This thesis presents a novel approach to modeling quantum molecular dynamics (QMD). Theoretical approaches to QMD are essential to understanding and predicting chemical reactivity and spectroscopy. We implement a method based on a trajectory-guided basis set. In this case, the nuclei are propagated in time using classical mechanics. Each nuclear configuration corresponds to a basis function in the quantum mechanical expansion. Using the time-dependent configurations as a basis set, we are able to evolve in time using relatively little information at each time step. We use a basis set of moving frozen (time-independent width) Gaussian functions that are well-known to provide a simple and efficient basis set for nuclear dynamics. We introduce a new perspective to trajectory-guided Gaussian basis sets based on existing numerical methods. The distinction is based on the Galerkin and collocation methods. In the former, the basis set is tested using basis functions, projecting the solution onto the functional space of the problem and requiring integration over all space. In the collocation method, the Dirac delta function tests the basis set, projecting the solution onto discrete points in space. This effectively reduces the integral evaluation to function evaluation, a fundamental characteristic of pseudospectral methods. We adopt this idea for independent trajectory-guided Gaussian basis functions. We investigate a series of anharmonic vibrational models describing dynamics in up to six dimensions. The pseudospectral sampling is found to be as accurate as full integral evaluation, while the former method is fully general and integration is only possible on very particular model potential energy surfaces. Nonadiabatic dynamics are also investigated in models of photodissociation and collinear triatomic vibronic coupling. Using Ehrenfest trajectories to guide the basis set on multiple surfaces, we observe convergence to exact results using hundreds of basis functions. The pseudospectral sampling of Gaussian basis functions introduces a new and efficient means of calculating the underlying quantum mechanics associated with trajectory-guided basis sets. We also discuss the conceptual connections to the quantum trajectory method and the benefits of solving quantum mechanics on a discrete grid. We include a chapter studying the strengths and weaknesses of the parametric two-electron reduced-density-matrix (p2-RDM) method for systems susceptible to delocalization error. Density matrix methods are known to overestimate the energetic effects of electron delocalization, including severe effects such as diatomic dissociation to fractionally charged atoms. We consider the role of delocalization error in p2-RDM and demonstrate that the p2-RDM is resistant to delocalization error in challenging cases.