Density Functional Theory (DFT) and Time-Dependent Density Functional Theory (TDDFT) are powerful methods for solving a variety of problems, including ground state electronic structure, electron dynamics, and the absorbance cross section of molecules and materials. DFT is used to calculate the ground state electron configuration, whereas TDDFT is used to solve for the absorption cross section of excited systems. These techniques are not without their challenges. DFT requires the solution of Kohn-Sham orbitals through the diagonalization of the one electron Hamiltonian, which scales as O(N^3) where N signifies the number of orbitals in a simulation. TDDFT has its challenges as well. Each orbital must be propagated every time step, but since a single TDDFT simulation requires thousands of time steps, it is very costly. In this dissertation, we present methods that were developed to circumvent the limitations of DFT and TDDFT.

One method for decreasing the cost of DFT and TDDFT is direct delocalization, which was used to calculate the electron transfer of rate of a fullerene derivative dimer. Specifically, a common way of determining the electron transfer rate is through the use of Marcus theory, which relies on the dimer having symmetric environments. In nature this is usually the case because the dimer is surrounded by other molecules, thus creating a locally homogeneous environment. In theoretical simulations this is much more difficult to achieve. One way is to add solvating molecules, but this can be extremely costly. Instead we were able to use, in Chapter 1, a modified version of Marcus theory, which applies a bias across the Fock-matrix. This modified version of the Marcus theory allows us to solve for the electron transfer rate using one DFT calculation, because it eliminates the need to solvate the dimer to balance out the environments. The electron transfer rate was calculated to qualitatively determine the factors which lead to a good acceptor in an organic solar cell, as is important for creating an efficient solar cell.

In Chapter 2, we present a method for solving for the coupling constant of a dimer without having to balance the environments with solvating molecules. The coupling constant is used in Marcus theory to determine the electron transfer rate. To avoid the balancing we apply an electric field to the system, mimicing the effect that the solvating molecules have on the dimer. The method presented in Chapter 2 is cheaper than the preceding approaches as it limits the size of the system.

The final method we developed was the stochastic paradigm for DFT and TDDFT. The stochastic methods that are described in Chapters 3 and 4 reduce the cost of large-scale calculations by replacing the Kohn-Sham orbitals with stochastic orbitals. The density function is determined by a statistical average of the stochastic orbitals. This method enables the calculation of the absorption cross section of large systems such as 9-(4-Mercaptaphenylehtylnyl)anthracene (MPEA) chemisorbed onto a gold surface, and large gold nanoclusters. Both these systems, which contain thousands of electrons, are expensive to simulate using conventional TDDFT, but the stochastic approach we use, stochastic TDDFT (TDsDFT) makes these calculations feasible since it scales moderately (sublinearly) with the number of electrons in the system.

Organic solar cells have gained much attention as an inexpensive alternative

to traditional inorganic cells. While experimental efforts have steadily improved

the efficiency of organic devices, a large portion of the improvements have been

the result of trial-and-error. Therefore, it would be ideal to be able to use theory to predict which types of materials would produce the most efficient devices.

This dissertation presents a series of theoretical studies designed to improve understanding of what makes certain solar cells perform well and to serve as a predictive

tool to screen potential new materials.

First, a study of electron transfer in pentacene dimers is presented. The study

compares several methods for calculating the electronic transfer integral, including

time-dependent density function theory, a time dependent semi-empirical method,

and several static calculations. The results demonstrate that at large separations,

static calculations can underestimate the strength of coupling.

Next, electronic coupling in fullerenes is calculated. In this section, a method

for mimicking bulk chemical environments in film through the use of solvation and

application of electric fields is developed. The method is a applied to a number

of fullerenes used in organic solar cells, and compared with experimental data on

local and bulk electron mobilities. Comparing the theory and experiment allows

one to identify beneficial self-assembly behavior in the fullerenes studied.

This method is then extended to a calculation method we have termed direct

delocalization. In this method, a field is applied directly to the Fock matrix in

order to delocalize frontier orbitals across a dimer. Once this is accomplished,

electronic transfer time is calculated within the standard Marcus theory framework. The results are compared to the more thorough methods described above,

and found to be in agreement.

Next, the formulation of a stochastic approach to the GW approximation is

presented. In this section, a method for calculating the polarization self-energy

with stochastic orbitals is introduced. The method is highly efficient, achieving

near linear scaling with respect to system size, compared with the theoretical

fourth order scaling. The method is applied to large silicon clusters and several

fullerenes to accurately calculate quasiparticle energies.

Finally, in the last two chapters, several methods for studying plasmons are

presented. The first presents a method for studying the interaction between

molecules and plasmonic materials. The method interfaces a semiempirical quantum mechanical calculation (to study the molecules) with a finite- ifference time-

domain (FDTD) calculation (to study the plasmonic material). The study shows

that plasmon propagation can be heavily influenced by the presence of a molecule.

In the last section, an alternative FDTD method is presented. This method, labeled near-field, is a time-dependent version of the quasistatic frequency-dependent

Poisson algorithm. This approach is advantageous in that it allows for much large

time steps in the propagation, greatly expediting the calculation.

Organic solar devices can provide a cheaper alternative to the current silicon-based solar cell devices. The main disadvantage of organic photovoltaic is their low efficiency. Therefore there is a great need to better understand the mechanism of electron transfer in order to improve the efficiency of these devices. The main goal in my dissertation is to find a more accurate measure of electron transfer in these devices. I have been using theoretical methods to study electron transfer in fullerene derivatives, a common component of organic solar devices.

One such method that we have investigated is time-dependent split (TD-Split) to study A to B electron transfer by a TD evaluation of the lowest energy transition from the ground state of the combined (AB)- system.

Another method that we have developed is the time-dependent ZINDO method (TD-ZINDO) to study absorption. ZINDO is a useful theoretical tool for systems of interest due to its capacity to handle large systems within reasonable times. We were able to perform explicit time calculations with a minimal basis set. The results were then compared with higher order DFT and TDDFT results.

We also used a DFT based method to calculate the charge transfer between very large solvated organic dimers like fullerenes from isolated dimer calculations. In this method, a delocalized bias is applied directly to the Fock matrix of the dimer until the extra electron is balanced between the two molecules in the dimers. Then the transfer rate can be calculated using Marcus theory. These theoretical methods differ in accuracy and speed. In my dissertation, I will present these different methods and compare them to each other and to experimental values.

Electronic structure theory has become a powerful predictive tool in chemistry. Results from calculations provide insights into understanding a variety of material properties, as well as chemical and biological processes. A central challenge in the field is to develop methods with better efficiency, such that simulations can be carried out on larger scale and realistic systems. Different strategies are employed to address this problem, including the development of composite methods, incorporation of novel numerical schemes, as well as search for better effective Hamiltonians. In this dissertation, we present three projects that we carried out in the quest for highly efficient electronic structure theory methods.

In Chapter 2, we give an introduction to our stochastic quantum chemsitry (sQC) framework. Central to the framework is a numerical technique called stochastic resolution of identity (sRI). It allows us to replace the expensive sum of states by an average over much fewer stochastic samples. In this way the computational cost of calculations is drastically reduced. We then discuss two methods under the sQC framework, where we separately combined sRI with density functional theory (DFT) and the GW method. The resulting stochastic DFT and stochastic GW methods produce results that are in good agreement with traditional deterministic implementations, but at a much lower computational cost.

Chapter 3 presents the stochastic embedding DFT method, which is an extention of the stochastic DFT method. It is designed to selectively reduce the stochasic error of results for a specific subsystem. We applied it to study a p-nitroaniline molecule in water, and indeed it managed to reduce the stochastic error of calculated forces on the p-nitroaniline molecule by 10-fold, without increasing the computational time required in the simulation.

Chapter 4 presents the stochastic GW/RSH method, aiming at finding an optimal DFT starting point for stochastic GW calculations. We applied the method to study a few solids, and results were in good agreement with those obtained from self-consistent GW calculations.

We will also introduce another project, where we developed a novel formulation of the projector augmented wave (PAW) method. The PAW method improves the efficiency of the calculations by eliminating explicit treatment of core electrons. However, traditional implementations of PAW destroys the orthogonality of wavefunctions. In our orthogonal PAW (OPAW) method, we set to restore this orthogonality. Chapter 5 provides a short introduction to pseudopotentials and PAW, and Chapter 6 gives a detailed account of OPAW. We applied OPAW in a DFT code and succesfully reproduced results from existing non-orthogonal PAW calculations from the ABINIT package.

Electronic structure theory seeks to describe the behavior of electrons in atomic and molecular systems.Due to the intractable nature of solving the molecular Schr\o dinger's equation, approximations are made. The main challenge is to create methods that are accurate enough to gain insight while also being efficient enough to run calculations in a reasonable amount of time. In this balancing act, many strategies have been developed to allow for electronic structure calculations of large systems. Much progress has been made from calculating the states of isolated one-electron systems to now being able to simulate dynamic processes in large extended systems. This dissertation seeks to contribute to the development of novel methods to enable more efficient large-scale electronic structure calculations. A major theme of the dissertation is the use of stochastic techniques to reduce the computational scaling of methods.

Chapter 2 discusses these techniques and highlights the improvement in computational scaling when implemented with density functional theory (DFT) and many-body perturbation theory within the GW approximation.Many improvements to stochastic DFT (sDFT) have been made over the years, incorporating techniques such as embedding to reduce the required number of statistical samples. Chapter 3 continues in the same line of work and introduces the concept of tempering and its application in sDFT. The core idea of tempering is to rewrite the electronic density into the sum of a cheaper "warm" term and a smaller more expensive "cold" term. This results in a significant reduction in the statistical fluctuations and systematic deviation compared to sDFT for the same computational effort.

Chapter 4 discusses the gapped filtering method and its application in the stochastic GW (sGW) approximation.In gapped-filtering, a short Chebyshev expansion accurately represents the density-matrix operator. The method optimizes the Chebyshev coefficients to give the correct density matrix at all energies except within the gapped region where there are no eigenstates. Gapped filtering reduces the number of required terms in the Chebyshev expansion compared to traditional expansion methods, as long as one knows or can efficiently determine the HOMO and LUMO positions such as in sGW. Another direction in this dissertation is laying the foundations to implement the projector augmented wave (PAW) method into stochastic quantum methods. Compared to norm-conserving pseudopotentials (NCPP), PAW has the advantage of lower kinetic energy cutoffs and larger grid spacing at the cost of having to solve for non-orthogonal wavefunctions. Orthogonal PAW (OPAW) was earlier developed with DFT to allow the use of PAW when orthogonal wavefunctions are desired. To make OPAW viable for post-DFT stochastic methods, time-dependent wavefunctions are required. For this purpose, chapters 5 and 6 detail OPAW and its implementation in the real-time time-dependent (TD) DFT framework.

Molecular excitons in large extended systems are often not well described by local time-dependent density functional theory (TDDFT) due to highly delocalized states with long range electronic coupling. The issue of long-range coupling is made exceptionally more difficult when we consider excitons delocalized over many large molecules in aggregates ranging up to micron scale. In this thesis, we develop a series of electronic structure theory methods leveraging stochastic techniques that enable us to perform higher quality calculations on molecular excitons, and enable us to study extremely large systems in the context of molecular aggregates. We have developed a linear scaling method that can study spectroscopic observables such as the density of states and participation ratio in systems of millions of coupled dye dipoles. For the study of single excitons in large molecular complexes, we have developed a stochastic formalism of the Bethe-Salpeter equation, the linear response formalism that arises from the GW approximation of many-body perturbation theory. Through a series of algorithmic improvements to the method, we have developed new approximations to capture the screened Coulomb interaction at lower computational cost, leading to the study of systems with several thousand electrons.

Surface plasmons, the coupling of photons to charges at metal interfaces, are widely used to improve efficiency of sensing, energy transfer, and catalysis. There has been much effort to optimize plasmonic systems and exploit their field enhancement property. However, the system structure, resonance frequencies, and field enhancement are all coupled, making characterization difficult. While Maxwell finite-domain time-difference (FDTD) simulations can handle ideal systems, measurement and characterization of realistic (imperfect) experimental systems is desired.

Recently, we developed a novel single molecule superresolution method to characterize plasmonic nanostructures. We use the field strength sensitivity of stochastic blinking in quantum dots (QDs) as an indirect measurement of the local field strength, allowing measurement of the localized plasmonic near-field with a far-field reporter. Using traditional confocal excitation with a wide field capture EMCCD camera, in conjunction with Maxwell FDTD simulations, metallic nanostructures were mapped out with high spatial and local field intensity precision. Our approach offers advantages such as low-cost, high-throughput, and superresolved mapping of localized plasmonic fields.

A hydrated electron is formed when an excess electron is captured and stablized by an aqueous solution. It is generally believed that the electron carves out a quasi-spherical cavity in water and resides in it, but recent simulation work has shown a different picture where the electron is softer and the wavefunction overlaps with water molecules in the inner solvation shell. This thesis explores predictions of these two models for both the transient absorption spectroscopy and the time-resolved photoelectron spectroscopy of photo-excited hydrated electrons, providing direct comparison with experimental results. It has been shown that the non-cavity model does a better job of explaining the non-adiabatic dynamics and temperature dependent behavior of a hydrated electron in its excited state.

This thesis also includes a new attempt using range-separated hybrid functional based ab initio molecular dynamics for small water clusters with an excess electron. The negatively charged water clusters have been studied as a nano-scale version of the hydrated electron and their properties can be extrapolated to bulk solutions. Here the \textit{ab initio} molecular dynamics scheme is demonstrated to perfectly reproduce the spectral signatures of small water cluster anions, thus paving the way for more detailed simulations for bulk hydrated electron.