Ab initio quantum mechanical studies employing basis sets approaching the quality of triple-zeta plus double-polarization on Al and N and single-polarization on C and H are reported for AlH3N(CH3)(3) (TMAA), N(CH3)(3) (TMA), and AlH3. The self-consistent-field method was employed for all basis sets; the single- and double-excitation configuration interaction and comparable coupled-cluster methods were utilized with the double-zeta plus polarization function basis set. The dissociation energy of AlH3-N(CH3)(3) is predicted to be 31 kcal/mol after correcting for zero-point vibrational energy. With the exception of the brans methyl hydrogens, the AlH3 and TMA moieties do not undergo large changes upon formation of TMAA. The hyperconjugative effect on the trans hydrogens is half that deduced experimentally, but is verified to be real. Rotations of methyl groups and alane groups influence the hyperconjugative effect. This work corrects earlier assignments of the Al-N stretching vibrational mode. TMAA structures examined include the minimum; one, two and three rotated methyls; and rotated alane. The minimum structure and a singly rotated methyl TMA structure were also studied. Equilibrium geometries, energies, harmonic vibrational frequencies, infrared intensities, atomic charges,:vertical ionization potentials, dipole moments, and hyperconjugative effects are reported.