  The confined variational method is applied to investigate the low-energy
elastic scattering of ortho-positronium from $\text{H}_2$ by first-principles
quantum mechanics. Describing the correlation effect with explicitly correlated
Gaussians, we obtain accurate $S$-wave phase shifts and pick-off annihilation
parameters for different incident momenta. By a least-squares fit of the data
to the effective-range theory, we determine the $S$-wave scattering length,
$A_s=2.06a_0$, and the zero-energy value of the pick-off annihilation
parameter, $^1\!\text{Z}_\text{eff}=0.1858$. The obtained
$^1\!\text{Z}_\text{eff}$ agrees well with the precise experimental value of
$0.186(1)$ (J.\ Phys.\ B \textbf{16}, 4065 (1983)) and the obtained $A_s$
agrees well with the value of $2.1(2)a_0$ estimated from the average
experimental momentum-transfer cross section for Ps energy below 0.3 eV (J.\
Phys.\ B \textbf{36}, 4191 (2003)).
