Electronic structure calculation is an essential approach for determining the structure and function of molecules and is therefore of critical interest to physics, chemistry, and materials science. Of the various algorithms for calculating electronic structure, the pseudospectral method is among the fastest. However, the trade-off for its speed is more up-front programming and testing, and as a result, applications using the pseudospectral method currently lag behind those using other methods. In Part I of this dissertation, we first advance the pseudospectral method by optimizing it for an important application, polarized Raman spectroscopy, which is a well-established tool used to characterize molecular properties. This is an application of particular importance because often the easiest and most economical way to obtain the polarized Raman spectrum of a material is to simulate it; thus, utilization of the pseudospectral method for this purpose will accelerate progress in the determination of molecular properties. We demonstrate that our implementation of Raman spectroscopy using the pseudospectral method results in spectra that are just as accurate as those calculated using the traditional analytic method, and in the process, we derive the most comprehensive formulation to date of polarized Raman intensity formulas, applicable to both crystalline and isotropic systems. Next, we apply our implementation to determine the orientations of crystalline oligothiophenes --- a class of materials important in the field of organic electronics --- achieving excellent agreement with experiment and demonstrating the general utility of polarized Raman spectroscopy for the determination of crystal orientation. In addition, we derive from first-principles a method for using polarized Raman spectra to establish unambiguously whether a uniform region of a material is crystalline or isotropic. Finally, we introduce free, open-source software that allows a user to determine any of a number of polarized Raman properties of a sample given common output from electronic structure calculations. In Part II, we apply the pseudospectral method to other areas of scientific importance requiring a deeper understanding of molecular structure and function. First, we use it to accurately determine the frequencies of vibrational tags on biomolecules that can be detected in real-time using stimulated Raman spectroscopy. Next, we evaluate the performance of the pseudospectral method for calculating excited-state energies and energy gradients of large molecules --- another new application of the pseudospectral method --- showing that the calculations run much more quickly than those using the analytic method. Finally, we use the pseudospectral method to simulate the bottleneck process of a solar cell used for water splitting, a promising technology for converting the sun's energy into hydrogen fuel. We apply the speed of the pseudospectral method by modeling the relevant part of the system as a large, explicitly passivated titanium dioxide nanoparticle and simulating it realistically using hybrid density functional theory with an implicit solvent model, yielding insight into the physical nature of the rate-limiting step of water splitting. These results further validate the particularly fast and accurate simulation methodologies used, opening the door to efficient and realistic cluster-based, fully quantum-mechanical simulations of the bottleneck process of a promising technology for clean solar energy conversion. Taken together, we show how both polarized Raman spectroscopy and the pseudospectral method are effective tools for analyzing the structure and function of important molecular systems.