Multi-electron atoms

A multi-electron atom consists of a nucleus of a charge of +Ze and Z electrons each of charge —e. 

In the helium atom, the eigenfunctions of two electrons overlap highly in any quantum states, and so the electrons cannot be distinguished. Indistinguishability of electrons must be taken into account when describing multi-electron atoms. 

Considering multi-electron atoms we should bear in mind the Pauli exclusion principle. For electrons in a single atom, it states that no two electrons can have all of the four corresponding quantum numbers equal. That is, for example, if n, I and m/ are the same, s = 1/2 must be different. So two electrons have opposite spins. In any atomic system, for two identical electrons the total eigenfunction tp is antisymmetric. The Pauli principle requires the sign of eigenfunction to change when any two electrons swap. 

Each of Z moving electrons is affected by the Coulomb attraction of nucleus and by the Coulomb repulsion of the other Z — 1 electrons. Due to the magnitude of 

Energy of an electron in the multi-electron atom is given by 

Arrows are added to an orbital diagram to show the distribution of electrons in the possible orbitals and the relative spin of each electron. The following is an orbital diagram for a helium atom. 

The orbital distribution of electrons can also be described with a shorthand notation that describes the arrangement of electrons in the sublevels without reference to their spin. This shorthand for helium s configuration is written s and is commonly called an electron configuration. The 1 represents the first principal energy level, the s indicates an electron cloud with a spherical shape, and the 2 shows that there are two electrons in that Ir sublevel. 

In the broadest sense, an electron configuration is any description of the complete distribution of electrons in atomic orbitals. Although this can mean either an orbital diagram or the shorthand notation, this text will follow the common convention of referring to only the shorthand notation as an electron configuration. For example, 

14 to remind you of the correct order of filling of the sublevels. The following steps explain how to write it and use it yourself 

An atomic orbital may contain two electrons at most, and the electrons must have different spins. Because each s sublevel has one orbital and each orbital contains a maximum of two electrons, each s sublevel contains a maximum of two electrons. Because each p sublevel has three orbitals and each orbital contains a maximum of two electrons, each p sublevel contains a maximum of six electrons. Using similar reasoning, we can determine that each 5 sublevel contains a maximum often electrons, and each/sublevel contains a maximum of 14 electrons (Table 11.2). 

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When multi-electron atoms are combined to form a chemical bond they do not utilize all of their electrons. In general, one can separate the electrons of a given atom into inner-shell core electrons and the valence electrons which are available for chemical bonding. For example, the carbon atom has six electrons, two occupy the inner Is orbital, while the remaining four occupy the 2s and three 2p orbitals. These four can participate in the formation of chemical bonds. It is common practice in semi-empirical quantum mechanics to consider only the outer valence electrons and orbitals in the calculations and to replace the inner electrons + nuclear core with a screened nuclear charge. Thus, for carbon, we would only consider the 2s and 2p orbitals and the four electrons that occupy them and the +6 nuclear charge would be replaced with a +4 screened nuclear charge. 

This same procedure may be used to explain, in a qualitative way, the chemical behavior of the elements in the periodic table. The application of the Pauli exclusion principle to the ground states of multi-electron atoms is discussed in great detail in most elementary textbooks on the principles of chemistry and, therefore, is not repeated here. 

The Auger process is somewhat more complicated than that of X-ray photoemission (see Section 5.1.2). Let us firstly consider the energies of the various energy levels in an isolated, multi-electron atom (Figure 5.28). 

Unfortunately, the Schrodinger equation for multi-electron atoms and, for that matter, all molecules cannot be solved exactly and does not lead to an analogous expression to Equation 4.5 for the quantised energy levels. Even for simple atoms such as sodium the number of interactions between the particles increases rapidly. Sodium contains 11 electrons and so the correct quantum mechanical description of the atom has to include 11 nucleus-electron interactions, 55 electron-electron repulsion interactions and the correct description of the kinetic energy of the nucleus and the electrons - a further 12 terms in the Hamiltonian. The analysis of many-electron atomic spectra is complicated and beyond the scope of this book, but it was one such analysis performed by Sir Norman Lockyer that led to the discovery of helium on the Sun before it was discovered on the Earth. 

Figure Al.l Approximate energy level diagram for electronic orbitals in a multi-electron atom. Each horizontal line can accommodate two electrons (paired as so-called spin-up and spin-down electrons), giving the rules for filling the orbitals - two in the s-levels, 6 in the p-levels, 10 in the d-levels. Note that the 3d-orbital energy is lower than the 4p, giving rise to the d-block or transition elements. (From Brady, 1990 Figure 7.10. Copyright 1990 John Wiley Sons, Inc. Reprinted by permission of the publisher.)...

The quantum numbers n and i. Multi-electron atoms can be characterized by a set of principal and orbital quantum numbers n, t which labels one-electron wave functions (orbitals). 

In the lithium atom, and for all other multi-electron atoms, orbitals in different energy sublevels differ in energy. 

Interelectronic interactions that alter how any particular electron in a multi-electronic atom interacts with the nucleus and vice versa. These effects lead to so-called chemical shifts in NMR experiments, thereby providing valuable structural information concerning a molecule s bonding and conformation. 

In a multi-electronic atom, the following quanmm numbers can also be used to describe the energy levels, and the relationships between the quantum number of electrons are as follows. 

In a multi-electron atom, the energy sublevels within a principal energy level have different energies. For example, the three 2p orbitals are of higher energy than the 2s orbital. 

There is no allowance in this model for more than one electron. Some researchers tried to accommodate the extra electrons by using elliptical orbits, but that didn t work. How do we know it failed We could use the same criterion we used for hydrogen. The question to ask is Can we reproduce the experimental spectrum For multi-electron atoms there are far too many lines of nearly the same wavelength. The Bohr model cannot predict the occurrence of these transitions. We need a new model the new quantum mechanics. We shall use a formulation due to Schrodinger. 

Friedrich Hund determined a set of rules to determine the ground state of a multi-electron atom in the 1920s. One particular rule is called Hund s Rule in introductory chemistry courses. Hund s rule states that every orbital in a subshell is singly occupied with one electron before any one orbital is doubly occupied, and all electrons in singly occupied orbitals have the same spin. 

Just as the emission spectrum of hydrogen has four characteristic lines that identify it, so the emission spectrum for each element has a characteristic set of spectral lines. This means that the energy levels within the atom must also be characteristic of each element. But when scientists investigated multi-electron atoms, they found that their spectra were far more complex than would be anticipated by the simple set of energy levels predicted for hydrogen. Figure 7.4 shows spectra for three elements. 

The specific structure of the states for Hp was described in detail in [79], where it is mentioned as a well-known physical effect. For example, it was noted in the theory of disordered semiconductors that a similar "ladder" structure of states is realized for the system where the Coulomb potential is modified within a sphere as a constant potential (see [86,87] for a qualitative discussion and analytical solution of the problem). For quantum chemistry, the situation is interesting, as was shown in a series of publications of Connerade, Dolmatov and others (see e.g. [19,88-91] note that the series of publications on confined many-electron systems by these authors is much wider). The picture described is realized to some extent for the effective potential of inner electrons in multi-electron atoms, as it is defined by orbital densities with a number of maximal points. The existence of a number of extrema generates a system of the type described above [89]. This situation was modeled and described for the one-electron atom in [88] it is similar to that one described in Sections 5.2 and 5.3. 

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