Principle Pauli

Charge of total electron cloud inside sphere of radius r = - qe 

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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. 

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