Electrons many-electron atoms

The true value of tk for a many-electron atom or a molecule is unknown. If we could set it equal ( expand it) to a linear combination of an infinite number of basis functions, each defined in a space of infinite dimensions, we could carry out an exact calculation of (k. Such a set of basis functions would be a complete set. 

Petersson and coworkers have extended this two-electron formulation of asymptotic convergence to many-electron atoms. They note that the second-order MoUer-Plesset correlation energy for a many-electron system may be written as a sum of pair energies, each describing the energetic effect of the electron correlation between that pair of electrons ... 

You are probably used to this idea from descriptive chemistry, where we build up the configurations for many-electron atoms in terms of atomic wavefunctions, and where we would write an electronic configuration for Ne as... 

Exact solutions to the electronic Schrodinger equation are not possible for many-electron atoms, but atomic HF calculations have been done both numerically and within the LCAO model. In approximate work, and for molecular applications, it is desirable to use basis functions that are simple in form. A polyelectron atom is quite different from a one-electron atom because of the phenomenon of shielding", for a particular electron, the other electrons partially screen the effect of the positively charged nucleus. Both Zener (1930) and Slater (1930) used very simple hydrogen-like orbitals of the form... 

In order to retain the orbital model for a many-electron atom, Hartree assumed that each electron came under the influence of the nuclear charge and an average potential due to the remaining electrons. He therefore retained the form of the radial equation for a one-electron atom, equation 12.2, but assumed that the mutual potential energy U was the sum of... 

For the following pairs of orbitals, indicate which is lower in energy in a many-electron atom. 

Analysis of the spectra of many-electron atoms shows the following similarities to the hydrogen atom case. 

Figure 15-11 shows a schematic energy level diagram of a many-electron atom. Blue patterns... 

It is possible to remove two or more electrons from a many-electron atom. Of course it is always harder to remove the second electron than the first because the second electron to come off leaves an ion with a double positive charge instead of a single positive charge. This gives an additional electrical attraction. Even so, the values of successive ionization energies have great interest to the chemist. 

Turn back to Figure 15-11, the energy level diagram of a many-electron atom, and consider the occupied orbitals of the element potassium. With 19 electrons placed, two at a time, in the orbitals of lowest energy, the electron configuration is... 

These days students are presented with the four quantum number description of electrons in many-electron atoms as though these quantum numbers somehow drop out of quantum mechanics in a seamless manner. In fact, they do not and furthermore they emerged, one at a time, beginning with Bohr s use of just one quantum number and culminating with Pauli s introduction of the fourth quantum number and his associated Exclusion Principle. 

The notion of electrons in orbitals consists essentially of ascribing four distinct quantum numbers to each electron in a many-electron atom. It can be shown that this notion is strictly inconsistent with quantum mechanics (7). Definite quantum numbers for individual electrons do not have any meaning in the framework of quantum mechanics. The erroneous view stems from the original formulation of the Pauli principle in 1925, which stated that no two electrons could share the same four quantum numbers (8), This version of the principle was superseded by a new formulation that avoids any reference to individual quantum numbers for separate electrons. The new version due to the independent work of Heisenberg and Dirac in 1926 states that the wave function of a many-electron atom must be antisymmetrical with respect to the interchange of any two particles (9,10). 

Soon after Bohr developed his initial configuration Arnold Sommerfeld in Munich realized the need to characterize the stationary states of the electron in the hydrogen atom by. means of a second quantum number—the so-called angular-momentum quantum number, Bohr immediately applied this discovery to many-electron atoms and in 1922 produced a set of more detailed electronic configurations. In turn, Sommerfeld went on to discover the third or inner, quantum number, thus enabling the British physicist Edmund Stoner to come up with an even more refined set of electronic configurations in 1924. 

This problem clearly did not worry Stoner, who just went ahead and assumed that three quantum numbers could be specified in many-electron atoms. In any case, Stoner s scheme solved certain problems present in Bohr s configurations. For example, Bohr had assigned phosphorus the configuration 2,4,4,41, but this failed to explain the fact that phosphorus shows valencies of three and five. Stoner s configuration for phosphorus was 2,2,2,4,2,2,1, which easily explains the valencies, since it becomes plausible that either the two or the three outermost subshells of electrons form bonds. 

In many electron atoms the maximum contributions to the polarizability and to London forces arise from configurations with more than one electron contributing to the net dipole moment of the atom. But in such configurations the electronic repulsion is especially high. The physical meaning to be attributed to the Qkl terms is just the additional electron repulsive energy which these configurations require. 

Boys, S. F., Proc. Roy. Soc. London) A207, 181, Electronic wave functions. IV. Some general theorems for the calculation of Schrodinger integrals between complicated vector-coupled functions for many-electron atoms."... 

A many-electron atom is also called a polyelectron atom. 

FIGURE 1.41 The relative energies of the shells, subshells, and orbitals in a many-electron atom. Each of the boxes can hold at most two electrons. Note the change in the order of energies of the 3d- and 4s-orbitals after Z = 20. 

As well as being attracted to the nucleus, each electron in a many-electron atom is repelled by the other electrons present. As a result, it is less tightly bound to the nucleus than it would be if those other electrons were absent. We say that each electron is shielded from the full attraction of the nucleus by the other electrons in the atom. The shielding effectively reduces the pull of the nucleus on an electron. The effective nuclear charge, Z lle, experienced by the electron is always less than the actual nuclear charge, Ze, because the electron-electron repulsions work against the pull of the nucleus. A very approximate form of the energy of an electron in a many-electron atom is a version of Eq. 14b in which the true atomic number is replaced by the effective atomic number ... 

In a many-electron atom, because of the effects of penetration and shielding, the order of energies of orbitals in a given shell is s < p < d < f. 

Describe the factors affecting the energy of an electron in a many-electron atom (Section l.f2). 

The observed structure of the spectra of many-electron atoms is entirely accounted for by the following postulate Only eigenfunctions which are antisymmetric in the electrons , that is, change sign when any two electrons are interchanged, correspond to existant states of the system. This is the quantum mechanics statement (26) of the Pauli exclusion principle (43). 

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