Structural optimizations and frequency analyses have been performed on free Ge-n(m-) and KGen- (n=3-10, m=2-4) Zintl anions and ionization potentials and electron affinities calculated for KGen using the density functional theory (DFT) of Becke's three-parameter hybrid functional with the Perdew/Wang 91 expression. The DFT results obtained for small clusters (n=3-5) are further checked with both the second-order Moller-Plesset perturbation theory (MP2) and the configuration interaction calculations with all single and double substitutions from the Hartree-Fock reference determinant (CISD). Free Ge-n(2-) anions are found to share the same geometries as naked Zintl anions observed in solids with a systematical expansion in bond lengths within about 5%. Intensive searches indicate that two isomers, a tricapped trigonal prism (D-3h) and a slightly distorted tricapped trigonal prism (C-2v), exist for Ge-9(2-) and Ge-9(3-), while nido-Ge-9(4-) clearly favors the monocapped antisquare prism (C-4v) structure. HOMO-LUMO energy gaps >2.23 eV are obtained for Ge-n(m-) series at the DFT level, except Ge-9(3-) which has a much narrower energy gap of 1.16 eV. The calculated Gibbs free energy change of Ge-9(2-)+Ge-9(4-)=2 Ge-9(3-) conversion reaction involving nonagermanides has the value of DeltaGdegrees=-2.91x10(5) J mol(-1), providing the first quantum chemistry evidence that the geometrically deduced mixed valent couple of Ge-9(2-) and Ge-9(4-) in a previous study is thermodynamically unstable compared to two Ge-9(3-) anions. The calculated stabilization energies of Ge-n(2-), Ge-n(-), and Ge-n exhibit similar variation trends, clearly indicating a maximum at n=7, a minimum at n=8, and an obvious recovery at n=9 and 10. The calculated normal vibrational frequencies reproduce the six observed Raman peaks of naked Ge-5(2-) with an averaged discrepancy of 11 cm(-1). Face-capped or edge-capped deltahedral structures are predicted for binary KGen-anions and KGen and K2Gen neutrals. The magic numbers at n=5, 9, and 10 obtained in both stabilization energies and ionization potentials well reproduce the abundance distributions of KGen- observed in time-of-flight mass spectra. The validity of the Zintl-Klemm-Busmann principle in KGen and K2Gen neutrals is supported by the finding that sizable electron transfers from K atoms to Ge-n nuclei occur in these clusters and the Ge-n nuclei approach corresponding structures of free closo-Ge-n(2-) anions.