Atomic Point-Like Nucleus Case

This is the charge density distribution for the point-like nucleus case (PNC), which we include for completeness and because of the importance of this model as a reference for any work with an extended model of the atomic nucleus (finite nucleus case, FNC). The charge density distribution can be given in terms of the Dirac delta distribution as... 

The eigenvalue problems, defined above for the radial functions through Eqs. (113) to (116), reduce to homogeneous problems in the case of one-electron atoms, since the various terms A (r) and j(r) are zero. Closed-form solutions are well known for the one-electron atom with a point-like nucleus, both in the non-relativistic and the relativistic framework, but do not exist for the large majority of finite nucleus models. A determination of the energy eigenvalue for a bound state i of the one-electron atom with a finite nucleus, is then possible in two ways, either by perturbation... 

Kiuc( ) continuous at the matching radius. The inner solution can be calculated, for any such finite nucleus model and for an axbitrary state i, from a short-range series expansion, whereas the outer solution is known as the irregular solution of the Coulomb problem of the hydrogen-like atom with a point-like nucleus, both in the non-relativistic and the relativistic case. The logarithmic derivative for the outer solution, is very sensi-... 

Similar correction cases, either in higher orders or on excited hydrogenic states, may be considered following the same line of analysis however with the same principally conclusion that they do not in fact contribute to the real shift (or perturbation) of the hydrogenic atoms within the point like nucleus framework this leads with the idea that indeed, for atomic and supra-atomic systems the point like hypothesis of the nucleus finely works and will be in next assumed as such (for instance when treating the chemical bonding by means of the Dirac theory, see next chapters). 

In the analysis that leads to Eq. (9.153) it is assumed that the electron-electron interaction potential functions do not behave like 1/r or are even more singular at the origin. Since in the relativistic case the point-like nucleus can only be applied for Z < c, the prefactor in Eq. (9.187) is always < 1. The term in parentheses in Eq. (9.187) is always larger than —1, such that jS > —1. Typical values for the Zn atom are —0.02. Therefore, in case of singular behavior... 

Many applications in chemistry require us to interpret—and even predict—the results of measurements where we have only limited information about the system and the process involved. In such cases the best we can do is identify the possible outcomes of the experiment and assign a probability to each of them. Two examples illustrate the issues we face. In discussions of atomic structure, we would like to know the position of an electron relative to the nucleus. The principles of quantum mechanics tell us we can never know the exact location or trajectory of an electron the most information we can have is the probability of finding an electron at each point in space around the nucleus. In discussing the behavior of a macroscopic amount of helium gas confined at a particular volume, pressure, and temperature we would like to know the speed with which an atom is moving in the container. We do not have experimental means to tag a particular atom. 

We can now consider the question of the electrical centroids of the positive and the negative charges apart, that is to say, the electrical centroid of the nuclei alone, and the electrical centroid of the electron cloud alone. It may happen that the two points coincide, just as they always do in individual atoms, where the positive centroid is identical with the nucleus, and where also the centroid of the negative charge cloud, on account of the central symmetry of its charge distribution, always coincides with the nucleus. In general, however, the two centroids will be distinct from each other consequently the external action of the molecule is like that of an electric dipole. In this case we speak of a permanent electric dipole moment, and denote it by the vector... 

Due to the discontinuity of p r) at r = R, the second and higher derivar tives of V r) do not exist at this point. Only in the non-relativistic case the radial functions for hydrogen-like atoms with this electron-nucleus potential are known analytically in closed form. The corresponding energy eigenvalues must be determined iteratively [57-59]. 

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