Ab initio all-electron fully relativistic molecular spinor (RMS) Dirac-Fock (DF) self-consistent field (SCF) and nonrelativistic limit Hartree-Fock (HF) calculations are reported at four Rf-Cl bond distances for the ground state of tetrahedral (T-d) rutherfordium tetrachloride (RfCl(4)) with our universal Gaussian basis set. The optimized Rf-Cl bond distance computed from our relativistic and nonrelativistic SCF wave functions for RfCl(4) (T-d) is 2.39 and 2.45 Angstrom, respectively. The;relativistic correction to the total electronic energy of RfCl(4) was calculated as similar to -4355 hartrees (-118 504 eV) at the Dirac-Fock level. The dominant magnetic part of the Breit interaction correction for RfCl(4), is estimated by perturbation method as 66.8509 hartrees. Our Hartree-Fock, Dirac-Fock, and Dirac-Fock-Breit calculations predict the tetrahedral RfCl(4) to be bound with the calculated dissociation energy of -14.14, -15.56, and -15.53 eV, respectively. Mulliken population analysis of our Dirac-Fock wave function indicates RfCl(4) (T-d) to be more volatile than that estimated from the corresponding Hartree-Fock wave function.