  An optimized Rayleigh-Schr\"{o}dinger expansion scheme of solving the
functional Schr\"odinger equation with an external source is proposed to
calculate the effective potential beyond the Gaussian approximation. For a
scalar field theory whose potential function has a Fourier representation in a
sense of tempered distributions, we obtain the effective potential up to the
second order, and show that the first-order result is just the Gaussian
effective potential. Its application to the $\lambda\phi^4$ field theory yields
the same post-Gaussian effective potential as obtained in the functional
integral formalism.
