The electronic structure of quantum dots containing N greater than or equal to 1000 atoms is difficult to calculate by conventional molecular methods since the effort scales as N-3. Our newly developed method allows calculation of eigenstates within a desired "energy window" and thus has a linear-in-N scaling. This method is applied here to Si quantum dots using a plane wave basis expansion and an empirical pseudopotential Hamiltonian. Hydrogen atoms passivate the surface dangling bonds using a realistic surface relaxation geometry. We investigate the dependences of energy gaps and radiative recombination rates on the size, shape, and orientation of the Si quantum dots. We find that (1) a unified curve exists for band gap vs size of quantum spheres, cubes, and rectangular boxes; (2) the band edge states of Si quantum dots are bulklike, not surfacelike; (3) the band gap is insensitive to the surface orientation and to the overall shape of the quantum dot as long as it is not too prolate; (4) the radiative lifetime is sensitive to the shape and orientation; and (5) effective mass and single band truncated crystal models describe inadequately the electronic structure of Si quantum dots in the size range (less than or similar to 40 Angstrom) studied here.