S-N rule

The incorporation of phosphorus yields fourfold-coordinated P atoms, which are positively charged, as phosphorus normally is threefold coordinated. This substitutional doping mechanism was described by Street [52], thereby resolving the apparent discrepancy with the so-called S N rule, with N the number of valence electrons, as originally proposed by Mott [53]. In addition, the incorporation mechanism, because charge neutrality must be preserved, leads to the formation of deep defects (dangling bonds). This increase in defect density as a result of doping explains the fact that a-Si H photovoltaic devices are not simple p-n diodes (as with crystalline materials) an intrinsic layer, with low defect density, must be introduced between the p- and n-doped layers. 

Links between atoms serve to compensate for the lack of the electrons which are necessary to attain the electron configuration of the next noble gas in the periodic table. With a common electron pair between two atoms each of them gains one electron in its valence shell. As the two electrons link two centers , this is called a two-center two-electron bond or, for short, 2c2c bond. If, for an element, the number of available partner atoms of a different element is not sufficient to fill the valence shell, atoms of the same element combine with each other, as is the case for polyanionic compounds and for the numerous organic compounds. For the majority of polyanionic compounds a sufficient number of electrons is available lo satisfy the demand for electrons with the aid of 2c2e bonds. Therefore, the generalized S—N rule is usually fulfilled for polyanionic compounds. 

Note that if we identify the sum over 8-fimctions with the density of states, then equation (A1.6.88) is just Femii s Golden Rule, which we employed in section A 1.6.1. This is consistent with the interpretation of the absorption spectmm as the transition rate from state to state n. 

Perhaps the most notable difference between S-N and N-O compounds is the existence of a wide range of cyclic compounds for the former. As indicated by the examples illustrated below, these range from four- to ten-membered ring systems and include cations and anions as well as neutral systems (1.14-1.18) (Sections 5.2-5.4). Interestingly, the most stable systems conform to the well known Htickel (4n -1- 2) r-electron rule. By using a simple electron-counting procedure (each S atom contributes two electrons and each N atom provides one electron to the r-system in these planar rings) it can be seen that stable entities include species with n = 1, 2 and 3. 

But I want to return to my claim that quantum mechanics does not really explain the fact that the third row contains 18 elements to take one example. The development of the first of the period from potassium to krypton is not due to the successive filling of 3s, 3p and 3d electrons but due to the filling of 4s, 3d and 4p. It just so happens that both of these sets of orbitals are filled by a total of 18 electrons. This coincidence is what gives the common explanation its apparent credence in this and later periods of the periodic table. As a consequence the explanation for the form of the periodic system in terms of how the quantum numbers are related is semi-empirical, since the order of orbital filling is obtained form experimental data. This is really the essence of Lowdin s quoted remark about the (n + , n) rule. 

These 20 cases do not represent anomalies to the order of orbital filling which is invariably governed by the n + ( rule but are anomalous in the sense that the s orbital is not completely filled before the corresponding d orbital begins to fill. 

An alternative stream came from the valence bond (VB) theory. Ovchinnikov judged the ground-state spin for the alternant diradicals by half the difference between the number of starred and unstarred ir-sites, i.e., S = (n -n)l2 [72]. It is the simplest way to predict the spin preference of ground states just on the basis of the molecular graph theory, and in many cases its results are parallel to those obtained from the NBMO analysis and from the sophisticated MO or DFT (density functional theory) calculations. However, this simple VB rule cannot be applied to the non-alternate diradicals. The exact solutions of semi-empirical VB, Hubbard, and PPP models shed light on the nature of spin correlation [37, 73-77]. 

Silver cyanate, AgNCO, consists of infinite chains of alternating Ag+ and NCO- ions. Ag+ has c.n. 2 and only one of the terminal atoms of the cyanate group is part of the chain skeleton, being coordinated to 2 Ag+. Decide with the aid of Pauling s second rule which of the cyanate atoms (N or O) is the coordinated one. (Decompose the NCO- to N3-, C4+ and O2-). 

The coupling of the spins of the electrons in an atom is accounted for by adding their magnetic spin quantum numbers. Since they add up to zero for paired electrons, it is sufficient to consider only the unpaired electrons. The spins of n unpaired electrons add up according to Hund s rule to a total spin quantum number S = n. The magnetic moment of these n electrons, however, is not the scalar sum of the single magnetic moments. The... 

This rule, which is an extension of the (8-N) rule for the elements (17), was developed on an empirical basis (18—21) and later shown (22) to be a mathematical formulation of the requirement that there be complete octets on all A s. Moreover, the rule can be extended (22) to allow configurations other than octets ... 

Before we close this MO/LCAO discussion of the generalized (8-N) rule, we note that a derivation of Eqn. II. 1 has been reported by Hulliger and Mooser (23) on a similar basis. However, a careful analysis of their treatment reveals that, in addition to features of general MO/LCAO theory (Thms. II. 1—II.3) and necessary assumptions (equivalents of Hyps. II.1—II.3), they also introduce some superfluous assumptions and specializations. This not only obscures the treatment, it also introduces new aspects which it may be instructive to dwell on in some detail. In order to keep the number of notational symbols to a minimum, the definitions already invoked in the preceding discussion will be utilized as far as possible. However, the disposition and layout of their paper differ significantly from ours since, moreover, many of Hulliger and Mooser s assumptions are to be classified as being only partly superfluous, some quotations are inescapable. 

It is evident from the above that the requirement for semiconductivity should also be added to the list of superfluous assumptions. However, the association of the generalized (8-N) rule with the question of semiconductivity is not in itself irrelevant. It is also worth noting that R (Def. 11.19) does not enter into Hulliger and Mooser s treatment. The reason for this is hidden in Stms. II.3 and II.4, according to which R =0. Surprisingly enough, Hulliger and Mooser do not draw this conclusion explicitly. 

Given the power of the concept of covalency and the deeper electronic implications that were realized by G. N. Lewis s octet-rule and shared-electron-pair concepts, it is natural to wonder whether these advances are limited to s/p-block elements or apply to the entire periodic table. 

The development of G. N. Lewis s octet rule for the s/p-block elements was strongly influenced by the stoichiometric ratios of atoms found in the common compounds and elemental forms (CH4, CCI4, CO2, CI2, etc.). Let us therefore begin analogously by examining the formulas of the common neutral binary chloride, oxide, and alkyl compounds of transition metals. (Here we substitute alkyl groups for hydrogen because only a small number of binary metal hydrides have been well characterized.)... 

Proof Let S be either a two dimensional abelian variety or a geometrically ruled surface over an elliptic curve over C. Let S be a good reduction of S modulo q, where gcd(q, n) = 1 such that the assumptions of lemma 2.4.7 hold. Then A 5n iis a good reduction of KSn- modulo q. (3) now follows by lemma 2.4.10 and remark 1.2.2. (1) and (2) follow from this by the formula of Macdonald for p(S n z) (see the proof of theorem 2.3.10). ... 

Exceptions to the E.A.N. rule do occur, particularly with d8 metal ions, where many examples of square-planar, 16-electron complexes are known. In these complexes, the high-lying pz orbital is nonbonding and remains empty. This deviation from the rule is often said to be due to the large s-to-p promotion energies found for the free atoms. As the atomic number increases across a given transition-metal series, the... 

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