The general scattering theoretic technique of localized Green functions is applied to the calculation of one-electron elastic tunneling current through nonseparable, microscopically localized barriers. The new technique is an exact theory of tunneling, not restricted, as the transfer-Hamiltonian method of Bardeen, to weakly coupled electrodes. For illustration purpose and application to scanning tunneling microscopy (STM), we consider a model hemisphere-plane junction for which the multiple-image barrier is taken into account for the first time. The full Green function is calculated at the points of a fine grid covering the tunnel region from the solution of Dyson's equation which treats the localized tunnel barrier to all orders of perturbation of the planar barrier Green function. The tunnel current is obtained at the grid points from the Lippman-Schwinger equation, starting from the solution of the strictly planar, separable junction. The principal results of the calculation are presented as a current distribution at the planar electrode from which the lateral resolution of the model STM can be discussed. We find that the tunnel current intensity is already reduced by a factor of 2 at a distance from the tip axis equal to the tip radius. This explains, from a calculation based on first principles, the observed lateral resolution of the microscope. © 1988, American Vacuum Society. All rights reserved.