Spherical potential

The metal cluster will be modeled as an infinitely deep spherical potential well with the represented by an infinitely high spherical barrier. Let us place this barrier in the center of the spherical cluster to simplify the calculations. The simple Schrodinger equation, containing only the interaction of the electrons with the static potential and the kinetic energy term and neglecting any electron-electron interaction, can then be solved analytically, the solutions for the radial wave functions being linear combinations of spherical Bessel and Neumann functions. 

Consider now the solutions of the spherical potential well with a barrier at the center. Figure 14 shows how the energies of the subshells vary as a function of the ratio between the radius of the C o barrier Rc and the outer radius of the metal layer R ui- The subshells are labeled with n and /, where n is the principal quantum number used in nuclear physics denoting the number of extrema in the radial wave function, and / is the angular momentum quantum number. 

Fig. 14. Energy levels calculated for an infinitely deep spherical potential well of radius with an infinitely high central potential barrier with a radius the zigzag line...

Finally we want to remark, that the theory presented applies for non-spherical potentials as well. Explicit expressions such as in Eq. (4) are replaced by expressions in terms of t matrices, which are non-diagonal in the angular momentum. Matrix expressions such as Eqs. (27) and (28) remain unchanged. 

These parameters cannot be compared with Eqs. 9 and 10, since pure carbon dioxide cannot be adequately represented by a spherical potential. Figure 18 shows that they give only a moderate... 

In the s-wave-tip model (Tersoff and Hamann, 1983, 1985), the tip was also modeled as a protruded piece of Sommerfeld metal, with a radius of curvature R, see Fig. 1.25. The solutions of the Schrodinger equation for a spherical potential well of radius R were taken as tip wavefunctions. Among the numerous solutions of this macroscopic quantum-mechanical problem, Tersoff and Hamann assumed that only the s-wave solution was important. Under such assumptions, the tunneling current has an extremely simple form. At low bias, the tunneling current is proportional to the Fermi-level LDOS at the center of curvature of the tip Pq. 

Fig. 1.25. The s-wave-tip model. The tip was modeled as a spherical potential well of radius R. The distance of nearest approach is d. The center of curvature of tip is To, at a distance (R + d) from the sample surface. Only the 5-wave solution of the spherical-potential-well problem is taken as the tip wavefunction. In the interpretation of the images of the reconstructions on Au(llO), the parameters used are R = 9 A, d = 6 A. The center of curvature of the tip is 15 A from the Au surface. (After Tersoff and Hamann, 1983.)...

The original 5-wave-tip model described the tip as a macroscopic spherical potential well, for example, with r 9 A. It describes the protruded end of a free-electron-metal tip. Another incarnation of the 5-wave-tip model is the Na-atom-tip model. It assumes that the tip is an alkali metal atom, for example, a Na atom, weakly adsorbed on a metal surface (Lang, 1986 see Section 6.3). Similar to the original 5-wave model, the Na-atom-tip model predicts a very low intrinsic lateral resolution. 

G. T. McConville. A consistent spherical potential function for para-hydrogen. J. Chem. Phys., 74 2201, 1981. 

Soft non-spherical potentials are one step towards a more realistic model. Luckhurst et al. [413] have used a potential having the prolate spheroidal symmetry discussed above but which is based on the well known Lennard-Jones or twelve-six potential. This involves an attractive 1/r6 potential based on London forces and a repulsive 1/r12 potential. Once again it is possible to predict the existence of a smectic phase. 

An alternative simple modeling of doped fullerenes, specifically, A C6o, was developed initially in [34]. It was then used extensively in a number of photoionization studies of thus encaged atoms [34 41], The method is based on approximating the C60 cage by a spherical potential V(r) which differs from zero only within an infinitesimally thin wall of a sphere of radius RC/ the latter being considered the C60 radius, Rc = 6.639 au [47] ... 

As stated earlier, coexistence of deformed and spherical shapes is characteristic of the lighter deformed A=100 nuclei. Both and have anomalous states (at 599 and 890 keV respectively) that are not associated with the lower energy bands. These states decay only to the 3/2[301] bandhead, hence could be the pjy2 state expected for the 39t 1 proton in a spherical potential. Spherical A=100 nuclei have a 2 + energy of -800 keV, and the anomalous states are fed only by -800 keV y rays, vMch supports our speculation that they may be p y2 particles ooupled to a spherical core. 

Approximating the Coulomb interaction between the electrons by a mean spherical potential V(r), it follows that... 

Pauly, H. (1979). Elastic scattering cross sections I Spherical potentials, in Atom-Molecule Collision Theory, ed. R.B. Bernstein (Plenum Press, New York). 

In the case of metal clusters, for example, valence electrons show the shell structure which is characteristic of the system consisting of a finite number of fermions confined in a spherical potential well [2]. This electronic shell structure, in turn, motivated some theorists to study clusters as atomlike building blocks of materials [3]. The electronic structure of the metallofullerenes La C60 [4] and K C60 [5] was investigated from this viewpoint. This theorists dream of using clusters as atomlike building blocks was first realized by the macroscopic production of C60 and simultaneous discovery of crystalline solid C60, where C60 fullerenes form a close-packed crystalline lattice [6]. 

This relation applies generally and specifically to spheres and cylinders. Empirical relations have been developed to convert the measured lifetime (x) to this mean free path ( ). In case of small pores (<2 nm) an earlier version is adequate [43, 44], The model is based on the assumptions of spherical potential wells with infinite depth and radius r that traps the positronium. 

Although these integrals cannot be evaluated in closed form they are well suited for numerical evaluation. To get the two-electron integrals into an equally acceptable form it is necessary to make a few simplifying approximations. Firstly it is noted that for p 1 the two integrals become identical. This integral is simplified by first evaluating the the spherical potential [139]... 

First of all, the existence of the surface establishes a certain correlation between different electronic states, particularly between incident and reflected electrons. In addition there are electron density variations both within the surface and along the surface normal. Since the screening radius increases with decreasing electron density, it is obviously unrealistic to third of a spherical potential hole accompanying its electron in the immediate surface region, where the electron density drops very rapidly to zero. At the very lowest densities, in particular, the potential must somehow go over in continuous fashion into the classical image potential applying far away from the metal surface. Finally, from the dynamic nature of these interactions it follows that the problem must be dealt with self-consistently each electron contributes to the holes of all the other electrons in a manner dependent on the detailed features of its own potential hole. 

This species is essentially an electron stabilised by the surrounding water molecules. It has been the subject of detailed theoretical studies(35), but can be considered as an electron in a spherical potential well consisting of solvent molecules. Specific short-range solvation effects are thought to be important as well as long-range polarization forces. 

Various refinements of the above model have been proposed for example, using alternative spherical potentials or allowing for nonspherical perturbations,and these can improve the agreement of the model with the abundance peaks observed in different experimental spectra. For small alkali metal clusters, the results are essentially equivalent to those obtained by TSH theory, for the simple reason that both approaches start from an assumption of zeroth-order spherical symmetry. This connection has been emphasized in two reviews,and also holds to some extent when considerations of symmetry breaking are applied. This aspect is discussed further below. The same shell structure is also observed in simple Hiickel calculations for alkali metals, again basically due to the symmetry of the systems considered. However, the developments of TSH theory, below, and the assumptions made in the jellium model itself, should make it clear that the latter approach is only likely to be successful for alkali and perhaps alkali earth metals. For example, recent results for aluminium clusters have led to the suggestion that symmetry-breaking effects are more important in these systems. ... 

Quantum chemical calculations on RE + /3-diketonate complexes are presently restricted to two approaches the effective core potential (ECP) and the SMLC/AMl method. The SMLC/AMl method is a very powerful addition to the semi-empirical molecular orbital method AMI in that it allows the prediction of geometric parameters of rare-earth /3-diketonate complexes of very difficult experimental determination. In this method the RE3+ ion is a sparkle represented by a -i-3e charge in the center of a repulsive spherical potential of the form exp(—ar). Recently, ab initio effective core potential calculations have also succeeded in reproducing the coordination polyhedron geometries of RE + /S-diketonate complexes with high accuracy . 

In the period 1940-1946, Ogg (132) developed the first quantitative theory for the solvated electron states in liquid ammonia. The Ogg description relied primarily on the picture of a particle in a box. A spherical cavity of radius R is assumed around the electron, and the ammonia molecules create an effective spherical potential well with an infinitely high repulsive barrier to the electron. It is this latter feature that does not satisfactorily represent the relatively weakly bound states of the excess electron (9,103). However, the idea of a potential cavity formed the basis of subsequent theoretical treatments. Indeed, as Brodsky and Tsarevsky (9) have recently pointed out, the simple approach used by Ogg for the excess electron in ammonia forms the basis of the modem theory (157) of localized excess-electron states in the nonpolar, rare-gas systems. [The similarities between the current treatments of trapped H atoms and excess electrons in the rare-gas solids has also recently been reviewed by Edwards (59).]... 

We consider a spherical potential which is zero up to r = a, where it is infinite. The radial boundary condition results in the radial equation (4.10) being an eigenvalue problem with eigenvalues which are positive with respect to the zero of energy. We define a wave number k by... 

Figure 1, Energy level diagram for high-spin ions with a D-free ion ground term, a) Free ion in a spherical potential b) splitting diagram for d and d ions in an octahedral fields and d and d ions in a tetrahedral field c) splitting diagram for d and d ions in an octahedral fields and for d and rf ions in a tetrahedral field.

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