecules during a molecular simulation. By scaling and shifting the LJ potential in the direction of the interatomic distance between particles, the method eliminates the problem of the creation or annihilation of a large repulsive LJ potential at the initial or final state of a TI. The optimal degree of shifting and scaling the LJ potential as a function of a control variable lambda was studied for the annihilation and creation of neon in aqueous solution. The method was further tested on the calculation of the free energy of aqueous solvation of a small molecule, ethanol. In contrast to linear scaling of the LJ potential during TI, the calculated free energies using the new separation-shifted scaling approach are reasonably well converged after 200-500 ps of simulation and show smaller hysteresis comparing forward and reverse TI.