Atoms multielectron

The quantum mechanical description of a multielectron system is in principle the same than that of hydrogen atom. However the complexity of differential equations arising from building a Hamilton operator appropriate for multielectron systems prevents us from obtaining exact solutions. Nonetheless approximated methods, based fundamentally on reformulations of the problem 

The scheme in Fig. 1.6 shows schematically the energies of the orbitals in neutral multielectron atoms. 

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. 

FIGURE 1.14 Set of five 3d orbitals, from left to right m = -2, -1,0,1,2. 

TABLE 1.1 Configurations of the Elements of the First Row of the Periodic Table  

Imagine this as seats on a bus—when the English board the bus, each person sits in a new seat, rather than next to someone who is already there. Only when every seat is half full are book bags put on the floor, and the other half of the seat is occupied. Table 1.1 shows the electron configurations of the first row of the periodic table. The electrons are represented by arrows, spin +1/2 up, and spin -1/2 down. 

Write the electronic configuration for the ground state of each of the following (show the contents of p, Py, and p orbitals where appropriate)  

As suggested above, the results obtained for the hydrogen atom provide the basis of a very useful conceptual model for describing electrons in a multielectron atom. However, we must anticipate that adjustments will need to be made because, in a multielectron atom, we have interactions of electrons not only with the nucleus but with other electrons. A wealth of evidence, experimental and theoretical, supports the validity of a conceptual model based on the following points.  

The quantum mechanical wave function for a multielectron atom can be approximated as a superposition of orbitals, each bearing some resemblance to those describing the quantum states of the hydrogen atom. Each orbital in a multielectron atom describes how a single electron behaves in the field of a nucleus under the average influence of all the other electrons. 

The order in which we assign electrons to specific orbitals is based on minimizing Eatojj,- Orbitals that minimize the value of F may not necessarily minimize Eatoji,- Therefore, we must be careful not to place too much emphasis on the energies of the orbitals themselves. 

The form of the equation Eatom F — G might look, at first, a little odd because it seems to suggest that the energy of an atom is lowest when electron-electron 

Orbital energy-level diagrams for the hydrogen atom and a multielectron atom 

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With this background, we show how electron arrangements in multielectron atoms and the monatomic ions derived from them can be described in terms of—... 

For reasons we will discuss later, a fourth quantum number is required to completely describe a specific electron in a multielectron atom. The fourth quantum number is given the symbol ms. Each electron in an atom has a set of four quantum numbers n, l, mi, and ms. We will now discuss the quantum numbers of electrons as they are used in atoms beyond hydrogen. 

A hydrogen atom or a helium cation contains Just one electron, but nearly all other atoms and ions contain collections of electrons. In a multielectron atom, each electron affects the properties of all the other electrons. These electron-electron interactions make the orbital energies of eveiy element unique. 

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.)... 

Extracting an electron from helium takes less energy than expected because of electron-electron repulsion. The helium nucleus actually does pull twice as a hard as a hydrogen nucleus does, but the two electrons in helium are also repelling one another. The net effect is to make an electron in a multielectron atom easier to remove than one would expect if the other electrons were not present. 

NUCLEAR MAGNETIC RESONANCE Multielectronic atom interactions with the nucleus,... 

For multielectron atoms, the term symbol for an energy state is represented as... 

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. 

The importance of the spin quantum number comes when electrons occupy specific orbitals in multielectron atoms. According to the Pauli exclusion principle,... 

The difference in energy between subshells in multielectron atoms results from electron-electron repulsions. In hydrogen, the only electrical interaction is the attraction of the positive nucleus for the negative electron, but in multielectron atoms there are many different interactions to consider. Not only are there the attractions of the nucleus for each electron, there are also the repulsions between every electron and each of its neighboring electrons. 

The ground-state electron configuration of a multielectron atom is arrived at by following a series of rules called the aufbau principle. 

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