A variational theory called free-complement (FC) ${s}_{ij}$ theory for solving the Schr\odinger equation of atoms and molecules using only one- and two-electron integrals over Slater or Gaussian functions is proposed. It is derived from the scaled Schr\odinger equation [Phys Rev. Lett. 93, 030403 (2004)] by replacing the two-electron part ${r}_{ij}$ of the scaling $g$ operator with ${s}_{ij}={r}_{ij}^{2}$, both of which solve the divergence difficulty inherent to the original Schr\odinger equation by avoiding the interelectron collisions. The ${s}_{ij}$ function can be rewritten with only one-electron functions so that when the initial wave function of the FC theory is composed of only one-electron functions, the FC wave function including ${s}_{ij}$ can be rewritten with only one-electron functions. Therefore, the variational calculations of the FC ${s}_{ij}$ theory can be performed with only one- and two-electron integrals. However, in comparison with the ${r}_{ij}$ function, the ${s}_{ij}$ function behaves less efficiently when two electrons come close to each other: the electron-electron cusp condition is not satisfied with the FC ${s}_{ij}$ theory, though the wave function has explicit ${r}_{ij}$ dependence. On the other hand, the electron-nuclear cusp condition is satisfied, even with the Gaussian functions, for the presence of the electron-nuclear function ${r}_{iA}$ in the $g$ operator. Test applications of the FC ${s}_{ij}$ theory were done to He, Li, and the $^{5}S^{\mathrm{o}} {\mathit{sp}}^{3}$ state of carbon atom and to hydrogen molecule using the local-molecular orbital (MO)- and valence bond (VB)-type initial functions. We examined both Slater and Gaussian functions. As the order of the FC theory increased, the wave functions were improved and the energies approached from a few kcal/mol to less than 1 kcal/mol accuracies relative to the known exact values. Thus, with the FC ${s}_{ij}$ theory, the Schr\odinger equation was solved to less than 1 kcal/mol accuracy for all the systems examined. The costs for the ${s}_{ij}$ calculations were much smaller than those of the ${r}_{ij}$ theory. However, for the $^{5}S^{\mathrm{o}} {\mathit{sp}}^{3}$ state of the carbon atom, we observed that the convergence rate became slow when the calculation came close to $\ensuremath{\sim}1$ kcal/mol accuracy. This suggests that for the FC ${s}_{ij}$ theory, the calculations would become easier if the required accuracy is within a few kcal/mol for the absolute energy. Actually, what we need in chemical studies are mostly related to the difference energy, whose accuracy could be better than that of the absolute energy with the robust variational method. Two lines of future studies were suggested.