Spherically symmetrical electronic configurations

Note that there are two anomalies in the first transition series [Ar]3d54s1 (instead of [Ar]3d44s2) for Cr and [Ar]3d104s1 (instead of [Ar]3d94s2) for Cu. These two configurations arise from the extra stability of a half-filled or completely filled subshell. Such stability comes from the spherically symmetric electron density around the nucleus for these configurations. Take the simpler case of p3 as an example. The angular portion of the density function is proportional to... 

Section 1.1 A review of some fundamental knowledge about atoms and electrons leads to a discussion of wave functions, orbitals, and the electron configurations of atoms. Neutral atoms have as many electrons as the number of protons in the nucleus. These electrons occupy orbitals in order of increasing energy, with no more than two electrons in any one orbital. The most frequently encountered atomic orbitals in this text are 5 orbitals (spherically symmetrical) and p orbitals ( dumbbell -shaped). 

The other mechanism is called the Fermi contact interaction and it produces the isotropic splittings observed in solution-phase EPR spectra. Electrons in spherically symmetric atomic orbitals (s orbitals) have finite probability in the nucleus. (Mossbauer spectroscopy is another technique that depends on this fact.) Of course, the strength of interaction will depend on the particular s orbital involved. Orbitals of lower-than-spherical symmetry, such as p or d orbitals, have zero probability at the nucleus. But an unpaired electron in such an orbital can acquire a fractional quantity of s character through hybridization or by polarization of adjacent orbitals (configuration interaction). Some simple cases are described later. 

Consider a doublet of electronic orbitals (fa,fp) localized around some reference atom taken at origin. We assume that the environment potential felt by an electron in the reference atom is the sum of atomic-like spherically symmetric potentials Va(r — R) centred at the other atomic positions R, and we take as z-axis the interatomic axis of the linear molecule in the equilibrium configuration. We consider thus the matrix... 

Another factor is that the configuration 3dw4s has a spherically symmetrical distribution of electron density, a stabilizing arrangement characteristic of all filled or half-filled subshells. On the other hand, the configuration 3d 4s2 has a hole (a missing electron) in the 3d subshell, destroying the symmetry and any extra stabilization. 

We can then make an even more drastic approximation and represent the molecular orbitals in these configurations by pure 2p atomic orbitals on the O atom. This approximation was called pure precession by Van Vleck [41] in this approximation the electrons in these outermost orbitals are in a spherically symmetric environment and they have a well defined value of the orbital angular momentum quantum number / (unity for a p orbital). In the pure precession approximation, we can derive very simple expressions for the g-factors [66], The values for OH predicted on the basis of this very simple model are given in table 9.4. The fact that they agree reasonably well with the experimental numbers suggests that the theoretical model is essentially correct. 

This result can be further generalized the charge distribution of a filled subdivision (/),asalso of each closed principal quantum state or shell (n) is spherically symmetrical. All inert gas atoms or ions, and also all ions with an 18-electron configuration, such as Cu+, Ag+, Zn2+, Ga3+ etc., (in which a d-sub-division is completely filled), therefore have spherical symmetry. 

Using a central field approximation in which it is assumed that each electron moves independently in an average spherically symmetric potential, it is possible to solve for the energies of the different configurations. Calculations of this type show that the / -configuration is the lowest energy configuration for the trivalent lanthanides and actinides. 

However, in quantum mechanics, as is well known, a particle cannot lie absolutely at rest on a certain point. That would contradict the uncertainty relation. According to quantum mechanics our isotropic oscillators, even in their lowest states, make a so-called zero-point motion which one can only describe statistically, for example, by a probability function which defines the probability with which any configuration occurs whilst one cannot describe the way in which the different configurations follow each other. For the isotropic oscillators these probability functions give a spherically symmetric distribution of configurations round the rest position. (The rare gases, too, have such a spherically symmetrical distribution for the electrons around the nucleus.)... 

The amount of tt back donation in a symmetric electronic system, as a atom, is still controversial. The d configuration, which in the gaseous state has spherical symmetry, is very stable. As a consequence, the ionization potential in the configuration is much higher than in the spin-paired d or d configuration 150). 

In the ground state, both atoms have three singly occupied p orbitals. With this configuration, which gives a state, is associated a spherically symmetrical distribution of electrons and a high ionisation potential. Since the compounds are essentially covalent, however, ionisation potentials are less important than electronegativities. 

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