Atom self-polarizability

For an even cyclopolyene (alternating hydrocarbon) the 7r-electron density is equal to unity at each atom. Therefore, from (12), an increase in /r-bond energy always follows the introduction of a heteroatom. But at the same time the cyclic delocalization energy, which is the difference in 7r-bond energy between the cyclic and the corresponding linear polyenes, will be decreased because the maximum atom self-polarizability of a linear polyene chain always has a higher value than that of the corresponding cyclopolyene 12°). 

The charge density index refers to the nature of the molecule before allowance is made for perturbing effects due to the approaching reactant. Such a method is often called a first-order method, a terminology that is discussed more fully in Chapter 12. For alternant molecules, it is necessary to proceed to a high-order method, one that reflects the ease with which molecular charge is drawn toward some atom, or pushed away from it, as approach by a charged chemical reactant makes that atom more or less attractive for electrons. An index which measures this is called atom self-polarizability, symbolized 7tr,r- The formulas for this and related polarizabilities are derived in Chapter 12. For now, we simply note that the formula is... 

In most isocyclic hydrocarbons (alternating hydrocarbons, see below) the charge distribution is uniform, so that qk = 1 for all atoms. Then there occurs in place of these quantities the so-called self-polarizability ... 

Po =ftcc (for the standard carbon-carbon bond in benzene), which appears to be the nearest SCF equivalent of the empirical Hiickel jSq. Another practice would be to express all polarizabilities in units of the self-polarizability of a carbon atom in benzene. 

Some quantities related to the self-polarizability, 7Tr, r are of practical utility in connection with heteroaromatic molecules. They are the mutual polarizabilities of atoms r and s and the polarizability of atom r by the bond s-t defined, respectively, as... 

The electrostatic energy is calculated using the distributed multipolar expansion introduced by Stone [39,40], with the expansion carried out through octopoles. The expansion centers are taken to be the atom centers and the bond midpoints. So, for water, there are five expansion points (three at the atom centers and two at the O-H bond midpoints), while in benzene there are 24 expansion points. The induction or polarization term is represented by the interaction of the induced dipole on one fragment with the static multipolar field on another fragment, expressed in terms of the distributed localized molecular orbital (LMO) dipole polarizabilities. That is, the number of polarizability points is equal to the number of bonds and lone pairs in the molecule. One can opt to include inner shells as well, but this is usually not useful. The induced dipoles are iterated to self-consistency, so some many body effects are included. 

The Drude oscillators are typically treated as isotropic on the atomic level. However, it is possible to extend the model to include atom-based anisotropic polarizability. When anisotropy is included, the harmonic self-energy of the Drude oscillators becomes... 

GY noticed that the polarizability of an atom is approximately an exponential function of charge, and that the polarizability correction provided by the CPE expansion for an isolated atom was equal to the inverse of the Coulomb self energy of the... 

---