Quantum corrections to the properties of a homogeneous interacting Bose gas
at zero temper ature can be calculated as a low-density expansion in power
s of root rho a(3), where rho is the number density and a is the S-wave sca
ttering length. We calculate the ground state energy density to second orde
r in root rho a(3). The coefficient of the rho a(3) correction has a logari
thmic term that was calculated in 1959. We present the first calculation of
the constant under the logarithm. The constant depends not only on a, but
also on an extra parameter that describes the low energy 3 --> 3 scattering
of the bosons. In the case of alkali atoms, we argue that the second order
quantum correction is dominated by the logarithmic term, where the argumen
t of the logarithm is rho al(V)(2), and l(V) is the length scale set by the
van der Waals potential.