This paper fills a gap in the literature by constructing examples of geometrically finite groups of divergent type which do not satisfy the parabolic gap condition (PGC). Previously, known examples of geometrically finite groups are either (1) convergent or (2) divergent and satisfy the PGC. The main result demonstrates the existence of Hadamard manifolds with pinched negative curvature whose group of isometries contains geometrically finite Schottky groups of divergent type which do not satisfy the parabolic gap condition. Furthermore, the Patterson-Sullivan measure may be finite or infinite. As a corollary, it follows that there exist negatively-curved geometrically-finite manifolds whose corresponding geodesic flow does not admit a finite measure of maximal entropy but is completely conservative and ergodic with respect to the associated Patterson-Sullivan measure. The proof of these results is constructive. In Section 2, convergent parabolic groups are constructed on \(\mathbb{R}^N \cong \mathbb{R}^{N-1} \times \mathbb R\) with help of a family of metrics with variable negative curvatures that tend to \(-1\) at infinity. With this construction, however, the group of isometries of \(\mathbb{R}^N\) will in general be elementary, hence the geometric construction is further elaborated upon in Section 3, in which the author constructs Hadamard manifolds containing convergent parabolic elements and whose group of isometries is non elementary. Finally, in Section 4, the author shows how to construct Schottky groups with convergent parabolic factor and shows how to choose the metric inside the corresponding cuspidal end to prove the main results.