Atoms the normal states of which are not $S$ states experience quadrupole-quadrupole repulsive and attractive forces which vary as the inverse fifth power of the separation distance. At large distances, exchange being negligible, the expression for the interaction energy contains as factors two atomic coefficients and a root of a secular equation determined by the molecular state of the system. For a given atom the coefficient is determined by the nature of its lowest level. If one uses the Hartree-Fock approximation, it is proportional to the average of the square of the radial atomic distance for those electrons with orbital angular momentum not in complete shells. Atomic coefficients have been calculated for most of the atoms of the periodic table having incomplete $p$ and $d$ shells and the secular equations have been solved for a number of cases. At distances of twice the sum of the atomic radii, diatomic molecules resulting from the combination of such atoms of the first row of the periodic table have quadrupole-quadrupole energies of a few tenths of a volt, such energies becoming rapidly larger for smaller distances because of the inverse fifth power dependence.