Quantum mechanical simulations of carrier transport in Si require an accurate model of the complicated Si bandstructure. Tight-binding models are an attractive method of choice since they bear the full electronic structure symmetry within them and can discretize a realistic device on an atomic scale. However, tight-binding models are not simple to parameterize and characterize. This work addresses two issues: (1) the need for an automated fitting procedure that maps tight-binding orbital interaction energies to physical observables such as effective masses and band edges, and (2) the capabilities and accuracy of the nearest and second-nearest neighbor tight-binding sp3s* models with respect to carrier transport in indirect bandgap materials. A genetic algorithm approach is used to fit orbital interaction energies of these tight-binding models in a nine-and 20-dimensional global optimization problem for Si. A second-nearest neighbor sp3s* parameter set that fits all relevant conduction and valence band properties with a high degree of accuracy is presented. No such global fit was found for the nearest neighbor sp3s* model and two sets, one heavily weighed for electron properties and the other for hole properties, are presented. Bandstructure properties relevant for electron and hole transport in Si derived from these three sets are compared with the seminal Vogl et al. [Journal of the Physics and Chemistry of Solids 44, 365 (1983)] parameters.