The atomization energy of a hypothetical vibrationless molecule in its ground state equilibrium nuclear configuration is partitioned into bonded contributions (bond energy, BE) with the use of properties of the electronic charge density at bond critical points. The method is applied in the framework of Kohn-Sham density functional theory (with a gradient-corrected exchange-correlation functional and a triple-zeta plus polarization AO basis) to various diatomics and strained and unstrained hydrocarbon molecules. A linear relation is established between the bond energies and a quantity termed bond electron energy which is the total energy density divided by a constant plus the charge density at the critical point. The use of one empirical parameter per atom pair and a bond path curvature term, which is the difference of the length of the curve of maximum electron density and the distance between the atoms, allows the determination of atomization and bond energies with errors below 0.5-1% in most cases. The bond energies of the CC (CH) bonds of ethane, ethene, benzene, and acetylene are found to be 86.1 (104.1), 140.5 (106.1), 120.3 (106.4), and 188.9 (111.4) kcal/mol, respectively. By comparison with the BE values of strainless reference compounds, the bond strain and total strain energies of some saturated and unsaturated hydrocarbons (cycloalkanes, cycloalkenes, tetrahedrane, cubane, benzocyclopropene, fenestrane) are calculated in good agreement with experimental data.