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In $(C2D2)2$ the hyperfine structure arises mainly from the quadrupole coupling of the four deuterium atoms. This dimer also exhibits a large amplitude motion consisting of a geared rotation of the two acetylene subunits, during which each deuterium atom is put in turn into the hydrogen bond, thus leading to a change of the electric field gradient tensor for any given deuterium nucleus. The hyperfine structure of $(C2D2)2$ should therefore give us important information on the interaction of the quadrupole coupling with the large amplitude motion. The theoretical approach involves setting up a complete quadrupole Hamiltonian for the four deuterium atoms, which is allowed to depend upon the large amplitude coordinate necessary to fully describe the various configurations of the molecule. The mean value of this operator is then evaluated for each of the tunneling levels, which for acetylene dimer1 take the form of nondegenerate A or B type levels, or doubly degenerate E type levels. The unexpected result is that for nondegenerate tunneling levels the calculation of the mean value leads to an effective hyperfine Hamiltonian which is the sum of four one-deuterium hyperfine Hamiltonians, each with the same effective coupling constant. The group $S4$ of the permutations of 4 identical particules can be used to classify the hyperfine energy levels, and symmetry adapted wavefuctions can be constructed prior to the diagonalisation of the hyperfine Hamiltonian. Such an approach is similar to the one used by Wolf, Williams and Weatherly$\mathrm{<mrow\; data-mjx-texclass="ORD">2</mrow>}$ for $CFCl3$ and $CHCl3$, except that the number of identical particules is increased by one in the present problem. The particular symmetry properties of the effective hyperfine Hamiltonian will be discussed, and we hope to be able to compare predicted hyperfine patterns with data. $\mathrm{<mrow\; data-mjx-texclass="ORD">1</mrow>}$ G. T. Fraser, R. D. Suenram, F. J. Lovas, A. S, Pine, J. T. Hougen, W. J. Lafferty, and J. S. Muenter, J. Chem. Phys. 89, 6028-6045 (1988). $\mathrm{<mrow\; data-mjx-texclass="ORD">2</mrow>}$ A. A. Wolf, Q. Williams, and T. L. Weatherly, J. Chem. Phys. 47, 5101-5109 (1967).