Electron Spin and the Exclusion Principle

If we accept the model of electron spin, then we can rationalize our experimental facts if we assume each electron is capable of being in one of but two possible states of opposite spin. This is done in the following way. If we attribute opposite spins to the two Is electrons in, say, silver, their spin moments should cancel. Similarly, aU other orbital-sharing electrons would contribute nothing if their spins were opposed. Only the outermost (5s) electron would have an uncanceled spin moment. Its two possible orientations would cause the beam to split into two con xtnents as is observed.  

The evident need for the introduction of the concept of electron spin means that our wavefunctions of earlier sections are incomplete. We need a wavefunction that tells us not only the probability that an electron will be found at given r, 6, f coordinates in three-dimensional space, but also the probability that it will be in one or the other spin state. Rather than seeking detailed mathematical descriptions of spin state functions, we will singly symbolize them a and Then the symbol / (l)a(l) will mean that electron number 1 is in a spatial distribution corresponding to space orbital / , and that it has spin O. In the independent electron scheme, then, we could write the spin orbital (includes space and spin parts) for the valence electron of silver either as 5s(l)a(l) or 5s(l )y6(l). These two possibilities both occur in the atomic beam and interact differently with the inhomogeneous magnetic field. 

We now focus on the maimer in which spin considerations affect wavefunction symmetry. The electrons are still identical particles, so our particle distribution must be 

In the independent electron approximation, the lowest-energy configuration for helium is Is. Let us write the various conceivable spin combinations for this configuration. They are 

It is easy to see that the common space term ls(l )ls(2) is symmetric for electron interchange. Likewise, a(l)a(2) and (1) (2) are each symmetric, so Eqs. (5-19) and (5-22) are totally symmetric wavefunctions. The spin parts of Eqs. (5-20) and (5-21) are unsymmetric (not antisymmetric) for interchange, so these wavefunctions are not satisfactory. However, we can take the sum and difference of Eqs. (5-20) and (5-21) to obtain 

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Only those periodic aspects of electronic configuration that depend on electron spin and the exclusion principle remain unaccounted for. 

For example, the brilliant observation that the electron should be seen not just as a particle charged with electric charge, but also as a carrier of an own magnetic moment , coupled with the Lewis interrogation, i.e., if the octet mle and of the doublet can be somehow reduced to a basic mle , i.e., by forming pairs of electrons - anticipated the fundamental concept of electronic spin and the Exclusion Principle (Pauh principle). 

Wolfgang Pauli is well recognized as an outstanding theoretical physicist, famous for his formulation of the two-valuedness of the electron spin, for the exclusion principle, and for his prediction of the neutrino. Less well known is the fact that Pauli spent a lot of time in different avenues of human experience and scholarship, ranging over fields such as the history of ideas, philosophy, religion, alchemy and Jung s psychology. Pauli s... 

Wolfgang Pauli helped to develop quantum mechanics in the 1920s by forming the concept of spin and the exclusion principle. According to Schrodinger s Equation, each electron is unique. The Pauli Exclusion Principle states that no two electrons may have the same set of quantum numbers. Thus, for two electrons to occupy the same orbital, they must have different spins so each has a unique set of quantum numbers. The spin quantum number was confirmed by the Stern-Gerlach experiment. 

Electron Spin and the Pauli Exclusion Principle Orbital Energy Levels in Multielectron Atoms Electron Configurations of Multielectron Atoms Electron Configurations and the Periodic Table... 

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 ... 

Consider first the two Is electrons. To satisfy the exclusion principle, one of these electrons must have J while the other has = —j. If M5 is the quantum number that specifies the z component of the total spin of the Is electrons, then the only possible value of is 5 — j = 0 [Eq. (11.34)]. This single value of Mg clearly means that the total spin of the two Is electrons is zero. Thus, although in general when we add the spins Si = j and j of two electrons according to the rule (11.39), we get the two possibilities 5 = 0 and 5=1, the restriction imposed by the Pauli principle leaves 5 = 0 as the only possibility in this case. Likewise, the spins of the 2s electrons add up to zero. The exclusion principle does not restrict the values of the 2p and 3d elec-... 

Note that the paramagnetic or diamagnetic behavior of atoms is a consequence of the quantization of electron spin and the Pauli exclusion principle, both of which are purely quantum-mechanical phenomena. Thus, the experimental observation of the magnetic behavior of atoms (paramagnetic or diamagnetic) represents further experimental confirmation of the quantum-mechanical nature of atoms and molecules. 

Opposing spins of the electron pair. As the exclusion principle (Section 8.2) prescribes, the space formed by the overlapping orbitals has a maximum capacity of two electrons that must have opposite spins. When a molecule of H2 forms, for instance, the two 1 electrons of two H atoms occupy the overlapping Is orbitals and have opposite spins (Figure 11.1 A). 

Because single-electron wave functions are approximate solutions to the Schroe-dinger equation, one would expect that a linear combination of them would be an approximate solution also. For more than a few basis functions, the number of possible lineal combinations can be very large. Fortunately, spin and the Pauli exclusion principle reduce this complexity. 

The simplest many-electron wave function that satisfies the Exclusion Principle is a product of N different one-electron functions that have been antisymmetrized, or written as a determinant. Here, N is the number of electrons (or valence electrons) in the molecule. HyperChem uses this form of the wave function for most semi-empirical and ab initio calculations. Exceptions involve using the Configuration Interaction option (see page 119). HyperChem computes one-electron functions, termed molecular spin orbitals, by relatively simple integration and summation calculations. The many-electron wave function, which has N terms (the number of terms in the determinant), never needs to be evaluated. 

In Pauli s model, the sea of electrons, known as the conduction electrons are taken to be non-interacting and so the total wavefunction is just a product of individual one-electron wavefuncdons. The Pauli model takes account of the exclusion principle each conduction electron has spin and so each available spatial quantum state can accommodate a pair of electrons, one of either spin. 

Brickstock, A., and Pople, J. A., Phil. Mag. 44, 705, The spatial correlation of electrons in atoms and molecules. IV. The correlation of electrons on a spherical surface." Two examples—four electrons of the same spin and eight paired electrons—have been studied to compare the effects of the exclusion principle and the interelectronic repulsion. 

In dealing with systems containing only two electrons we have not been troubled with the exclusion principle, but have accepted both symmetric and antisymmetric positional eigenfunctions for by multiplying by a spin eigenfunction of the proper symmetry character an antisymmetric total eigenfunction can always be obtained. In the case of two hydrogen atoms there are three... 

The phenomenon of electron pairing is a consequence of the Pauli exclusion principle. The physical consequences of this principle are made manifest through the spatial properties of the density of the Fermi hole. The Fermi hole has a simple physical interpretation - it provides a description of how the density of an electron of given spin, called the reference electron, is spread out from any given point, into the space of another same-spin electron, thereby excluding the presence of an identical amount of same-spin density. If the Fermi hole is maximally localized in some region of space all other same-spin electrons are excluded from this region and the electron is localized. For a closed-shell molecule the same result is obtained for electrons of p spin and the result is a localized a,p pair [46]. 

Here, the summation goes over all the individual electron wave functions that are occupied by electrons, so the term inside the summation is the probability that an electron in individual wave function ijx((r) is located at position r. The factor of 2 appears because electrons have spin and the Pauli exclusion principle states that each individual electron wave function can be occupied by two separate electrons provided they have different spins. This is a purely quantum mechanical effect that has no counterpart in classical physics. The point of this discussion is that the electron density, n r), which is a function of only three coordinates, contains a great amount of the information that is actually physically observable from the full wave function solution to the Schrodinger equation, which is a function of 3N coordinates. 

The single-Slater determinant includes correlation between the motion of two electrons with parallel spins that avoid each other because of the exclusion principle (Szabo and Ostlund 1989), but correlation between the motion of electrons with opposite spin is neglected. The wave function of Eq. (3.2) does not prevent the two electrons from being at the same point in space, which is physically impossible. The Slater determinant wave function is therefore described as uncorrelated. 

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. 

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. 

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