The Pauli Principle

In the above treatment of the hydrogen molecule ion and the hydrogen molecule, the effect of electronic spin has been excluded, but the complete wave function of an electron must include not only the orbital motion, with which we have been concerned so far, but also a contribution for the spin. With single atoms it was possible to introduce a fourth quantum number s in addition to the three quantum numbers , I and m in order to account for the spin of the electron, and for polyelectronic molecules it is possible to proceed in an analogous manner. The complete wave function of an electron is considered to be the product of the orbital wave function, i,e. the wave function that we have been considering so far, and a wave function representing the orientation of the spin axis of the electron. 

If iff represents the orbital wave function of an atom for a given value of the quantum numbers n, / and m, then the spin wave function may be represented by two functions a and P whose values are determined by the following conditions  

These conditions mean that the wave function a refers to the state of the electron in which the spin quantum number s, has the value + /2 consequence the probability of it having the value — V2 zero. Similarly the function j8 describes those states corresponding to a value for of — and consequently the probability of it having the value -h V2 zero. The complete wave function of the electron is then represented by 

For two electrons, the complete wave function of the first electron will be (l)a(l) which denotes that the first electron is attached to the nucleus a and the solution is given by 

The function given in equation 3.103 is, however, a special case in which the spins of electrons i and 2 are antiparallel. In the general case each electron may have any value of the spin quantum number, so that in place of the two wave functions a(l) (2) and a(2) (l), representing the 

Tlie wave function specifying the state of an electron depends not only on the coordinates and z but dso on the spin state of the electron. What effect does this have on 

To a very good approximation, the Hamiltonian for a system of electrons does not involve the spin variables but is a function only of spatial coordinates and derivatives with respect to spatial coordinates. As a result, we can separate the stationary-state wave function of a single electron into a product of space and spin parts  

In quantum mechanics the uncertainty principle tells us that we cannot follow the exact path taken by a microscopic particle. If the microscopic particles of the system all have different masses or charges or spins, we can use one of these properties to distinguish the particles from one another. But if they are all identical, then the one way we had in classical mechanics of distinguishing them, namely by specifying their 

We now derive the restrictions on the wave function due to the requirement of indistinguishability of identical particles in quantum mechanics. The wave function of a system of n identical microscopic particles depends on the space and spin variables of the particles. For particle 1, these variables are x-i, y, Zj, Let qx stand for all four of these variables. Thus i/ = q2, - -, )- 

We define the exchange or permutation operator Px2 as the operator that interchanges all the coordinates of particles 1 and 2  

We can construct a total electronic wavefunction as the product of a spatial part and a spin part. For the electronic ground state of H2 we can consider combinations of the two spatial terms 

In the limit of infinite atom separations, or if we switch off the Coulomb repui. sion between two electrons, all four wavefunctions have the same energy. But they correspond to different eigenvalues of the electron spin operator the first combination describes the singlet electronic ground state, and the other three combinations give an approximate description of the components of the first triplet excited state. 

The magnetic forces between electrons are negligibly small compared to the electrostatic forces, and they are of no importance in determining the distribution of the electrons in a molecule and therefore in the formation of chemical bonds. The only forces that are important in determining the distribution of electrons in atoms and molecules, and therefore in determining their properties, are the electrostatic forces between electrons and nuclei. Nevertheless electron spin plays a very important role in chemical bonding through the Pauli principle, which we discuss next. It provides the fundamental reason why electrons in molecules appear to be found in pairs as Lewis realized but could not explain. 

The wave function for any system is a function of both the spatial coordinates of the electrons and of the spins of the electrons. It is convenient to describe the two possible values of the spin angular momentum of an electron as the two possible values of its spin coordinate, 

The requirement that electrons have antisymmetrical wave functions is called the Pauli principle, which can be stated as follows  

An electronic wave function must be antisymmetric with respect to the interchange of any 

The Pauli principle is usually first met in a more restricted form called the Pauli exclusion principle, which states that 

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The Hartree approximation is usefid as an illustrative tool, but it is not a very accurate approximation. A significant deficiency of the Hartree wavefiinction is that it does not reflect the anti-synnnetric nature of the electrons as required by the Pauli principle [7], Moreover, the Hartree equation is difficult to solve. The Hamiltonian is orbitally dependent because the siumnation in equation Al.3.11 does not include the th orbital. This means that if there are M electrons, then M Hamiltonians must be considered and equation A1.3.11 solved for each orbital. 

Semiconductors are poor conductors of electricity at low temperatures. Since the valence band is completely occupied, an applied electric field caimot change the total momentum of the valence electrons. This is a reflection of the Pauli principle. This would not be true for an electron that is excited into the conduction band. However, for a band gap of 1 eV or more, few electrons can be themially excited into the conduction band at ambient temperatures. Conversely, the electronic properties of semiconductors at ambient temperatures can be profoundly altered by the... 

In Chapter VIII, Haas and Zilberg propose to follow the phase of the total electronic wave function as a function of the nuclear coordinates with the aim of locating conical intersections. For this purpose, they present the theoretical basis for this approach and apply it for conical intersections connecting the two lowest singlet states (Si and So). The analysis starts with the Pauli principle and is assisted by the permutational symmetry of the electronic wave function. In particular, this approach allows the selection of two coordinates along which the conical intersections are to be found. 

Qualitatively, the first term of Eq. (27) represents the electron exchange repulsion as a result of the Pauli principle, and the second long-range term accounts for the attractive dispersion interaction. The [12-6] formulation is only qualitatively... 

Slater showed that spinorbitals, arrayed as a determinant, change sign on election exchange so as to obey the Pauli principle. If we wi ite a linear combination of two spinorbitals as a determinant where we assume the space parts are the same but the spin parts are not the same... 

The most general statement of the Pauli principle for electrons and other fermions is that the total wave function must be antisymmetric to electron (or fermion) exchange. For bosons it must be symmetric to exchange. 

So, the states that arise from the ground configuration of oxygen are 2g, Ig and Ag. One of Hund s mles (the first on page 212) tells us thsAX I g is the ground state. The Pauli principle forbids the Eg, Ig and Ag states. 

Exclusion Principle, also called the Pauli Principle. 

I will refer to the Hartree model from time to time in the text. Hartree s energies were in poor agreement with experiment. With the benefit of hindsight he should have allowed for indistinguishability and the Pauli principle. This was Fock s contribution to the field he wrote the wavefunction as what we would now recognize as a Slater determinant. Such a wavefunction automatically satisfies the Pauli principle. 

Don t confuse the state wavefunction with a molecular orbital we might well want to build the state wavefunction, which describes all the 16 electrons, from molecular orbitals each of which describe a single electron. But the two are not the same. We would have to find some suitable one-electron wavefunctions and then combine them into a slater determinant in order to take account of the Pauli principle. 

There is actually a further problem to do with the Pauli principle. Suppose that we had been able to calculate a wavefunction for the a-electron and the ar-electron parts, written... 

Each of them will have to satisfy the Pauli principle. We might be tempted to write a total wavefunction for the 16 electrons as... 

Since the coiTelation between opposite spins has both intra- and inter-orbital contributions, it will be larger than the correlation between electrons having the same spin. The Pauli principle (or equivalently the antisymmetry of the wave function) has the consequence that there is no intraorbital conelation from electron pairs with the same spin. The opposite spin correlation is sometimes called the Coulomb correlation, while the same spin correlation is called the Fermi correlation, i.e. the Coulomb correlation is the largest contribution. Another way of looking at electron correlation is in terms of the electron density. In the immediate vicinity of an electron, here is a reduced probability of finding another electron. For electrons of opposite spin, this is often referred to as the Coulomb hole, the corresponding phenomenon for electrons of the same spin is the Fermi hole. 

The Dirac equation automatically includes effects due to electron spin, while this must be introduced in a more or less ad hoc fashion in the Schrodinger equation (the Pauli principle). Furthermore, once the spin-orbit interaction is included, the total electron spin is no longer a good quantum number, an orbital no longer contains an integer number of a and /) spin functions. The proper quantum number is now the total angular momentum obtained by vector addition of the orbital and spin moments. 

The diagonal elements may be larger than 2. This implies more than two electrons in an orbital, violating the Pauli principle. 

We have not explained why two, but no more than two, electrons can occupy each orbital. This is not known and is accepted because the facts of nature require it. This assumption is called the Pauli Principle. 

The notion of electrons in orbitals consists essentially of ascribing four distinct quantum numbers to each electron in a many-electron atom. It can be shown that this notion is strictly inconsistent with quantum mechanics (7). Definite quantum numbers for individual electrons do not have any meaning in the framework of quantum mechanics. The erroneous view stems from the original formulation of the Pauli principle in 1925, which stated that no two electrons could share the same four quantum numbers (8), This version of the principle was superseded by a new formulation that avoids any reference to individual quantum numbers for separate electrons. The new version due to the independent work of Heisenberg and Dirac in 1926 states that the wave function of a many-electron atom must be antisymmetrical with respect to the interchange of any two particles (9,10). 

The use of the older restricted version of the Pauli principle has persisted, however, and is routinely employed to develop the electronic version of the periodic table. Modern chemistry appears to be committing two mistakes. Firstly, there is a rejection of the classical chemical heritage whereby the classification of elements is based on the accumulation of data on the properties and reactions of elements. Secondly, modem chemistry looks to physics with reverence and the false assumption that therein lies the underlying explanation to all of chemistry. Chemistry in common with all other branches of science appears to have succumbed to the prevailing tendency that attempts to reduce everything to physics (11). In the case of the Pauli principle, chemists frequently fall short of a full understanding of the subject matter, and... 

This vanishing of the probability density for rx — r2 and Ci == C2 means that it is unlikely for two electrons having parallel spins to be in the same place (rx = r2). The phenomenon is called the Fermi hole and we note that it is a direct consequence of the Pauli principle for electrons with the same spin. 

In addition to the Schrodinger equation we have the antisymmetry requirement (Eq. II.2) connected with the Pauli principle and, by means of the antisymmetrization operator (Eq. 11.16), the Hartree product (Eq. 11.37) is then transformed into a Slater determinant ... 

If we take the Pauli principle into account and instead start from the Slater determinant (Eq. 11.38), we obtain... 

The correlation error can, of course, be defined with reference to the Hartree scheme but, in modem literature on electronic systems, one usually starts out from the Hartree-Fock approximation. This means that the main error is due to the neglect of the Coulomb correlation between electrons with opposite spins and, unfor-tunetely, we can expect this correlation error to be fairly large, since we force pairs of electrons with antiparallel spins together in the same orbital in space. The background for this pairing of the electrons is partly the classical formulation of the Pauli principle, partly the mathematical fact that a single determinant in such a case can... 

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