Variational determination of the two-particle density matrix : the case of doubly-occupied space

The world at the level of the atom is described by the branch of science called quantum mechanics. The crown jewel of quantum mechanics is given by the Schrödinger equation which describes a system of indistinguishable particles, that interact with each other. However, an equation alone is not enough: the solution is what interests us. Unfortunately, the exponential scaling of the Hilbert space makes it unfeasible to calculate the exact wave function. This dissertation concerns itself with one of the many ab initio methods that were developed to solve this problem: the variational determination of the second-order density matrix. This method already has a long history. It is not considered to be on par with best ab initio methods. This work tries an alternative approach. We assume that the wave function has a Slater determinant expansion where all orbitals are doubly occupied or empty. This assumption drastically reduces the scaling of the N-representability conditions. The downside is that the energy explicitly depends on the used orbitals and thus an orbital optimizer is needed. The hope is that by using this approximation, we can capture the lion's share of the static correlation and that any missing dynamic correlation can be added through perturbation theory. We developed an algorithm based on Jacobi rotations. The scaling is much more favorable compared to the general case. The method is then tested on a array of benchmark systems

Simultaneous modelling and clustering of visual field data

This thesis was submitted for the award of Doctor of Philosophy and was awarded by Brunel University LondonIn the health-informatics and bio-medical domains, clinicians produce an enormous amount of data which can be complex and high in dimensionality. This scenario includes visual field data, which are used for managing the second leading cause of blindness in the world: glaucoma. Visual field data are the most common type of data collected to diagnose glaucoma in patients, and usually the data consist of 54 or 76 variables (which are referred to as visual field locations). Due to the large number of variables, the six nerve fiber bundles (6NFB), which is a collection of visual field locations in groups, are the standard clusters used in visual field data to represent the physiological traits of the retina. However, with regard to classification accuracy of the data, this research proposes a technique to find other significant spatial clusters of visual field with higher classification accuracy than the 6NFB. This thesis presents a novel clustering technique, namely, Simultaneous Modelling and Clustering (SMC). SMC performs clustering on data based on classification accuracy using heuristic search techniques. The method searches a collection of significant clusters of visual field locations that indicate visual field loss progression. The aim of this research is two-fold. Firstly, SMC algorithms are developed and tested on data to investigate the effectiveness and efficiency of the method using optimisation and classification methods. Secondly, a significant clustering arrangement of visual field, which highly interrelated visual field locations to represent progression of visual field loss with high classification accuracy, is searched to complement the 6NFB in diagnosis of glaucoma. A new clustering arrangement of visual field locations can be used by medical practitioners together with the 6NFB to complement each other in diagnosis of glaucoma in patients. This research conducts extensive experiment work on both visual field and simulated data to evaluate the proposed method. The results obtained suggest the proposed method appears to be an effective and efficient method in clustering visual field data and 3 improving classification accuracy. The key contributions of this work are the novel model-based clustering of visual field data, effective and efficient algorithms for SMC, practical knowledge of visual field data in the diagnosis of glaucoma and the presentation a generic framework for modelling and clustering which is highly applicable to many other dataset/model combinations