Electron spin and the Pauli exclusion principle

Marie Sklodowska Curie, born in Warsaw, Poland, began her doctoral work with Henri Becquerel soon after he discovered the spontaneous radiation emitted by uranium salts.She found this radiation to be an atomic property and coined the word radioactivity for it. In 1903 the Curies and Becquerel were awarded the Nobel Prize in physics for their discovery of radioactivity.Three years later, Pierre Curie was killed in a carriage accident.Marie Curie continued their work on radium and in 1911 was awarded the Nobel Prize in chemistry for the discovery of polonium and radium and the isolation of pure radium metal.This was the first time a scientist had received two Nobel awards. (Since then two others have been so honored.) 

Mendeleev s arrangement of the elements, the periodic table, was originally based on the observed chemical and physical properties of the elements and their compounds. We now explain this arrangement in terms of the electronic structure of atoms. In this chapter we will look at this electronic structure and its relationship to the periodic table of elements. 

In Chapter 7 we found that an electron in an atom has four quantum numbers—n, I, mi, and —associated with it. The first three quantum numbers characterize the orbital 

I Protons and many nudei also have spin. See the Instrumental Methods essay at the end of this section. 

The two possible spin orientations are indicated by the models. By convention,the spin direction is given as shown by the large arrow on the spin axis. Electrons behave as tiny bar magnets,as shown in the figure. 

We have now seen that we can use hydrogen-like orbitals to describe many-electron atoms. What, however, determines which orbitals the electrons occupy That is, how do the electrons of a many-electron atom populate the available orbitals To answer this question, we must consider an additional property of the electron. 

Electron spin is crucial for understanding the electronic structures of atoms. In 1925 the Austrian-bom physicist Wol%ang PauU (1900-1958) discovered the principle that governs the arrangements of electrons in many-electron atoms. The Pauli exclusion principle states that no two dectrons in an atom can have the same set of four quantum numbers n, 1, mj, andrUg. For a given orbital, the values of , Z, and Wj are fixed. Thus, if we 

Atoms having unpaired electron with spin quantum number m, = + Vz deflect in one direction those having unpaired electron with = — Vz deflect in opposite direction 

MRI has none of the disadvantages of X-rays. Diseased tissue appears very different from healthy tissue, resolving overlapping structures at different depths in the body is much easier, and the radio frequency radiation is not harmful to humans in the doses used The technique has had such a profound influence on the modern practice of medicine that Paul Lauterbur, a chemist, and Peter Mansfield, a physicist, were awarded the 2003 Nobel Prize in Physiology or 

A FIGURE 6.28 MRI Image. This image of a human head, obtained using magnetic resonance imaging, shovtrs a normal brain, airways, and faciai tissues. 

From this figure, why are there only two possible values for the spin quantum number  

The foundation of MRI is a phenomenon called nuclear magnetic resonance (NMR), which was discovered in the mid-1940s. Today NMR has become one of the most important spectroscopic methods used in chemistry. NMR is based on the observation that, like electrons, the nuclei of many elements possess an intrinsic spin. Like electron spin, nuclear spin is quantized. For example, the nucleus of H has two possible magnetic nuclear spin quantum numbers, -I-j and - y 

We can represent the electron configuration of hydrogen (l ) in a slightly different way with an orbital diagram, which gives similar information but symbolizes the electron as an arrow and the orbital as a box. The orbital diagram for a hydrogen atom is  

Helium is the first element on the periodic table that contains two electrons. Its two electrons occupy the li orbital. 

How do the spins of the two electrons in helium align relative to each other The answer to this question is addressed by the Pauli exclusion principle, formulated by Wolfgang Pauli in 1925. 

Pauli exclusion principle no two electrons in an atom can have the same four quantum numbers. 

Since two electrons occupying the same orbital have three identical quantum numbers (n, I, and mi), they must have different spin quantum numbers. Since there are only two possible spin quantum numbers (-1-5 and -5), the Pauli exclusion principle implies that each orbital can have a maximum of only two electrons, with opposing spins. By applying the exclusion principle, we can write an electron configuration and orbital diagram for helium as follows  

When atoms are placed in a magnetic field, the energy levels of the electrons split into more than one component These splittings are small (no more than 10 3 eV, even in strong magnetic fields), but can be seen in the line spectra of atoms this is called the Zeeman effect. There are other manifestations. For example, in an inhomogeneous (i.e. non-uniform) field, a beam of atoms can be deflected, and splits into several components this is the Stem-Gerlach experiment, and is illustrated in Fig. 5.5. 

Also by analogy with orbital angular momentum, the number of possible orientations of the spin in a magnetic field is 

The two orientations are specified by a quantum number ms. taking the values 

Often these two values are described as spin-up and spin-down electrons. 

This attempt to incorporate the spin of the electron, by using a halfintegral quantum number in a theory which seems to require integral values appears very artificial. It does nevertheless agree with the experimental observations. In 1928, Dirac developed a theory of the electron wavefunction which incorporated the principles of Einstein s theory of relativity. Very remarkably, the spin appears as a natural prediction of that theory, although the mathematical details are much too complicated to discuss here. 

FJectron correlations are intimately associated with two assumptions (1) a fourth quantum number, the electron-spin quantum number s, and ( ) the Pauli exclusion principle. In order to account for spectral data, it is necessary to postulate that electrons spin about their own axis to create a magnetic moment (G2o). Whereas the magnetic moment associated with the angular momentum may have (2Z + 1) components mi in the direction of an external magnetic field H, the spin moment may have only two components corresponding to s = ms = 1/2. Classically the magnitude of the moment fia associated with an angular momentum p is 

RUSSKLL-SAUNDERS COUPLING, SPECTROSCOPIC NOTATION, AND MULTIPLICITY 

In this volume, principal consideration is given to the lighter elements, so that the Russell-Saunders (549) vector model of the atom is used. In this model a multielectron atom is assumed to have the quantum numbers n, L = lif Ml, 8 = siy (or n, L, J = L + S, Mj). This implies stronger and Si-Sj coupling than U-Si coupling. It follows from Pauli s principle that for a closed shell = 

In order to interpret magnetic susceptibility data, it is necessary to know the ground state of an atom with more than one electron 

Hund s rule is a consequence of the Pauli principle plus electrostatic interactions between the electrons. Since any dual occupation of a spatial orbital must involve large electrostatic electron-electron repulsions, the energy of the multielectron state is lowered if such dual occupations can be minimized. From Pauli s principle, this dual occupation is minimized if equivalent electrons have as many like spins as possible, (liven several possible terms with the same (25 +1), electrons that orbit in the same sense (U have the same sign) collide less frequently than electrons orbiting oppositely. Therefore of the terms with maximum (25 +1), minimum electrostatic electron-electron repulsion is achieved in the term with greatest L. 

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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 ... 

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. 

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. 

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. 

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. 

Wolfgang Pauli helped develop quantum mechanics in the 1920s by developing the concept of spin and the Pauli exclusion principle, which states that if two electrons occupy the same orbital, they must have different spin (intrinsic angular momentum). This principle has been generalized to other quantum particles. 

Fermions include electrons, protons, neutrons, and He. They have half-integral units of spin and the Pauli exclusion principle prevents more than one particle from having the same quantum state. As a consequence, such particles must be treated using the F-D distribution fimction, given by... 

Orbital diagrams assign electrons to individual orbitals so the energy state of individual electrons may be found. This requires knowledge of how electrons occupy orbitals within a subshell. Hund s rule states that before any two electrons occupy the same orbital, other orbitals in that subshell must first contain one electron each with parallel spins. Electrons with up and down spins are shown by half-arrows, and these are placed in lines of orbitals (represented as boxes or dashes) according to Hund s rule, the Aufbau principle, and the Pauli exclusion principle. Below is the orbital diagram for vanadium ... 

THE SPINNING ELECTRON AND THE PAULI EXCLUSION PRINCIPLE, WITH A DISCUSSION OF THE HELIUM ATOM... 

The orbital concept and the Pauli exclusion principle allow us to understand the periodic table of the elements. An orbital is a one-electron spatial wave function. We have used orbiteils to obteiin approximate wave functions for many-electron atoms, writing the wave function as a Slater determinant of one-electron spin-orbitals. In the crudest approximation, we neglect all interelectronic repulsions and obtain hydrogenlike orbitals. The best possible orbitals are the Heu tree-Fock SCF functions. We build up the periodic table by feeding electrons into these orbitals, each of which can hold a pair of electrons with opposite spin. 

The theorem just proved shines in its simplicity. People thought that the wave function, usually a very complicated mathematical object (that depends on 3N space and N spin coordinates) is indispensable for computing the properties of the system. Moreover, the larger the system, the worse the difficulties in calculating it (recall Chapter 10 with billions of excitations, nonlinear parameters, etc.). Besides, how can we interpret such a complex object This is a horribly complex problem. And it turns out that everything about the system just sits in p(r), a function of position in our well-known 3-D space. It turns out that information about nuclei is hidden in such a simple object. This seems trivial (cusps), but it also includes much more subtle information about how electrons avoid each other due to Coulombic repulsion and the Pauli exclusion principle. 

It has been found that electrons behave as if they spin on an axis, and only electrons spinning in opposite directions (indicated by t and ) can occupy the same orbital. This principle, known as the Pauli exclusion principle, explains why orbitals can contain a maximum of two electrons. Hund s rule and the Pauli exclusion principle can be combined Electrons will pair with other electrons in an orbital only if there is no empty orbital of the same energy available and if there is one electron with opposite spin already in the orbital. 

When the species A and B approach so closely that their electron clouds overlap, the Pauli exclusion principle keeps electrons of the same spin away from each other. Therefore, electron density is removed fi om the overlap region. The positively charged nuclei are thus incompletely shielded from each other and mutually repel. This interaction is described as exchange-repulsion or just exchange (EX). 

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