There are two goals of this thesis, the first one is to understand the reactivity of noble metal nanoparticles for CO oxidation reaction. The second goal is to gain understanding to the second derivative (Hessian matrix) of the potential energy surfaces (PES) of adsorption systems, especially its eigenmodes and eigenvalues, and improving algorithms for geometry optimization in electronic structure calculations. The catalytic activity of gold nanoparticles has received wide attention since the discovery of their activity on CO oxidation by Professor Haruta in 1987. By using density functional theory (DFT) and microkinetic modeling, we study CO oxidation reaction pathway on a number of transition and noble metals, i.e. Au, Ag, Pt, Pd, Cu, Ni, Rh, Ru, with different surface morphologies, close packed surfaces, stepped surfaces, kinked surfaces, as well as 12␣atom corner model of a larger nanoparticle. The upper bound of the catalytic activity (Sabatier activity) is then obtained and shows that at room temperature gold nanoparticle is the best catalyst for CO oxidation among all the metals considered. Under high temperature reaction condition, however, close packed Pt surface become most catalytically active. We show also that the catalytic activity changes with the coordination number of metal atoms at the active sites. This effect is shown to be electronic in nature, since low coordinated metal atoms, which bind reactants most strongly, have the highest energy metal d states. We compare our theoretical study of CO oxidation with experimental studies. The latter shows promoted catalytic activity when gold particle size decreases to 5 nm. Oxidizing CO by N2O was found to involve a CO␣O transition state, with atomic O adsorbed on the gold B5 sites and CO on the corners. On the other hand, CO oxidation by molecular O2 occurs via a different reaction pathway, which instead involves a meta-stable intermediate CO-O2. However, although the two oxidizing agents used proceeded via different reaction pathways on different active sites, the apparent overall activation barriers obtained from both theory and experiment were found to be the same. The experiment findings are in good agreement with our theoretical calculations. The second part of the thesis focuses on improving the convergence property of Quasi-Newton algorithm. The eigenvalues of the Hessian matrix of 54 atoms bulk Cu model are calculated, and the sizes of eigenvalues follow power-law distribution. It is found that the anharmonicity of the weak modes lead to poor Newton step and poor Hessian update in BFGS type Quasi-Newton algorithm, which slow down the geometry optimization. Line search that fulfills Wolff conditions is then applied to improve the quality of Hessian update. We parameterized the optimizer and the parameter spaces for different test cases are scanned to find the optimum parameter set for surface adsorption type of problems. The test cases show that the BFGS algorithm with line search scheme with the optimized parameter set greatly improves the convergences of geometry optimization. The scanning of the parameter space of the algorithm shows that the value of preconditioner around the middle of the eigenvalue spectrum gives faster convergence rate. An adaptive update method (AUM) for adjusting the preconditioners of the unupdated modes is then proposed, so that they are set to be in the middle of the eigenvalue spectrum dynamically. Test results shown that the AUM is able to adjust poorly set preconditioners in several steps and improve the convergence rate. Finally, we use a model potential that describes bond stretching to calculate the Hessian matrix. Comparison with the exact Hessian shows that the model Hessian reproduces the vibrational modes in a decent manner, despite its simplicity. For homogeneous systems, preconditioning the optimizer with the model Hessian reduces the condition number by 14 times and largely improves the convergence rate.