Comparison of variational and CSE methods to polaron ground-state energy

Taking the same trial wave function, the ground-state energy of Fr\"{o}hlich polaron is investigated by variational method and coherent-state expansion (CSE) one, respectively. Within the accuracy to $\alpha^{2}$(the electron-phonon coupling constant), both methods can give the same analytic expressions of polaron ground-state energy as the function of electron-phonon coupling constant. We find that the CSE method can much simplify the calculation and shows more advantages in higher order approximations.