Multielectron atoms energy levels

This energy-level diagram shows the relative energy levels of atomic orbitals in a multielectron atom (in this case rubidium, Rb, atomic number 37). 

Electron Spin and the Pauli Exclusion Principle Orbital Energy Levels in Multielectron Atoms Electron Configurations of Multielectron Atoms Electron Configurations and the Periodic Table... 

The three quantum numbers n, l, and wi/ discussed in Section 5.7 define the energy, shape, and spatial orientation of orbitals, but they don t quite tell the whole story. When the line spectra of many multielectron atoms are studied in detail, it turns out that some lines actually occur as very closely spaced pairs. (You can see this pairing if you look closely at the visible spectrum of sodium in Figure 5.6.) Thus, there are more energy levels than simple quantum mechanics predicts, and a fourth quantum number is required. Denoted ms, this fourth quantum number is related to a property called electron spin. 

Which two of the four quantum numbers determine the energy level of an orbital in a multielectron atom ... 

The equation is used to describe the behaviour of an atom or molecule in terms of its wave-like (or quantum) nature. By trying to solve the equation the energy levels of the system are calculated. However, the complex nature of multielectron/nuclei systems is simplified using the Born-Oppenheimer approximation. Unfortunately it is not possible to obtain an exact solution of the Schrddinger wave equation except for the simplest case, i.e. hydrogen. Theoretical chemists have therefore established approaches to find approximate solutions to the wave equation. One such approach uses the Hartree-Fock self-consistent field method, although other approaches are possible. Two important classes of calculation are based on ab initio or semi-empirical methods. Ah initio literally means from the beginning . The term is used in computational chemistry to describe computations which are not based upon any experimental data, but based purely on theoretical principles. This is not to say that this approach has no scientific basis - indeed the approach uses mathematical approximations to simplify, for example, a differential equation. In contrast, semi-empirical methods utilize some experimental data to simplify the calculations. As a consequence semi-empirical methods are more rapid than ab initio. 

It is well to realize that in a multielectron atom, one with say twenty or more electrons, the energies of all of the levels are more or less dependent on the populations of all the other levels. Hence the diagram is rather complicated. 

The shape of the belt of stability can be understood by recognizing that the protons and neutrons can, to a good approximation, be described as if they occupy individual quantum energy levels in much the same way that electrons in multielectron atoms are described as occupying individual elecbon orbitals (Chapter 2). This shell model of the nucleus was developed independently by Maria Goeppert-Mayer and by Hans Jensen and coworkers. The shell model has been successful at explaining a number of general trends related to nuclear stability ... 

The Schrodinger equation for multielectron atoms has terms to account for the interactions of the electrons with one another that make it too complicated to solve exactly. However, approximate solutions indicate that the orbitals in multielectron atoms are hydrogen-like— they are similar to the s, p, d, and/orbitals that we examined in Chapter 7. In order to see how the electrons in multielection atoms occupy these hydrogen-Uke orbitals, we must examine two additional concepts the effects of electron spin, a fundamental property of all electrons that affects the number of electrons allowed in any one orbital and sublevel energy splitting, which determines the order of orbital filling within a level. 

Why are the sublevels within a principal level spUt into different energies for multielectron atoms but not for the hydrogen atom ... 

Table 17.2 gives the R i functions for n = 1,2, and 3. The entries in the table are for the hydrogen-like atom, which is a hydrogen atom with the nuclear charge equal to a number of protons denoted by Z. The He+ ion corresponds to Z = 2, the Li + ion corresponds to Z = 3, and so on. This modification to the R i functions will be useful when we discuss multielectron atoms in the next chapter. To obtain the radial factors and the energy levels of a hydrogen-like atom we need to replace the variable p by... 

The energy levels of multielectron atoms can be characterized by orbital and spin angular momentum values in the Russell-Saunders approximation. 

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