Algorithms which permit the construction of fully symmetry adapted Fock matrices are presented for cases where the basis functions and integrals are obtained using only the symmetry of a subgroup of the full point group of the system. In this way fully symmetry-adapted SCF MOs may be obtained even for open shell systems. Computationally, the algorithms are practical and fast. The symmetry information about the full point group is contained implicitly in the one-electron parts of the Fock matrices: no specific symmetry data need be provided. The use of these algorithms has been found to be of considerable help in obtaining convergence for excited state SCF wave functions. An algorithm, due to P. O. Löwdin, for removing near linear dependence from basis sets is reviewed. This method used the same matrix transformation techniques which are required for the symmetrization of the Fock matrices. The application of this algorithm to obtain a set of virtual orbitals suitable for configuration interaction calculations is discussed. © 1976.