The recently proposed (explicitly correlated) r(12)-MR-CI and r(12)-MR-ACPF (averaged coupled-pair functional) methods are applied to the computation of the clamped-nuclei nonrelativistic ground-state energies of the first-row atoms and their positive ions. For the neutral atoms we obtain accuracies of -0.05 (He and Li), -0.013 (Be), +0.12 (B), -0.1 (C and N), +0.3 (O) and +0.6 (F and Ne) mE(h). Our energies of B-F are by far the best available. In;all cases, the energy eigenvalues of the Schrodinger equation are calculated to better than chemical accuracy (1 kcal/mol). Since our method is completely general, this, for the first time, implies the possibility of performing quantum chemical calculations of general many-electron systems where the error of the computed energy is not any more very large compared to the desired accuracy.