We present Gaussian basis sets of quadruple zeta valence quality with a segmented contraction scheme for atoms H to Kr. This extends earlier work on segmented contracted split valence (SV) and triple zeta valence (TZV) basis sets. Contraction coefficients and orbital exponents are fully optimized in atomic Hartree-Fock (HF) calculations. As opposed to other quadruple zeta basis sets, the basis set errors in atomic ground-state HF energies are less than 1 mE(h) and increase smoothly across the Periodic Table, while the number of primitives is comparably small. Polarization functions are taken partly from previous work, partly optimized in atomic MP2 calculations, and for a few cases determined at the HF level for excited atomic states nearly degenerate with the ground state. This leads to basis sets denoted QZVP for HF and density functional theory (DFT) calculations, and for some atoms to a larger basis recommended for correlated treatments, QZVPP. We assess the performance of the basis sets in molecular HF, DFT, and MP2 calculations for a sample of diatomic and small polyatomic molecules by a comparison of energies, bond lengths, and dipole moments with results obtained numerically or using very large basis sets. It is shown that basis sets of quadruple zeta quality are necessary to achieve an accuracy of 1 kcal/mol per bond in HF and DFT atomization energies. For compounds containing third row as well as alkaline and earth alkaline metals it is demonstrated that the inclusion of high-lying core orbitals in the active space can be necessary for accurate correlated treatments. The QZVPP basis sets provide sufficient flexibility to polarize the core in those cases. All test calculations indicate that the new basis sets lead to consistent accuracies in HF, DFT, or correlated treatments even in critical cases where other basis sets may show deficiencies. (C) 2003 American Institute of Physics.