  Consider $N$ bosons in a finite box $\Lambda= [0,L]^3\subset \mathbf R^3$
interacting via a two-body smooth repulsive short range potential. We construct
a variational state which gives the following upper bound on the ground state
energy per particle \[\bar\lim_{\rho\to0} \bar \lim_{L \to \infty,
  N/L^3 \to \rho} (\frac{e_0(\rho)- 4 \pi a \rho}{(4 \pi
a)^{5/2}(\rho)^{3/2}})\leq \frac{16}{15\pi^2}, \] where $a$ is the scattering
length of the potential. Previously, an upper bound of the form $C 16/15\pi^2$
for some constant $C > 1$ was obtained in \cite{ESY}. Our result proves the
upper bound of the the prediction by Lee-Yang \cite{LYang} and Lee-Huang-Yang
\cite{LHY}.
