In a series of earlier articles [B. Poirier, J. Theor. Comput. Chem. 2, 65 (2003);10.1142/S0219633603000380 B. Poirier and A. Salam, J. Chem. Phys. 121, 1690 (2004);10.1063/1.1767511 B. Poirier and A. Salam, J. Chem. Phys. 121, 1704 (2004)10.1063/1.1767512], a new method was introduced for performing exact quantum dynamics calculations. The method uses a "weylet" basis set (orthogonalized Weyl-Heisenberg wavelets) combined with phase space truncation, to defeat the exponential scaling of CPU effort with system dimensionality-the first method ever able to achieve this long-standing goal. Here, we develop another such method, which uses a much more convenient basis of momentum-symmetrized Gaussians. Despite being non-orthogonal, symmetrized Gaussians are collectively local, allowing for effective phase space truncation. A dimension-independent code for computing energy eigenstates of both coupled and uncoupled systems has been created, exploiting massively parallel algorithms. Results are presented for model isotropic uncoupled harmonic oscillators and coupled anharmonic oscillators up to 27 dimensions. These are compared with the previous weylet calculations (uncoupled harmonic oscillators up to 15 dimensions), and found to be essentially just as efficient. Coupled system results are also compared to corresponding exact results obtained using a harmonic oscillator basis, and also to approximate results obtained using first-order perturbation theory up to the maximum dimensionality for which the latter may be feasibly obtained (four dimensions).