Dexterous robotic hands enable versatile interactions due to the flexibility and adaptability of multi-fingered designs, allowing for a wide range of task-specific grasp configurations in diverse environments. However, to fully exploit the capabilities of dexterous hands, access to diverse and high-quality grasp data is essential -- whether for developing grasp prediction models from point clouds, training manipulation policies, or supporting high-level task planning with broader action options. Existing approaches for dataset generation typically rely on sampling-based algorithms or simplified force-closure analysis, which tend to converge to power grasps and often exhibit limited diversity. In this work, we propose a method to synthesize large-scale, diverse, and physically feasible grasps that extend beyond simple power grasps to include refined manipulations, such as pinches and tri-finger precision grasps. We introduce a rigorous, differentiable energy formulation of force closure, implicitly defined through a Quadratic Program (QP). Additionally, we present an adjusted optimization method (MALA*) that improves performance by dynamically rejecting gradient steps based on the distribution of energy values across all samples. We extensively evaluate our approach and demonstrate significant improvements in both grasp diversity and the stability of final grasp predictions. Finally, we provide a new, large-scale grasp dataset for 5,700 objects from DexGraspNet, comprising five different grippers and three distinct grasp types. Dataset and Code:https://graspqp.github.io/