Energy level spacings, atomic clusters

Discrete energy levels are to be observed for position (a) as well as for position (b) at exactly the same values, in case (b) somewhat better expressed than in (a). The level spacing is 135 mV. This spectrum clearly identifies the Au55 cluster as a quantum dot in the classical sense, having discrete electronic energy levels, though broader than in an atom, but nevertheless existent. The description of such quantum dots as artificial, big atoms seems indeed to be justified. 

Energy levels of heavy and super-heavy (Z>100) elements are calculated by the relativistic coupled cluster method. The method starts from the four-component solutions of the Dirac-Fock or Dirac-Fock-Breit equations, and correlates them by the coupled-cluster approach. Simultaneous inclusion of relativistic terms in the Hamiltonian (to order o , where a is the fine-structure constant) and correlation effects (all products smd powers of single and double virtual excitations) is achieved. The Fock-space coupled-cluster method yields directly transition energies (ionization potentials, excitation energies, electron affinities). Results are in good agreement (usually better than 0.1 eV) with known experimental values. Properties of superheavy atoms which are not known experimentally can be predicted. Examples include the nature of the ground states of elements 104 md 111. Molecular applications are also presented. 

Fig. 7.20 Interaction of eight 2s orbitals of eight lithium atoms. The spacing of the energy levels depends upon the geometry of the cluster...

The dependence of the electronic structure of three-dimensional Ag clusters on size is shown in Fig. 7. Several trends are comparable to the trends observed for linear Ag particles. The HOMO and LUMO converge to 6 eV at the size range 30-50 atoms with a level spacing of 0.1 to 0.2 eV. The lowest 5s MO drops in energy with increasing size, unlike the behavior observed for linear Ag clusters. The occupied band width at 55 atoms is 5 eV. This lowest state (-11 eV) has dropped to an energy that should overlap d-orbital states since the latter are spread to almost the required degree in Fig. 6. This property would not be found in the linear clusters. 

In any finite-sized metal particle, the energy levels are split. Since we expect the atoms to be mobile within the clusters, the statistical level spacing 5 4 p/3iV should be applic-... 

At 0 K, the highest occupied molecular orbital (HOMO) in the valence band is known as Fermi level, which is named after Enrico Fermi, the physicist who first proposed it. The value of the Fermi level at absolute zero (-273.15°C) is called the Fermi energy and is a constant for each solid. The effect of size and shape of particle, temperature, electrophile, and nucleophile have an effect on this Fermi level shift. The mean level spacing d(e ) near the Fermi level is implied by the Kubo model by the following equation, where is the number of atoms in the cluster, Z is the valence of the atom, and is the... 

Fermi energy of the metal. Equation 13.15 clearly implies that when the number of atoms in the cluster increases, the mean level spacing diminishes (Fig. 13.12). 

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