We have developed prolapse free Gaussian basis sets which can be used for H-1 to Bi-83, imposing the condition that the Dirac-Fock-Roothaan (DFR) total energy (TE) decreases monotonically toward the numerical DF (NDF) TE as the expansion term increases. An even-tempered basis set was assumed. The resulting sets gave \TE(DFR)-TE(NDF)\less than or equal to1x10(-6) hartree for any atoms less or equal to Bi-83; TE(NDF)=-21 565.638 345, and TE(DFR)=-21 565.638 345+/-0.000 001 hartree for Bi when the expansion terms are in the range (58, 58, 58, 36, 36, 36, and 36) and (72, 72, 72, 36, 36, 36, and 36) for (s(+), p(-), p(+), d(-), d(+), f(-), and f(+)) symmetries, respectively. A practical set with 44, 44, 44, 36, 36, 32, and 32 for the respective symmetries is also proposed where \TE(DFR)-TE(NDF)\less than or equal to4x10(-5). (C) 2004 American Institute of Physics.