Pauli, generally Principle

W. Pauli, General Principles of Quantum Mechanics, Springer-Verlag, Berlin, 1980. 

The state F) is such that the particle states a, b, c,..., q are occupied and each particle is equally likely to be in any one of the particle states. However, if two of the particle states a, b, c,...,q are the same then F) vanishes it does not correspond to an allowed state of the assembly. This is a characteristic of antisynmietric states and it is called the Pauli exclusion principle no two identical fennions can be in the same particle state. The general fimction for an assembly of bosons is... 

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. 

The properties of the Slater determinant demonstrate immediately the Pauli exclusion principle, as usually taught. It reads No two electrons can have all four quantum numbers equal, that is to say that they cannot occupy the same quantum state. It is the direct result of the more general argument that the wavefunction must be antisymmetric under the permutation of any pair of (identical and indistinguishable) electrons. 

The connection between the Pauli exclusion principle and the more general Pauli principle can be understood as follows. If two electrons with the same spin a were to occupy the same orbital ip, the total wave function for the system would be written yi, zO (/ "to, yi,... 

The Pauli exclusion principle requires that no two electrons can occupy the same spin-orbital is a consequence of the more general Pauli antisymmetry principle ... 

Of course, the Coulomb interaction appears in the Hamiltonian operator, H, and is often invoked for interpreting the chemical bond. However, the wave function, l7, must be antisymmetric, i.e., must satisfy the Pauli exclusion principle, and it is the only fact which explains the Lewis model of an electron pair. It is known that all the information is contained in the square of the wave function, 1I7 2, but it is in general much complicated to be analyzed as such because it depends on too many variables. However, there have been some attempts [3]. Lennard-Jones [4] proposed to look at a quantity which should keep the chemical significance and nevertheless reduce the dimensionality. This simpler quantity is the reduced second-order density matrix... 

One of the pedagogically unfortunate aspects of quantum mechanics is the complexity that arises in the interaction of electron spin with the Pauli exclusion principle as soon as there are more than two electrons. In general, since the ESE does not even contain any spin operators, the total spin operator must commute with it, and, thus, the total spin of a system of any size is conserved at this level of approximation. The corresponding solution to the ESE must reflect this. In addition, the total electronic wave function must also be antisymmetric in the interchange of any pair of space-spin coordinates, and the interaction of these two requirements has a subtle influence on the energies that has no counterpart in classical systems. 

When systems are pushed together, nonbonded electrons, on the other hand, tend to retreat toward the system that has the more diffuse orbitals. In this case that is C, B, or Be. Since the nonbonded electrons are generally in orbitals less far out, this effect occurs at closer distances and, according to our calculations, wins out at equilibrium distances for CO and BF. This is the only effect for BeNe, and the moment is in the same direction at all of the distances we show. This retreat of electrons is definitely a result of the Pauli exclusion principle. 

Notice the similarities between the two parts of Table 1.1 and the form of the Periodic Table given in Figure 1.3. The quantum rules, the Pauli exclusion principle and the aufbau principle combine to explain the general structure of the Periodic Table. 

The Pauli Exclusion Principle states that no two electrons of any single atom may simultaneously occupy a slate described by only a single set of quantum numbers. Five such numbers arc needed to describe fully the quantum-mechanical conditions of an electron. For j-j coupling this set is generally ti. I., v. j. iij. and for l.-S it is /t. /. j. u(. nr,. From die coupling of the angular momentum associated with the latter sets a full description of the multielectron stale, described by it, L. S, J. Mis determined. 

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. 

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