Multielectron atoms more than three electrons

A multielectron atom can lose more than one electron, but ionization becomes more difficult as cationic charge increases. The first three ionization energies for a magnesium atom in the gas phase provide an illustration. (Ionization energies are measured on gaseous elements to ensure that the atoms are isolated from one another.)... 

To date, there is no known analytic solution to the second-order differential Schrodinger equation for the helium atom. This does not mean that there is no solution, or that wavefunctions do not exist. It simply means that we know of no mathematical function that satisfies the differential equation. In fact, for atoms and molecules that have more than one electron, the lack of separability leads directly to the fact that there are no known analytical solutions to any atom larger than hydrogen. Again, this does not mean that the wavefunctions do not exist. It simply means that we must use other methods to understand the behavior of the electrons in such systems. (It has been proven mathematically that there is no analytic solution to the so-called three-body problem, as the He atom can be described. Therefore, we must approach multielectron systems differently.)... 

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. 

Putting electrons into orbitals in multielectron atoms is governed by three rules, the Aufbau principle, the Pauli exclusion principle, and Hund s rule. The Aufbau, or building-up, principle tells us to put the electrons in the lowest-energy orbital that is available. The Pauli principle restricts the contents of the orbital to two electrons, with spins, s, +1/2 and -1/2. Hund s rule of maximum multiplicity (the law of antisocial electrons... ) means that where there is more than one orbital of equivalent energy, the electrons distribute between them in order to keep apart. 

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