We have developed "soft" ab initio Hartree-Fock pseudopotentials for Be-Ne and Al-Ar that avoid singularities at the electron-nuclear cusp. The absence of these singularities is a desired feature for quantum Monte Carlo (QMC) calculations, because it considerably reduces local energy fluctuations when sampling the QMC wave function. A modified procedure for obtaining soft pseudopotentials is introduced. Stability of QMC calculations with the developed pseudopotentials is demonstrated by the ability to use larger time steps for comparable accuracy with the previously available pseudopotentials. As expected, ionization potentials and electron affinities computed using the present approach compare very favorably with other Hartree-Fock pseudopotential methods.