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We establish an upper bound for the ground state energy per unit volume of adilute Bose gas in the thermodynamic limit, capturing the correct second orderterm, as predicted by the Lee-Huang-Yang formula. This result has been firstestablished in J. Stat. Phys. 136(3) by H.-T. Yau and J. Yin. Our proof, whichapplies to repulsive and compactly supported $V \in L^3 (\mathbb{R}^3)$, givesbetter rates and, in our opinion, is substantially simpler.

A new second order upper bound for the ground state energy of dilute Bose gases