In this paper, we solve different theoretical problems associated with the calculation of the kernel occurring in the Hill-Wheeler integral equations within the framework of generator coordinate method. In particular, we extend the Wick's theorem to nonorthogonal Bogoliubov states. Expressions for the overlap between Bogoliubov states and for the generalized density matrix are also derived. These expressions are valid even when using an incomplete basis, as in the case of actual calculations. Finally, the Hill-Wheeler formalism is developed for a finite range interaction and the Skyrme force, and evaluated for the latter. 20 refs., 1 fig., 4 tabs.