Pauli bond number

Another application of the concept of fractional valence bonds has been made in the field of metals and alloys. In the usual quantum mechanical discussion of metals, initiated by W. Pauli (Z. Physik, 41, 81 (1927)) and Sommerfeld (Naturwiss., 15, 825 (1927)), the assumption was made that only a small number of electrons contribute significantly to the binding together of the metal... 

Here 1, 2, represent the electrons. P is the operation of permuting the electrons among the spin-orbit functions, for example, interchanging 1 and 2 between afi and ba. There are (2n) of these operations in the permutation group 2n is the number of electrons for n bonds. The symbol (—l)p is 1 if P involves an even number of interchanges of pairs of electrons and — 1 if it involves an odd number. The function in the brackets satisfies the Pauli exclusion principle. R represents the 2 operations of interchanging the spin functions a and 0 for orbitals (such as a and 6) that are bonded together. 

Covalent bonded electrons are bosons and a large number of bonded electrons can occupy the same energy level. Free electron band, on other hand, consists of fermions and therefore according to Pauli Exclusion Principle no two electrons can occupy the same energy level. This is the reason they exists in a band... 

The main effect is already taken into account if symmetry numbers are included in the densities of states. The symmetry number is a correction to the density of states that allows for the fact that indistinguishable atoms occupy symmetry-related positions and these atoms have to obey the constraints of the Pauli principle (i.e. the wave function must have a definite symmetry with respect to any permutation), whereas the classical density of states contains no such constraint. The density of states is reduced by a factor that is equal to the dimension of the rotational subgroup of the molecule. When a molecule is distorted, its symmetry is reduced, and so its symmetry number changes by a proportion that is equivalent to the number of indistinguishable ways in which the distortion may be produced. For example, the rotational subgroup of the methane molecule is T, whose dimension is 12, whereas the rotational subgroup of a distorted molecule in which one bond is stretched is C3, whose dimension is 3. The ratio of these symmetry numbers, 4, is the number of ways in which the distortion can occur, i.e. the reaction path degeneracy. 

---