The total energy in tight-binding theory is obtained to second order in the ratio of the width of the bonding band to the bonding-antibonding splitting. This is the reciprocal of the expansion parameter appropriate to metals. No other important approximation on the minimal-basis, nearest-neighbor, tight-binding Hamiltonian is required for the periodic lattice. This leads to a simple theory of covalent bonding that is more accurate and much more general than bond-orbital methods. The lowest-order term is a bonding term that is a square root of a sum over neighbors performed at each atom. Writing the total bonding energy as a sum of such terms evaluated locally becomes an approximation in nonperiodic systems, but gives the total-energy estimate directly in terms of local interactions. The interesting second-order term, a chemical ``grip,'' is a sum over pairs of neighbors to each atom, depending upon the angle they subtend. A radial overlap repulsion of the form A/${\mathit{d}}^{3}$+B/${\mathit{d}}^{12}$ is added, and fitted to the observed equilibrium spacing and bulk modulus. The resulting form is used for a number of covalent systems to predict spacings and relative energies in competing structures. The bonding term always favors high coordination, but the grip, larger for small atoms and nonpolar systems, determines the tetrahedral structure for semiconductors and the graphite structure for carbon. An elastic shear constant in the tetrahedral structure is also obtained. The method generalizes directly to other systems such as transition-metal compounds and ${\mathrm{SiO}}_{2}$. It also gives directly short-ranged interatomic forces, which could be used in molecular dynamics.