The major source of errror in most ab initio calculations of molecular energies is the truncation of the one-electron basis set. A complete basis set model chemistry is defined to include corrections for basis set truncation errors. This model uses double zeta plus polarization level atomic pair natural orbital basis sets to calculate molecular self-consistent-field (SCF) energies and correlation energies. The small corrections to give the complete basis set SCF energies are then estimated using the l−6 asymptotic convergence of the multicenter angular momentum expansion. The calculated correlation energies of the atoms He, Be, and Ne, and of the hydrides LiH, BH3, CH4, NH3, H2O, and HF, using the double zeta plus polarization basis sets vary from 83.0% to 91.2% of the experimental correlation energies. However, extrapolation of each of the pair energies and pair-coupling terms to the complete basis set values using the asymptotic convergence of pair natural orbital expansions retrieves from 99.5±0.7% to 101.1±0.6% of the experimental correlation energies for these atoms and molecules. With the exception of ammonia which gave 101.1%, the calculated correlation energies agree with the experimental values to within the error limits of the experiments for all these atoms and molecules with more than four electrons. The total extrapolated energies (ESCF+ECorrelation) are then in agreement with experiment to within ±0.0014 hartree (root mean square deviation) and represent the most accurate total energy calculations yet reported for the molecules. The largest discrepancies with experiment occur for methane, where we obtain ETotal =−40.5112 hartree compared to EExpt =−40.514±0.002 hartree, and ammonia, where we obtain ETotal =−56.5659 hartree compared to EExpt =−56.563±0.002 hartree.