Atomic orbitals Pauli 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which stated that no two electrons could share the same four quantum numbers (8), This version of the principle was superseded by a new formulation that avoids any reference to individual quantum numbers for separate electrons.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). 

The wave function, constructed from the atomic orbitals must be antisymmetric with respect to interchange of electrons in order to satisfy the Pauli exclusion principle, having different spin quantum numbers (a and J3) for two electrons which are in the same orbital. 

The spins of two electrons are said to be paired if one is T and the other 1 (Fig. 1.43). Paired spins are denoted Tl, and electrons with paired spins have spin magnetic quantum numbers of opposite sign. Because an atomic orbital is designated by three quantum numbers (n, /, and mt) and the two spin states are specified by a fourth quantum number, ms, another way of expressing the Pauli exclusion principle for atoms is... 

Before estabiishing the connection between atomic orbitals and the periodic table, we must first describe two additionai features of atomic structure the Pauli exclusion principle and the aufbau principle. 

C08-0054. The following are hypothetical configurations for a beryllium atom. Which use nonexistent orbitals, which are forbidden by the Pauli principle, which are excited states, and which is the... 

Exchanging the coordinates of the two electrons changes the wave function to tlf(x2, yi, z2) Pa(x, yi, Zi), which is the same as before. Since the wave function does not change sign, it is forbidden by the Pauli principle. Hence two electrons with the same spin cannot be described by the same wave function, or, in other words, an orbital cannot contain two electrons with the same spin. As we shall see, this form of the Pauli principle is used in describing atoms and molecules in terms of orbitals. 

In the 1920s it was found that electrons do not behave like macroscopic objects that are governed by Newton s laws of motion rather, they obey the laws of quantum mechanics. The application of these laws to atoms and molecules gave rise to orbital-based models of chemical bonding. In Chapter 3 we discuss some of the basic ideas of quantum mechanics, particularly the Pauli principle, the Heisenberg uncertainty principle, and the concept of electronic charge distribution, and we give a brief review of orbital-based models and modem ab initio calculations based on them. 

The relative size of atomic orbitals, which is found to increase as their energy level rises, is defined by the principal quantum number, n, their shape and spatial orientation (with respect to the nucleus and each other) by the subsidiary quantum numbers, Z and m, respectively. Electrons in orbitals also have a further designation in terms of the spin quantum number, which can have the values +j or — j. One limitation that theory imposes on such orbitals is that each may accommodate not more than two electrons, these electrons being distinguished from each other by having opposed (paired) spins, t This follows from the Pauli exclusion principle, which states that no two electrons in any atom may have exactly the same set of quantum numbers. 

Figure 1.12 The hypothetical formation of methane from an sp -hybridized carbon atom. In orbital hybridization we combine orbitals, not electrons. The electrons can then be placed in the hybrid orbitals as necessary for bond formation, but always in accordance with the Pauli principle of no more than two electrons (with opposite spin) in each orbital. In this illustration we have placed one electron...

The filling of atomic orbitals follows an n + , n) orbital scheme known as the Madelung [75-77] or Klechkovskii [78] rule. In this orbital scheme, the electron occupies free states with the smallest value of the sum A = + of the principal quantum number n and the azimuthal quantum number ( according to the Pauli exclusion principle. In the presence of several states with identical N, the state with... 

For two and three dimensions, it provides a crude but useful picture for electronic states on surfaces or in crystals, respectively. Free motion within a spherical volume gives rise to eigenfunctions that are used in nuclear physics to describe the motions of neutrons and protons in nuclei. In the so-called shell model of nuclei, the neutrons and protons fill separate s, p, d, etc orbitals with each type of nucleon forced to obey the Pauli principle. These orbitals are not the same in their radial shapes as the s, p, d, etc orbitals of atoms because, in atoms, there is an additional radial potential V(r) = -Ze2/r present. However, their angular shapes are the same as in atomic structure because, in both cases, the potential is independent of 0 and (f>. This same spherical box model has been used to describe the orbitals of valence electrons in clusters of mono-valent metal atoms such as Csn, Cu , Na and their positive and negative ions. Because of the metallic nature of these species, their valence electrons are sufficiently delocalized to render this simple model rather effective (see T. P. Martin, T. Bergmann, H. Gohlich, and T. Lange, J. Phys. Chem. 95, 6421 (1991)). 

That the numbers of atomic orbitals in an atom are dependent upon the Pauli exclusion principle... 

The treatment of atoms with more than one electron (polyelectronic atoms) requires consideration of the effects of interelectronic repulsion, orbital penetration towards the nucleus, nuclear shielding, and an extra quantum number (the spin quantum number) which specifies the intrinsic energy of the electron in any orbital. The restriction on numbers of atomic orbitals and the number of electrons that they can contain leads to a discussion of the Pauli exclusion principle, Hund s rules and the aufbau principle. All these considerations are necessary to allow the construction of the modern form of the periodic classification of the elements. 

The general structure of the Periodic Table, based on atomic orbital energies, the aufbau principle, the Pauli exclusion principle and Hund s rules. 

In ab initio theory, ECPs are considerably more complex. They properly represent not only Coulomb repulsion effects, but also adherence to the Pauli principle (i.e., outlying atomic orbitals must be orthogonal to core orbitals having the same angular momentum). This being said, we will not dwell on the technical aspects of their construction. Interested readers are referred to the bibliography at the end of the chapter. 

Molecular-orbital theory treats molecule formation from the separated atoms as arising from the interaction of the separate atomic orbitals to form new orbitals (molecular orbitals) which embrace the complete framework of the molecule. The ground state of the molecule is then one in which the electrons are assigned to the orbitals of lowest energy and are subject to the Pauli exclusion principle. Excited states are obtained by promoting an electron from a filled molecular orbital to an orbital which is normally empty in the ground state. The form of the molecular orbitals depends upon our model of molecule formation, but we shall describe (and use in detail in Sec. IV) only the most common, viz., the linear combination of atomic orbitals approximation. 

If each of the six n electrons in benzene occupied a single atomic n orbital and there were no interaction, each would have an energy of a. The total energy would then be 6a, which is zero if we assume, as above, that a is the zero of our energy scale. However, when the atomic orbitals interact to produce the MOs, the six electrons will now occupy these MOs according to Hund s rule and the Pauli exclusion principle. The first two will enter the A orbital, and the remaining four occupy the E orbitals. The total energy of the system is then... 

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