A theoretical expression is derived for the harmonic (in the Leibfried and Ludwig <1961> sense) Gr¿neisen &ggr; function by means of a Legendre polynomial expansion of the frequency spectrum (Thirring series). At high temperatures the harmonic Gr¿neisen function is found to be rigorously independent of temperature; the postulate definition made by Born is also given a mathematical proof, and it is proven that the acoustical branches of the spectrum give an irrelevant contribution to &ggr;, as was first suggested by Knopoff and Shapiro <1969>. The explicit expression of the harmonic high temperature &ggr; function, its volume derivatives, and its asymptotic behavior are then derived also for three of the most used kinds of intermolecular pair potentials: the Lennard-Jones potential, the Morse potential, and the Rydberg potential. A practical application to sodium, copper, and iron is also presented. The expression found for the Gr¿neisen function is simple, compact, and consistent, however, a major problem arises from he theory: an evaluation of this function requires the knowledge of the intermolecular potential function up to the fourth derivative; since the pair potentials commonly used generally agree with the 'true' function at most on the second derivative, the accuracy which can be achieved is poor. This is clearly shown by the inconsistent results found in the practical applications presented. The same objection is obviously valid for any other theoretical expression of of &ggr; involving high-order derivatives of the intermolecular potential.