Eigenvalue problems from electronic structure theory
Degree GrantorUniversity of Canterbury
Degree NameMaster of Science
Eigenvalue problems from quantum chemistry are looked at. The topic is approached in such a way that a mathematician can understand not only the techniques used to solve these eigenproblems, but also their derivation which makes the meaning and usefulness of the results clearer. Various algorithms from both chemistry and mathematics are looked at. A short review of eigenvalue problems from various areas of quantum chemistry is given and recent references are cited. Two particular eigenvalue problems are looked at in detail. Both come from looking at the electronic energy levels in molecules and are known as molecular orbital methods. The first of these is in the self-consistent field procedure where Roothaan's equations are solved. The derivation of these equations is given along with the derivation of the Hartree-Fock equations which are needed to get Roothaan's equations. Level-shifting and direct inversion in the iterative space can both be used to improve the convergence of the procedure. Shepard's second-order SCF method for parallel implementation also improves convergence. The second eigenvalue problem is in the configuration interaction method. The most common method used to solve this problem is Davidson's method. The Lanczos method is looked at and its relationship to Davidson's method is discussed. The convergence of Davidson's method and recent CI modifications are also explored.