We study the (001) surface of the semiconductors Si and Ge using the total energy algorithm due to Chadi. The total energy minimization theory is described together with the LCAO theory and the Special Points Method. The electronic bands are calculated using the Slater-Koster simplified LCAO theory in its first-near-neighbour approximation. The Special Points Method is used to compute the electronic contribution to the total energy. We describe and illustrate a method for deriving special points for a distorted lattice. Our method is quicker and easier than first principles derivation of special points for distorted lattices. The accuracy of the total energy scheme is tested by computing bulk phonon frequencies and elastic moduli. The calculated results are in reasonable accord with experimental values. The Chadi formalism is used to deduce the (2X1) reconstructed geometries of Si and Ge (001) surfaces. Techniques for calculating projected bulk bands and surface states are detailed and applied to the dimer geometry of the (2X1) reconstruction. The symmetric dimer is found to give metallic surface states. The asymmetric dimer has the lower energy and semiconducting surface states. It is found that the calculated positions and dispersions of the semiconducting states do not agree with photoemission results. High order reconstructions on the Si and Ge (001) surface are examined and found to be within a few meV in energy of the (2X1) reconstruction. The Si (001) - (2X1) surface structure resulting from the Chadi formalism is compared with that deduced from LEED analysis and from an energy minimization process using an ab initio pseudopotential calculation. The Chadi structure is found to be the lowest in energy but we argue, by considering other theories, that the Chadi formalism is qualitative at best. We propose some improvements of the Chadi formalism to make the theory more quantitative for surface structural geometries.