A large number of one-range addition theorems for first and second derivatives of complete orthonormal sets of psi(alpha)-ETOs (alpha = 1,0, - 1, -2....) through the new basic auxiliary functions is presented. The basic auxiliary functions are constructed from the finite linear combinations of psi(alpha)-ETOs. These theorems are especially useful in the study of,electric field and its gradient induced at the nuclei of a molecule by its electrons. The relationships obtained are valid for the arbitrary values of parameters and location of orbitals. (c) 2005 Elsevier B.V. All rights reserved.