The Pauli Exclusion Principle

The Pauli principle states that a maximum of two electrons can reside in each orbital, and we distinguish them by means of a spin quantum number. Two electrons in one orbital must then have 

The total wave function for an electron is therefore (orbit) (spin). Further, the two spin functions corresponding to a and /3 spin are normalized and orthogonal to each other that is, 

The ground state of carbon presents a new problem Which orbital does the sixth electron go into This question can be answered by doing either an experiment or a calculation. The answers are summarized in Hund s first rule  

When electrons go into orbitals that have the same energy (degenerate orbitals), the state that has the highest number of equal spin quantum numbers will have the lowest energy. Thus electrons prefer to occupy different orbitals, if possible, since in many cases it is energetically favorable to avoid spin pairing. 

If the difference in energy between orbitals that are to receive electrons is small, as for example between the 4s and the 3d orbitals, it sometimes is more energetically favorable (refer to Hund s rule) to distribute the electrons in both orbitals. As an example, we find the lowest configurations for V, Cr, and Mn to be ([Ar] stands for a closed 18-electron shell) 

To a very good approximation, the Hamiltonian does not contain spin terms, so the energy is unaffected by inclusion of the spin factor in the ground-state wave function. Also, the ground state of helium is still nondegenerate when spin is considered. 

To further demonstrate that the spin factor does not affect the energy, we shall assume we are doing a variational calculation for the He ground state using the trial function f = /(ri, T2, ri2)2 / [a(l)j8(2) — j8(l)a(2)], where/is a normalized function symmetric in the coordinates of the two electtons. The variational integral is 

Since the spin function (10.25) is normalized, the variational integral reduces to /ff Hfdvi dvi, which is the expression we used before we introduced spin. 

Now consider the excited states of helium. We found the lowest excited state to have the zeroth-order spatial wave function 2 / [l5 (l)25 (2) - 2s(l)ls(2)] [Eq. (9.103)]. Since this spatial function is antisymmetric, we must multiply it by a symmetric spin function. We can use any one of the three symmetric two-electron spin functions, so instead of the nondegenerate level previously found, we have a triply degenerate level with the three zeroth-order wave functions 

For the next excited state, the requirement of antisymmetry of the overall wave function leads to the zeroth-order wave function 

We look for an operator U that has the property that I7 4 (t)) obeys (3.110) when H is r L. That is we require 

Putting J = in the angular momentum commutation rules (3.75) we can verify that 

We find the effect of 0 on the spin eigenstates (3.79) by using their representation as vectors in spin space. 

The effect of 6 on the angular momentum state of an electron with spin is seen by using the relations (3.91,3 93) and the symmetry of the 3-j symbol under change of sign of all the m indices. As we did in (3.98) we introduce an arbitrary phase factor independent of j and m, obtaining 

In general, time reversal changes the sign of the m index and multiplies by a phase factor. 

According to the Pauti exclusion principle, no two electrons in an atom can have the same four quantum numbers. If two electrons in an atcxn have the same n, (, and m values (meaning that they occupy the same orbital), then they must have different values of 01 that is, one must have nis = +i and the other must have m,= i. Because there are only two possible values for /Mj, and no two electrons in the same orbital may have the same value for a maximum of two electrons may occupy an atomic orbital, and these two electrons must have opposite spins. 

We can indicate the arrangement of electrons in atomic orbitals with labels that identify each orbital (or subshell) and the number of electrons in it. Thus, we could describe a hydrogen atom in the ground state using I5.  

The ground state for a many. ectron atom is the one m which all the electrons occupy orbitals of the lowest possible energy 

We can also represent the arrangement of electrons in an atom using orbital diagrams, in which each orbital is represented by a labeled box. The orbital diagram for a hydrogen atom in the ground state is 

The upward arrow denotes one of the two possible spins (one of the two possible m, values) of the electron in the hydrogen atom (the other possible spin is indicated with a downward arrow). Under certain circumstances, as we will see shortly, it is useful to indicate the explicit locations of electrons. The orbital diagram for a helium atom in the ground state is 

Wolfgang Pauli (1900-1958). Austrian physicist. One of the founders of quantum mechanics, Pauli was awarded the Nobel Prize in Physics in 1945. 

We have already discussed the Pauli exclusion principle in Chapter 5. In its most general form, this is  

Postulate vn must be antisymmetric (symmetric) for the exchange of identical fermions (bosons). 

A fundamental principle of quantum theory is the Pauli Exclusion Principle, which was formulated by the German physicist Wolfgang Pauli (1900-58), which states that no two electrons in the same atom can have the same set of quantum numbers. Because an individual orbital is specified by values of the first three quantum numbers—n, /, and m —the effect of the Pauli Exclusion Principle is that an individual orbital can be occupied by at most two electrons, one with spin up and one with spin dovm. 

The discovery of the Exclusion Principle won Wolfgang Pauli the 1945 Nobel Prize in physics. 

Why No More than Tliro Electrons Can Occupy the Same Orbital 

If two electrons in the same orbital were either both spin up or both spin down, then all four of their quantum numbers would be the same, which would violate the Pauli Exclusion Principle. We can offer a qualitative explanation for why this must be so. Remember that electrons are all negatively charged, so they tend to repel each other. An orbital defines an extremely tiny volume of space, so the repulsive force between two electrons in a single orbital is quite large. 

The concept of spin tends to emphasize the particie nature of the electrons, yet the name wave mechanical model is meant to emphasize the wave nature of the electrons. This is an unresolved dilemma in quantum physics. With no analog of electron spin in our everyday macroscopic world, any attempt we make to try to explain spin will be inadequate. 

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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 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 sum over n. can now be perfonned, but this depends on the statistics that the particles in the ideal gas obey. Fenni particles obey the Pauli exclusion principle, which allows only two possible values n. = 0, 1. For Bose particles, n. can be any integer between zero and infinity. Thus the grand partition fiinction is... 

The average kinetic energy per particle at J= 0, is of the Fenni energy p. At constant A, the energy increases as the volume decreases smce fp Due to the Pauli exclusion principle, the Fenni energy gives... 

Themiodynamic stability requires a repulsive core m the interatomic potential of atoms and molecules, which is a manifestation of the Pauli exclusion principle operating at short distances. This means that the Coulomb and dipole interaction potentials between charged and uncharged real atoms or molecules must be supplemented by a hard core or other repulsive interactions. Examples are as follows. 

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. 

In addition to being negatively charged electrons possess the property of spin The spin quantum number of an electron can have a value of either +5 or According to the Pauli exclusion principle, two electrons may occupy the same orbital only when... 

Because of the quantum mechanical Uncertainty Principle, quantum mechanics methods treat electrons as indistinguishable particles. This leads to the Pauli Exclusion Principle, which states that the many-electron wave function—which depends on the coordinates of all the electrons—must change sign whenever two electrons interchange positions. That is, the wave function must be antisymmetric with respect to pair-wise permutations of the electron coordinates. 

Again, for the filled orbitals L = 0 and 5 = 0, so we have to consider only the 2p electrons. Since n = 2 and f = 1 for both electrons the Pauli exclusion principle is in danger of being violated unless the two electrons have different values of either or m. For non-equivalent electrons we do not have to consider the values of these two quantum numbers because, as either n or f is different for the electrons, there is no danger of violation. 

Intrinsic Semiconductors. For semiconductors in thermal equiHbrium, (Ai( )), the average number of electrons occupying a state with energy E is governed by the Fermi-Dirac distribution. Because, by the Pauli exclusion principle, at most one electron (fermion) can occupy a state, this average number is also the probabiHty, P E), that this state is occupied (see Fig. 2c). In equation 2, K... 

Electrons act as if they were spinning around an axis, in much the same way that the earth spins. This spin can have two orientations, denoted as up T and down i. Only two electrons can occupy an orbital, and they must be of opposite spin, a statement called the Pauli exclusion principle. 

The four quantum numbers that characterize an electron in an atom have now been considered. There is an important rule, called the Pauli exclusion principle, that relates to these numbers. It requires that no two electrons in an atom can have the same set of four quan-... 

The Pauli exclusion principle has an implication that is not obvious at first glance. It requires that only two electrons can fit into an orbital, since there are only two possible values of m,. Moreover, if two electrons occupy the same orbital, they must have opposed spins. Otherwise they would have the same set of four quantum numbers. 

Hund s rule, like the Pauli exclusion principle, is based on experiment It is possible to determine the number of unpaired electrons in an atom. With solids, this is done by studying their behavior in a magnetic field. If there are unpaired electrons present the solid will be attracted into the field. Such a substance is said to be paramagnetic. If the atoms in the solid contain only paired electrons, it is slightly repelled by the field. Substances of this type are called diamagnetic. With gaseous atoms, the atomic spectrum can also be used to establish the presence and number of unpaired electrons. 

The fundamental laws which determine the behavior of an electronic system are the Schrodinger equation (Eq. II. 1) and the Pauli exclusion principle expressed in the form of the antisymmetry requirement (Eq. II.2). We note that even the latter auxiliary condition introduces a certain correlation between the movements of the electrons. 

In formulating the second-quantized description of a system of noninteracting fermions, we shall, therefore, have to introduce distinct creation and annihilation operators for particle and antiparticle. Furthermore, since all the fermions that have been discovered thus far obey the Pauli Exclusion principle we shall have to make sure that the formalism describes a many particle system in terms of properly antisymmetrized amplitudes so that the particles obey Fermi-Dirac statistics. For definiteness, we shall in the present section consider only the negaton-positon system, and call the negaton the particle and the positon the antiparticle. 

The wave function, constructed from the atomic orbitals must be antisymmetric with respect to interchange of electrons in order to satisfy the Pauli exclusion principle, having different spin quantum numbers (a and J3) for two electrons which are in the same orbital. 

The spins of two electrons are said to be paired if one is T and the other 1 (Fig. 1.43). Paired spins are denoted Tl, and electrons with paired spins have spin magnetic quantum numbers of opposite sign. Because an atomic orbital is designated by three quantum numbers (n, /, and mt) and the two spin states are specified by a fourth quantum number, ms, another way of expressing the Pauli exclusion principle for atoms is... 

Electrons occupy orbitals in such a way as to minimize the total energy of an atom by maximizing attractions and minimizing repulsions in accord with the Pauli exclusion principle and Hund s rule. 

Add Z electrons, one after the other, to the orbitals in the order shown in Figs. 1.41 and 1.44 but with no more than two electrons in any one orbital (the Pauli exclusion principle). 

We account for the ground-state electron configuration of an atom by using the building-up principle in conjunction with Fig. 1.41, the Pauli exclusion principle, and Hund s rule. 

According to the Pauli exclusion principle, each molecular orbital can accommodate up to two electrons. If two electrons are present in one orbital, they must be paired. 

VIII. THE PAULI EXCLUSION PRINCIPLE. THE INTERACTION OF TWO HELIUM ATOMS... 

The observed structure of the spectra of many-electron atoms is entirely accounted for by the following postulate Only eigenfunctions which are antisymmetric in the electrons , that is, change sign when any two electrons are interchanged, correspond to existant states of the system. This is the quantum mechanics statement (26) of the Pauli exclusion principle (43). 

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)]. 

Before estabiishing the connection between atomic orbitals and the periodic table, we must first describe two additionai features of atomic structure the Pauli exclusion principle and the aufbau principle. 

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