Electronic energy levels in small metal clusters are necessarily discrete due to the quantization of motion. The statistics of level spacings and its related implications are classical problems that have attracted renewed attention in recent years because of their relevance to quantum chaos and quantum dots. This thesis is motivated by two sets of experimental results which challenge the predictions derivable from the conventional view, that the level spacings should obey the same statistics as the eigenvalues of a random matrix. In one set of experiments, the measured far-infrared absorption by small metal particles is shown to be order(s) of magnitude larger than the theoretical prediction(s). In another, the level spacing distribution as measured by the Coulomb blockade experiments i...[ Read more ]

Electronic energy levels in small metal clusters are necessarily discrete due to the quantization of motion. The statistics of level spacings and its related implications are classical problems that have attracted renewed attention in recent years because of their relevance to quantum chaos and quantum dots. This thesis is motivated by two sets of experimental results which challenge the predictions derivable from the conventional view, that the level spacings should obey the same statistics as the eigenvalues of a random matrix. In one set of experiments, the measured far-infrared absorption by small metal particles is shown to be order(s) of magnitude larger than the theoretical prediction(s). In another, the level spacing distribution as measured by the Coulomb blockade experiments is found to be nearly Gaussian, in apparent violation of the random matrix theory. The gist of our approach is the inclusion of electron-electron interactions, both the Coulomb and the Breit interactions, in the electronic energy level calculation. Here by Breit interaction we mean the part of electron-electron interaction pertaining to the spin, such as the spin-spin and spin-orbit interaction between different electrons. By approximating the positive background potential by the jellium model, we have derived from the N-electron Dirac-Breit Hamiltonian the non-relativistic Breit-Hartree-Fock (BHF) equation, and numerically solved it for clusters with up to 276 electrons. Our main results are: (1) Level spacing statistics in randomly shaped jellium clusters is shown to differ qualitatively from the Wigner distribution and better correspond with the Poisson distribution. This is found to arise from the existence and overlap of three different energy scales: shape quantization, Coulomb interaction and Breit interaction. Prediction of significantly enhanced far-infrared absorption is in excellent accord with experimental observations. Breit interaction is identified as key to the semi-metallic behavior persistent in small metal clusters at energy scales much lower than the average level spacing. (2) The physics of Coulomb blockade excitations is found to differ qualitatively from that of the far-infrared excitations. Whereas the latter involves only neutral clusters, the Coulomb blockade excitations involve added electron(s). As a function of additional electrons, the Coulombic excitation level spacings are found to exhibit a linearly decreasing trend as a result of increased effective electron radius of the cluster, due to the "spillover" of the electronic wavefunctions beyond the jellium radius. Fluctuations in the level spacings is nearly Gaussian with a width that is 6% of the mean level spacing. These features are again in excellent agreement with experimental observations. (3) We also find the random matrix assumption, that the elements of the Hamiltonian matrix vary randomly, to be invalid in the small cluster context. Since the Wigner distribution is known to be valid in the regime where the electronic kinetic energy is large so that the electronic wavelength is small (compared with the cluster-boundary irregularities) and the electron-electron interaction is relatively unimportant, it is suggested that the predictions of the random matrix theory should be viewed as operating in a regime different from that of the small clusters.