Mirror-image theorem

From this mirror-image theorem it follows that MCD spectra of uncharged alternant systems that are paired with themselves should be zero. This is only true within the confines of the above approximations, in which the y." contributions to the MCD are neglected. The y contributions are in fact zero for uncharged alternant hydrocarbons since from the pairing theorem it follows that AHOMO = ALUMO. 

The originally on [y theoretically derived mirror-image theorem for MCD... 

The fact that the benzyl anion and benzyl cation should have opposite MCD signs according to the mirror-image theorem makes it easy to understand why benzene derivatives with mesomeric donor substituents, which are isoelectronic with the benzyl anion, and benzene derivatives with mesomeric acceptor substituents, which are isoelectronic with the benzyl cation, show opposite MCD signs. This fact can be used for a qualitative and even a quantitative characterization of mesomeric substituent effects. (See also Section 2.4 and Example 3.10.)... 

Theorem. Let G be an embedded graph. If there is an element of T(G) which is topologically chiral and which cannot be deformed to the mirror image of any other element of T(G), then G is topologically chiral [9]. 

There are 18 different radiation pathways that can be identified. Nine of them are shown in Fig. 13.14, and the other nine are mirror-images of those shown, using the right vertical border as the position of the mirror. For each of the radiation pathways, Hottel cross-string theorem is applied. The final view factor F. 2 is the sum of the eighteen view factors corresponding to the eighteen radiation paths. 

Charged alternant hydrocarbons occur as pairs of negatively and positively charged ions, and the pairing theorem relates the states of the former to the states of the latter. While at the standard PPP level of theory the absorption spectra of the members of a pair are predicted to be identical, and indeed experimentally almost exactly are, their MCD spectra are expected to be mirror images of each other and indeed nearly are (e.g., diphenylmethyl anion and cation). ... 

Stereoisomerism can refer to either enantiomerism (mirror-image nonsuperimposability) or diastereoisomerism. Both constitutional isomerism and stereoisomerism depend upon the symmetry operations of the molecular skeleton. The famous theorem of Pdlya derives an isomer-counting series by substitution of the so-called figure-counting series into a cycle index analogous to equation (5) for the symmetric groups S . 

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