Pauli exclusion principle, application

The resolution of this issue is based on the application of the Pauli exclusion principle and Femii-Dirac statistics. From the free electron model, the total electronic energy, U, can be written as... 

The application of the quantum mechanics to the interaction of more complicated atoms, and to the non-polar chemical bond in general, is now being made (45). A discussion of this work can not be given here it is, however, worthy of mention that qualitative conclusions have been drawn which are completely equivalent to G. N. Lewis s theory of the shared electron pair. The further results which have so far been obtained are promising and we may look forward with some confidence to the future explanation of chemical valence in general in terms of the Pauli exclusion principle and the Heisenberg-Dirac resonance phenomenon. 

Introduction of the half-integral spin of the electrons (values h/2 and —fe/2) alters the above discussion only in that a spin coordinate must now be added to the wavefunctions which would then have both space and spin components. This creates four vectors (three space and one spin component). Application of the Pauli exclusion principle, which states that all wavefunctions must be antisymmetric in space and spin coordinates for all pairs of electrons, again results in the T-state being of lower energy [equations (9) and (10)]. 

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 application of the Pauli exclusion principle is necessary for the understanding of the normal states of atoms. There is a simple why of... 

The previously described theory in its original form assumes that the classical kinetic theory of gases is applicable to the electron gas, that is, electrons are expected to have velocities that are temperature dependent according to the Maxwell-Boltzmann distribution law. But, the Maxwell-Boltzmann energy distribution has no restrictions to the number of species allowed to have exactly the same energy. However, in the case of electrons, there are restrictions to the number of electrons with identical energy, that is, the Pauli exclusion principle consequently, we have to apply a different form of statistics, the Fermi-Dirac statistics. 

Fermi-Dirac statistics — Fermi-Dirac statistics are a consequence of the extension of the application of Pauli s exclusion principle, which states that no two electrons in an atom can be in the same quantum state, to an ensemble of electrons, i.e., that no two could have the same set of quantum numbers. Mathematically, in a set of indistinguishable particles, which occupy quantum states following the Pauli exclusion principle, the probability of occupancy for a state of energy E at thermal equilibrium is given by f(E) = —(A)—, where E is the... 

Some general aspects related to the derivation, and interpretations of ELF analysis, as well as some representative applications have been briefly discussed. The ELF has emerged as a powerful tool to understand in a qualitative way the behaviour of the electrons in a nuclei system. It is possible to explain a great variety of bonding situations ranging from the most standard covalent bond to the metallic bond. The ELF is a well-defined function with a nice pragmatic characteristic. It does not depend neither on the method of calculation nor on the basis set used. Its application to understand new bond phenomenon is already well documented and it can be used safely. Its relationship with the Pauli exclusion principle has been carefully studied, and its consequence to understand the chemical concept of electron pair has also been discussed. A point to be further studied is its application to transition metal atoms with an open d-shell. The role of the nodes of the molecular orbitals and the meaning of ELF values below 0.5 should be clarified. 

Fermi-Dirac distribution - A modification of the Boltzmann distribution which takes into account the Pauli exclusion principle. The number of particles of energy E is proportional to [e > +l] , where p is a normalization constant, k the Boltzmann constant, and T the temperature. The distribution is applicable to a system of fermions. 

Although the four quantum numbers n, 1, m, and s, the Pauli Exclusion Principle, and Hund s rules were developed in the context of the Bohr-Sommerfeld model, they all found immediate application to Schrodinger s new quantum mechanics. The first three numbers specified atomic orbitals (replacing Bohr s orbits). Physicist Max Bom (1882-1970) equated the square of the wave functions, to regions of probability for finding electrons in each orbital. Werner Heisenberg (1901-76), whose mathematics provide the foundation of quantum mechanics, developed the uncertainty principle the product of the uncertainty in position (Ax) of a tiny particle such as an atom (or an electron) and the uncertainty in its momentum (Ap) is larger than the quantum (h/47t) ... 

Note that based on the Pauli exclusion principle, this is not applicable to electron pairs with the same quantum numbers except that of spin. This indicates that the wavefunction E is considered to have a hole at rn = 0, which is called a Coulomb hole. By excluding close electrons, this Coulomb hole decreases the Coulomb interactions, thus lowering the total energies. This energy gain corresponds to the electron correlation. 

Consider the Is orbital = 1, / = 0 and m, = 0, and there are no possibilities for changes in these values (any electron in a Is orbital must be associated with them). One electron could have a value of m of A, but the second electron must have the alternative value of m of - A. The two electrons occupying the same orbital must have opposite spins . Since there are no other combinations of the values of the four quantum numbers, it is concluded that only two electrons may occupy the Is orbital and that there can only be one Is orbital in any one atom. Similar conclusions are valid for all other orbitals. The application of the Pauli exclusion principle provides the necessary framework for the observed electronic configurations of the elements. 

A key concept in LCAO-MO bonding theory is the formation of the same number of molecular orbitals as the number of atomic orbitals that are combined (e.g., there are 12 M.O.s formed when 4s and 3d atomic orbitals combine in the Ti2 molecule, see Figure 2.68a). As the number of atoms increases to infinity within a crystal lattice, the AE 0 between energy levels within bonding and antibonding regions (Figure 2.68b). This is an application of the Pauli exclusion principle, which states for N electrons, there must be N/2 available states to house the electron density. ... 

Repulsive interactions are important when molecules are close to each other. They result from the overlap of electrons when atoms approach one another. As molecules move very close to each other the potential energy rises steeply, due partly to repulsive interactions between electrons, but also due to forces with a quantum mechanical origin in the Pauli exclusion principle. Repulsive interactions effectively correspond to steric or excluded volume interactions. Because a molecule cannot come into contact with other molecules, it effectively excludes volume to these other molecules. The simplest model for an excluded volume interaction is the hard sphere model. The hard sphere model has direct application to one class of soft materials, namely sterically stabilized colloidal dispersions. These are described in Section 3.6. It is also used as a reference system for modelling the behaviour of simple fluids. The hard sphere potential, V(r), has a particularly simple form ... 

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