We present a unified description of the position-space wave functions, the momentum-space wave functions, and the phase-space Wigner functions for the bound states of a Morse oscillator. By comparing with the functions for the harmonic oscillator the effects of anharmonicity are visualized. Analytical expressions for the wave functions and the phase space functions are given, and it is demonstrated how a numerical problem arising from the summation of an alternating series in evaluating Laguerre functions can be circumvented. The method is applicable also for other problems where Laguerre functions are to be calculated. The wave and phase space functions are displayed in a series of curves and contour diagrams. An Appendix discusses the calculation of the modified Bessel functions of real, positive argument and complex order, which is required for calculating the phase space functions for the Morse oscillator.
Crystalline cadmium sulfide is a semiconductor for which the wurtzite and zinc blende structures are energetically almost degenerate. Due to quantum-confinement effects, it is possible to tune the optical properties of finite cadmium sulfide clusters by varying their size. Here, we report results of a theoretical study devoted to the properties of stoichiometric Cd n S n clusters as a function of their size n. We have optimized the structure, whereby our initial structures are spherical parts of either of the two crystal structures, and we have studied systems with up to almost 200 atoms. The calculations were performed by using a simplified LCAO-DFT-LDA scheme. The results include the structure, electronic energy levels (in particular the frontier orbitals HOMO and LUMO), and stability as a function of size. The results allow for a unique definition of a surface region. The Mulliken populations indicate that the bonds within this region are more ionic than in the bulk. Furthermore, whereas the HOMO is delocalized over major parts of the nanoparticle, the LUMO is a surface state, which confirms recent experimental findings. Finally, the relative stability of the zinc blende and wurtzite structures is strongly dependent on the size of the system, and there is a close connection between the HOMO-LUMO energy gap and stability.
The Pauli exclusion principle gives an upper bound of 1 on the natural occupation numbers. Recently there has been an intriguing amount of theoretical evidence that there is a plethora of additional generalized Pauli restrictions or (in)equalities, of kinematic nature, satisfied by these numbers [1]. Here for the first time a numerical analysis of the nature of such constraints is effected in real atoms. The inequalities are nearly saturated, or quasi-pinned. For rank-six and rank-seven approximations for lithium, the deviation from saturation is smaller than the lowest occupancy number. For a rank-eight approximation we find well-defined families of saturation conditions.
The four most stable structures of Ni N clusters with N from 2 to 150 have been determined using a combination of the embedded-atom method in the version of Daw, Baskes and Foiles, the variable metric/quasi-Newton method, and our own Aufbau/Abbau method. A systematic study of energetics, structure, growth, and stability of also larger clusters has been carried through without more or less severe assumptions on the initial geometries in the structure optimization, on the symmetry, or on bond lengths. It is shown that cluster growth is predominantly icosahedral with islands of fcc, tetrahedral and decahedral growth. For the first time in unbiased computations it is found that Ni 147 is the multilayer (third Mackay) icosahedron. Further, we point to an enhanced ability of fcc clusters to compete with the icosahedral and decahedral structures in the vicinity of N = 79. In addition, it is shown that conversion from the hcp/anti-Mackay kind of icosahedral growth to the fcc/Mackay one occurs within a transition layer including several cluster sizes. Moreover, we present and apply different analytical tools in studying structural and energetic properties of such a large class of clusters. These include means for identifying the overall shape, the occurrence of atomic shells, the similarity of the clusters with, e.g., fragments of the fcc crystal or of a large icosahedral cluster, and a way of analysing whether the N -atom cluster can be considered constructed from the (N − 1)-atom one by adding an extra atom. In addition, we compare in detail with results from chemical-probe experiment. Maybe the most central result is that first for clusters with N above 80 general trends can be identified.
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