Globular proteins fold to create compact structures rich in alpha-helices and beta-sheets. While studies of cubic lattice models of simplified polypeptide chains have concluded that secondary structure is a necessary consequence of chain compactness, different conclusions have been reached from studies of off-lattice models of simplified chains. In an attempt to resolve this controversy, we study an all-atom off-lattice model of a protein subject to a variety of simplified energy functions. A Monte Carlo simulated annealing algorithm is used to search conformational space quickly. The algorithm uses pivot-type moves in which a residue is selected at random and the values of its main-chain dihedral angles are changed. The energy function used to accept or reject moves is taken to be either a term proportional to the volume occupied by a structure (to mimic the hydrophobic effect), a term proportional to the energy of main-chain hydrogen bonding, or a combination of these two terms. Secondary structure content is evaluated using several different definitions. For all the definitions used, compactness alone produces a 10% increase in secondary structure content. However, this is a small fraction of the secondary structure observed in native protein structures. Structures produced by minimizing the hydrogen bond energy have extensive secondary structure but are not densely packed. Structures having both the high density of native structures and extensive secondary structure are produced by minimizing combinations of the volume and hydrogen bond energy terms. Our results emphasize the close relationship between secondary structure and the geometry of main-chain hydrogen bonding. The results are consistent with a description of protein folding in which the hydrophobic effect favors dense packing while hydrogen bonding determines the specific local geometry which generates secondary structure. To make an analogy with lattice studies of packing density and secondary structure, it seems that hydrophobicity provides the packing density while hydrogen bonding provides the lattice.