Quantum theory Pauli exclusion principle

See quantum number orbital theory Pauli exclusion principle. 

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. 

In this chapter we give a brief review of some of the basic concepts of quantum mechanics with emphasis on salient points of this theory relevant to the central theme of the book. We focus particularly on the electron density because it is the basis of the theory of atoms in molecules (AIM), which is discussed in Chapter 6. The Pauli exclusion principle is also given special attention in view of its role in the VSEPR and LCP models (Chapters 4 and 5). We first revisit the perhaps most characteristic feature of quantum mechanics, which differentiates it from classical mechanics its probabilistic character. For that purpose we go back to the origins of quantum mechanics, a theory that has its roots in attempts to explain the nature of light and its interactions with atoms and molecules. References to more complete and more advanced treatments of quantum mechanics are given at the end of the chapter. 

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. 

Lewis and many other chemists saw in the Pauli exclusion principle and the Uhlenbeck-Goudsmit spin hypothesis firm physical support for the chemical valence theory of the electron pair. In fact, the Pauli exclusion principle has to be postulated within the physics of the quantum theory.23 Accepting the Nobel Prize in physics for 1945, Pauli expressed regret that the principle cannot be derived ab initio. 

There is an implicit assumption contained in all of the above The two bonding electrons are of opposite spin. If two electrons are of parallel spin, no bonding occurs, but repulsion instead curve /, Fig. 5.1). This is a result of the Pauli exclusion principle. Because of the necessity for pairing in each bond formed, the valence bond theory is often referred to as the electron pair theory, and it forms a logical quantum-mechanical extension of Lewis s theory of electron pair formation. 

PAULI, WOLFCANC ERNST (1900-1958). Pauli was an Austrian theorerical physicist, After WWT1. he became an American citizen. When just 20 years of age he wrote The Theory of Relativity." Later he wrote articles on Quantum Theoiy and Principles of Wave Mechanics." He is most remembered for formulating the Pauli exclusion principle- . This principle says that two electrons in an atom can never exist in the same state, This is important concept for modern physics. Pauli was awarded the Nobel Prize in physics in 1945 for this discovery. 

Nonrelativistic quantum mechanics, extended by the theory of electron spin and by the Pauli exclusion principle, provides a reliable theory for the computation of atomic spectral frequencies and intensities, of cross sections for scattering or capture of electrons by atomic systems, of chemical bonds and many properties of solids, including magnetic properties, although with much more complicated systems it has not always proved possible to develop with adequate accuracy the consequences of the theory. Quantum mechanics has also had a limited success in nuclear theory although m this field it is possible that a more fundamental system of mechanics is required. 

The development of quantum theory, particularly of quantum mechanics, forced certain changes in statistical mechanics. In the development of the resulting quantum statistics, the phase space is divided into cells of volume hf. where h is the Planck constant and / is the number of degrees of freedom. In considering the permutations of the molecules, it is recognized that the interchange of two identical particles does not lead to a new state. With these two new ideas, one arrives at the Bose-Einstein statistics. These statistics must be further modified for particles, such as electrons, to which the Pauli exclusion principle applies, and the Fermi-Dirac statistics follow. 

In its most general physical use, occupation number is an integer denoting the number of particles that can occupy a well-defined physical state. For fermions it is 0 or 1, and for bosons it is any integer. This is because only zero or one fermion(s), such as an electron, can be in the state defined by a specified set of quantum numbers, while a boson, such as a photon, is not so constrained (the Pauli exclusion principle applies to fermions, but not to bosons). In chemistry the occupation number of an orbital is, in general, the number of electrons in it. In MO theory this can be fractional. 

Even the layout of the periodic table of the elements cannot be derived from quantum theory without assuming an empirical concept, known as the Pauli exclusion principle. An alternative derivation (4.6.1) through number theory predicts the correct periodicity, without assuming the exclusion principle. In fact, the operation of an exclusion principle can be inferred from this periodic structure and reduced to a property of space, but it remains impossible to reconstruct or predict from more basic principles. 

Physical chemistry of the positron and Ps is unique in itself, since the positron possesses its own quantum mechanics, thermodynamics and kinetics. The positron can be treated by the quantum theory of the electron with two important modifications the sign of the Coulomb force and absence of the Pauli exclusion principle with electrons in many electron systems. The positron can form a bound state or scatter when it interacts with electrons or with molecules. The positron wave function can be calculated more accurately than the electron wave function by taking advantage of simplified, no-exchange interaction with electrons. However, positron wave functions in molecular and atomic systems have not been documented in the literature as electrons have. Most researchers perform calculations at certain levels of approximation for specific purposes. Once the positron wave function is calculated, experimental annihilation parameters can be obtained by incorporating the known electron wave functions. This will be discussed in Chapter 2. 

In the independent-partlcle-model (IPM) originally due to Bohr [1], each particle moves under the Influence of the outer potential and the average potential of all the other particles in the system. In modem quantum theory, this model was first Implemented by Hartree 12], who solved the corresponding one-electron SchrSdlnger equation by means of an iterative numerical procedure, which was continued until there was no change in the slgniflcant figures associated with the electric fields involved so that these could be considered as self-consistent this approach was hence labelled the Self-Conslstent-Fleld (SCF) method. In order to take the Pauli exclusion principle into account. Waller and Hartree [3] approximated the total wave function for a N-electron system as a product of two determinants associated with the electrons of... 

It has been realized in recent years that the lanthanide contraction is only part of the explanation for the behavior of the heavier elements. An equally important factor is relativity. On a fundamental level, relativity actually plays an integral role in quantum theory, beginning with the space-time and momentum-energy symmetric which suggested the form of the time-dependent Schrixlinger equation [cf. Sei tion 2.3]. Electron spin and the Pauli exclusion principle are, in fact, implication ... 

For a many-eicctron system, the Hartree-Fock wave function Fhi. defined as the product of spin orbitals Xi ss outlined in Equation 28-SI. where A(n) is an antisymmetrirer for the electrons, provides good answers. This is the starting point for either semiempirical or ab initio theory. It is necessary to have 4(n) to make the wave function antisymmetric. thus obeying the Pauli exclusion principle, which asserts that two electrons cannot be in the same quantum state. 

A second approach to bonding in molecules is known as the molecular orbital (MO) theory. The assumption here is that if two nuclei are positioned at an equilibrium distance, and electrons are added, they will go into molecular orbitals that are in many ways analogous to the atomic orbitals discussed in Chapter 2. In the atom there are s, p, d, f,. . . orbitals determined by various sets of quantum numbers and in the molecule we have a, tt, 5,. . . orbitals determined by quantum numbers. We should expect to find the Pauli exclusion principle and Hund s principle of maximum multiplicity obeyed in these molecular orbitals as well as in the atomic orbitals. 

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