A new type of potential operator has several kinds of applications in electronic structure calculations. Three uses are envisaged here. First when some special region of a covalently bonded solid or very large molecule is modeled by a modest sized cluster, each dangling bond at the cluster surface can be saturated in a way that exactly reproduces the bond in the complete system. Second a similar approach can be used at the matching surface in an embedding scheme for calculations on the same type of systems. The third application is to atomic pseudopotentials where the new potential operator avoids the possibility of "ghost" states that sometimes plague the widely used Kleinman-Bylander form of the pseudopotential. The theory of the new separable potential and its application to the dangling bond problem are the main subjects of the present paper. Starting from a given potential or pseudopotential, the new separable operator modifies some of the required eigenfunctions and eigenvalues in a controlled way while conserving all other eigenvalues. The method has been tested on the molecules X-SiH3 where X=H, F, Cl, Br, or I.