Correlation rules

Correlation rules relate the symmetry of reactants to the symmetry of products. More precisely, they give the symmetry of the fragments which can result when a molecule or transition state is distorted in the direction of reactants or products32,33. A familiar example is the correlation of the states of a diatomic molecule with those of its constituent atoms. Within the Bom-Oppenheimer separation we can deal with strictly electronic correlation rules, valid when there is negligible coupling between electronic and vibrational wave functions. When such coupling is important, correlations forbidden on a strictly electronic basis may be allowed, so the validity of purely electronic correlation rules is hard to assess for polyatomic molecules with strongly excited vibration. 

It has been emphasized32 that an association or atom transfer reaction will in general go through configurations of lowest possible symmetry one plane for 

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The Woodward-Hoffmann rules have intellectual roots that can be traced back to Wigner-Witmer correlation rules (E. Wigner and E. E. Witmer, Z. Phys. 51 [1928], 859) and general correlation-diagram concepts (R. S. Mulliken, Rev. Mod. Phys. 4 [1932], 1) as employed, e.g., by K. F. Herzfeld, Rev. Mod. Phys. 41 (1949), 527. Alternative MO... 

In contrast to binaphthyls, chiral biphenyl derivatives are challenging systems because their twist ability shows a strong dependence on the molecular structure, which does not conform to the empirical correlation rule described above. In fact, homochiral biphenyls 33-40 are characterized by P helicity along the biphenyl axis. In spite of this common feature, the twisting power spans from a highly positive value for 33-45 to a relatively negative value... 

It should be emphasized that the deviations from the logarithmic correlation obey the normal distribution statistics, allowing one to apply convenient statistical procedures to analytical measurement results. As will be shown below, the logarithmic linear correlation rule was observed for all types of geo-chemical samples, i. e., snow, air, water, and soil. The correlation analysis of the elemental composition of melted-snow fractions showed with confidence level 95% that no significant discrepancy exists between the element composition within the correlation curves (see Table 1) and the corresponding variances are thus homogeneous for any two randomly chosen samples (i. e., points of the territory). 

Fig. 13.1 The solubility of C60 and C70 fullerene is maximum in vegetable oils with lower level of unsaturation (i.e. number of double bonds) and decreases as the insaturation level grows. Having a peculiar chemical structure, castor oil cannot be included in the correlation rule (See Color Plates)...

To identify the diatomic potentials we consider the correlation rules for molec-... 

In other words the H -h OH correlation depends strictly on the shape of the triatomic and not just on the three bond lengths treated as independent variables. It is in this sense that the restriction (54) on Vj introduces an approximation to the potential. In effect by making it we are allowing for the spin correlation rules but not spatial correlation rules. 

Valuable insight, particularly with regard to the effects of electronic excitation on reaction cross sections and reaction dynamics, has also been achieved without accurate knowledge of the actual potential surfaces, through the use of molecular-orbital correlation diagrams. Adiabatic correlation rules for neutral reactions involving polyatomic intermediates were developed by Shuler 478 These were adapted and extended for ion-neutral interactions by Mahan and co-workers.192,45 479,480 Electronic-state correlation diagrams have been used to deduce the qualitative nature of the potential surfaces that control ion-neutral reaction dynamics. The dynamics of the reaction N+(H2,H)NH+ and in particular the different behavior of the N + (3P) and N + ( Z)) states,123 for example, have been rationalized from such considerations (see Fig. 62). In this case the... 

Silver8 adopted a similar approach, and proposed a further category of unfeasible reactions in which neither individual orbital correlation rules nor total symmetry correlation rules (Wigner-Witmer rules) were obeyed. Silver and Karplus87 have shown how to derive W-H symmetry rules using a simple valence-bond formalism. 

Unfortunately, most of these tools are extremely expensive, and are fairly complex to deploy, requiring a database backend for alert storage. Surprisingly, they also have limited correlation capabilities, only providing a dozen or so rules as example for the development of environment-specific correlation rules. They should be viewed as a development framework for writing correlation rules. Since our correlation needs are very diverse, we could not find a platform that would allow us to run multiple correlation processes in parallel, from dynamic statistical analysis to vulnerability assessment. Also, manipulation of contextual data with interfaces to the inventory and configuration databases of the companies, was a strong requirement that no commercial tool satisfied at the time we launched the project. 

The factors that contribute to the reactivity of an excited state include not only the energy excess, but also the intrinsic reactivity of the specific electronic arrangement and the relative efficiencies of the different competing pathways. The factors are all correlated and therefore the reaction pathways allowed by the correlation rules are often different from those of the ground-state partners [1]. 

We will see in due course that there are important correlation rules between atomic term symbols and molecular electronic states, rules that are important in understanding both the formation and dissociation of diatomic molecules. Elementary accounts of the theory of atomic structure are to be found in books by Softley [3] and Richards and Scott [4], Among the more comprehensive descriptions of the quantum mechanical aspects, that by Pauling and Wilson [5] remains as good as any whilst group theoretical aspects are described by Judd [6],... 

We have described the orbital approaches to the electron configurations of diatomic molecules, both the molecular orbital and the united atom models. We now turn to the question of what types of molecular states result from given states of the separate atoms. If Russell Saunders coupling is valid for the separate atoms, the correlation rules, due to Wigner and Witmer [16] provide a valid and complete summary of the molecular states. This information is extremely important for an understanding of both the formation and dissociation of diatomic molecules. 

Similar correlation rules exist for atoms which follow jj coupling, rather than Russell-Saunders coupling, but such cases are relatively rare and will not be discussed here. 

Eq. (6.4) has been applied to the available literature data on Tg s of polymers, in all nearly 600 from this study the correlation rules for Yg have been derived. It appeared that the Ygi-values of the relevant groups are not independent of some other groups present in the structural unit. 

Excitation of the atomic product of an exchange reaction involving three atoms is certainly rare. When excited states are accessible energetically, their formation is often precluded by the spin correlation rules. For example, the important reaction... 

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