Atomistic modeling of fracture is intended to illuminate the complex response of atoms in the very high stressed region just ahead of a sharp crack. Accurate modeling of the atomic scale fracture is crucial for describing the intrinsic nature of a material (intrinsic ductility/brittleness), chemical effects in the crack-tip vicinity, the crack interaction with different defects in solids such as grain boundaries, solutes, precipitates, dislocations, voids, etc. Here, different methods for atomistic modeling of fracture are compared in their ability to obtain quantitatively useful results that are in agreement with the basic principles of linear elastic fracture mechanics (LEFM). We demonstrate that the complicated atomic crack-tip behavior is precisely described in simulations of semi-infinite cracks, where the loading is uniquely controlled by the applied stress intensity factor K. Such 'K-test' simulations are shown to be equally applicable in crystalline and amorphous materials, and to be suitable for quantitative evaluation of various critical stress intensity factors, the overall material fracture toughness, and quantitative comparison with theories. We further demonstrate that the simulation of a nanoscale center-crack tension (CCT) specimen often leads to the results that do not satisfy the conditions for application of LEFM. The simulated intrinsic fracture toughness, one of the basic material properties, using CCT test geometry is shown to be dependent on the crack size and far-field loading. In general, this study resolves quantitative differences between several methods for atomistic modeling of fracture and recommends that application of simulations based on nanoscale finite size cracks not be pursued.