The infinite-order regular approximation (IORA) and IORA with modified metric (IORAmm) is used to develop an algorithm for calculating relativistically corrected isotropic hyperfine structure (HFS) constants. The new method is applied to the calculation of alkali atoms Li-Fr, coinage metal atoms Cu, Ag, and Au, the Hg+ radical ion, and the mercury containing radicals HgH, HgCH3, HgCN, and HgF. By stepwise improvement of the level of theory from Hartree-Fock to second-order Moller-Plesset theory and to quadratic configuration interaction theory with single and double excitations, isotropic HFS constants of high accuracy were obtained for atoms and for molecular radicals. The importance of relativistic corrections is demonstrated. (C) 2004 American Institute of Physics.