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The potential energy surfaces governing the $H2F2$ system have been investigated using a priori electronic structure theory. The problem has been divided into five distinct parts:\begin{eqnarray}HF + HF \rightarrow (HF)_2 \ F + H_2 \rightarrow FH + H \ H + FH \rightarrow HF + H \ H + F_2 \rightarrow HF + F \ F + HF \rightarrow FH + F\end{eqnarray}Basis sets of double-zeta-plus-polarization quality have been used throughout and both self-consistent-field (SCF) and configuration interaction (CI) wave functions have been obtained. The orbitals appearing in the correlated wavefunctions were optimeized via the iterative natural orbital method. Predicted dissociateion energies and barrier heights agree well with available experimental data. However, electron correlation is shown to paly a crucial role in barrier height predictions. For the exchange reactions (3) and (5), experimental activation energies are not known. The surprisingly high values (18 and 40 kcal/mole, respectively) obtained here suggest the inadequacy of semi-empirical potential surfaces in pridicting such barriers. This work was performed under the auspices of the Energy Research and Development Administration.