Electron spin and the Pauli principle

Energy ordering of the hydrogenic orbitals in a polyelectronic atom. [Reproduced from http //en.wikibooks.org/ wiki/High SchooLChemistry/ Families on the Periodic Table (accessed November 30, 2013).] 

Moeller s rubric for the general order of orbital filling using the one-electron hydrogenic orbitals. 

Periodic tabie showing the correiation with Moeiier s rubric for orbital filling. 

In 1928, Paul Dirac developed a relativistic theory of quantum mechanics from which the concept of spin arose naturally. The inclusion of a fourth variable (time) required the presence of a fourth quantum number. According to Dirac s derivation, an electron possesses both orbital (L) and spin (S) angular momentum. The total angular momentum (J) is a linear combination of the two, as shown in Equation (4.13). 

The spin angular momentum vector S can take values of mj till, where can take values of +1/2 (a) or -1/2 (P), depending on whether it aligns against or with the external magnetic field, respectively. The usual classical picture of electron spin, where the electron can be considered as a top spinning on its axis either in the 

For our purposes, the main significance of electron spin is connected with the postulate of Austrian physicist Wolfgang Pauli (1900-1958) In a given atom no two electrons can have the same set of four quantum numbers (n, (, rri(, and m ). This is called the Pauli exciusion principie. Since electrons in the same orbital have the same values of n, i, and m, this postulate says that they must have different values of m. Then, since only two values of nis are allowed, an orbital can hold only two electrons, and they must have opposite spins. This principle will have important consequences as we use the atomic model to account for the electron arrangements of the atoms in the periodic table. 

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We now reconsider the helium atom from the standpoint of electron spin and the Pauli principle. In the perturbation treatment of helium in Section 9.3, we found the zeroth-order wave function for the ground state to be 15(1)15(2). To take spin into account, we must multiply this spatial function by a spin eigenfunction. We therefore consider the possible spin eigenfunctions for two electrons. We shall use the notation a(l)a(2) to indicate a state where electron 1 has spin up and electron 2 has spin up a(l) stands for a(Wji). Since each electron has two possible spin states, we have at first sight the four possible spin functions ... 

So far, we have not seen any very spectacular consequences of electron spin and the Pauli principle. In the hydrogen and heUum atoms, the spin factors in the wave functions and the antisymmetry requirement simply affect the degeneracy of the levels but... 

The application of the new quantum theory to the problem of the valence has, in the hands of London, Heitler and others, led to an extensive understanding of the nature of chemical valence. It related the valence values of atoms to certain group properties of the Schrodinger chfferential equations that describe the atoms, using the existence of electron spin and the Pauli principle. 

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