For small internuclear distances, the quadrupole moment of the hydrogen molecule is quadratic in the internuclear distance, with a coefficient determined by the quadrupole shielding factor of the helium atom. Similarly the hexadecapole moment is quartic in the internuclear distance for small internuclear distances, with a coefficient which is determined by the properties of the helium atom. We give the results of a machine computation of the hexadecapole moment of the hydrogen molecule using up to 54 term variational wave functions and its matrix elements between rotational and vibrational states of the molecule. The intensity of the ΔJ = 4 induced spectrum of hydrogen gas is also discussed.