Development of Methods for Reducing the Cost of Density Functional Theory and Time-Dependent Density Functional theory

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.