Time-dependent density functional theory (TDDFT) is a powerful and efficient method for calculating excitation energies and properties of electronic excited states. It has found wide applications in various scientific fields due to its accuracy and computational efficiency. However, solving the TDDFT equations involves large eigenvalue and linear problems, which canbe computationally challenging. To address this, matrix-free iterative subspace algorithms have been developed. The first half of the thesis demonstrates the use of density functional theory (DFT) to elucidate the electronic structure of the first synthesis of Neodymium(II) encapsulated in a 2.2.2-cryptand ligand. The comparison between experimental results and the calculated molecular structure, as well as the UV-Vis spectrum obtained through TDDFT, provided strong evidence supporting the discovery of the traditional 4f 4 electron configuration. In the second part of this thesis, I introduce libkrylov, which is a versatile and open-source Krylov subspace library designed for performing large-scale matrix computations on-the-fly. The main goals of libkrylov are to provide a versatile application programming interface (API) design and a modular structure that allows seamless integration with specialized matrix-vector evaluation “engines.” The library is designed to offer pluggable preconditioning, orthonormalization, and tunable convergence control, making it highly flexible and easily adaptable to various computational scenarios and requirements. By providing these features, libkrylov enables users to customize and optimize their calculations, thereby enhancing the efficiency and accuracy of large-scale matrix computations in computational chemistry and other related fields. I extend libkrylov to Hamiltonian structured problems often encountered in TDDFT. The implementation is based on preserving the full SO(1,1) symmetry of the TDDFT response equations, which in the absence of magnetic fields reduces to the use of split-complex numbers. In Krylov subspace methods, this preservation is achieved by utilizing symmetry-adapted basis vectors that maintain the orthonormality condition of the symplectic problem. The calculation of excitation energies for some test molecules show improved convergence over the Olsen algorithm and the TURBOMOLE implementation. The improved convergence highlights the importance of the correct algebraic approach to the problem.