3-dimensional one-electron wave functions of molecules with cylindrical symmetry (linear molecules) are evaluated on the basis of the wellknown variational and orthogonality principles. Before application of these principles the testfunctions Φ p (x, ρ, φ) are split in two factors : an analytical one, m=magnetic quantum number, ZK= nuclear charge of the K-th atom and r K, e =distance of the K-th nucleus from the electron under consideration) that is common for all states with the same magnetic quantum number, i. e. all σ-, π-, δ- ... states respectively, and that generates cusps at the loci of the nuclei, and an unknown one, Fp , that is individual for the different states and is submitted to variation. The product inserted into the original 3-dimensional SCHRÖDINGER equation (expressed in cylindrical coordinates) yields a number of terms, of which all that cause singularities at the nuclei or on the cylindrical axis compensate mutually themselves and the potential of the nuclei. It remains a pseudo-potential which is generally discontinuous at the nuclei, but everywhere finite, contrary to the true potential. The expressions for the expectation value of the energy and is variation appear to be formally completely symmetric in the coordinates x and ρ, save an analytical factor e -2R ρ 2m+1 . Defining the unknown functions F p in a 2-dimensional lattice with constant mesh, it is possible to gain best approximations to the true eigenfunctions by means of a digital computer. A program based on these special conditions and written in FORTRAN II is shortly described. The size of the field for the function F p (x, ρ) is 90 × 50 mesh points. The results for some σ- and π-states of Η and H 2 + are satisfactory (error in energy less than 1.5%), compared to the time needed.