Three-Atom Expansions

A novel example of a three-atom expansion is found in the work of Liebeskind in which rhodium catalysis is used to expand a cy-clobutenone to provide a cycloheptadienone product (Eq. 22) via ring 

Research by Dowd illustrates another approach to seven-membered rings based on expansion of a cyclobutanone (Eq. 23) mediated by 

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Bell R J 1970 Multipolar expansion for the non-additive third-order interaction energy of three atoms J. 

Three atom ring-expansion of cyclopropenyl azides and diazo-compounds provides a useful route to triazines and pyridazines respectively 278). 

We wish to end this section by saying that similarly as in the two-body case, nonadditive induction, induction-dispersion, and dispersion terms have well defined asymptotic behaviors from the multipole expansions of the intermolecular interaction operators. For instance, the leading term in the multipole expansion of the three-body dispersion energy for three atoms in a triangular geometry is given by the famous Axilrod-Teller-Muto formula311,312,... 

Chapter VII.3). The conversion of VII/31to VII/32 (Scheme VII/8) however, goes in very good yield. Compound VII/33 is formed by transesterification [17]. One, two, and three carbon atom expansions, following the same reaction principle, have been published quite recently [19] [19a], By this method, ethyl l-methoxycarbonylmethyl-2-oxocyclohexancarboxylate, for example, can be transformed to ethyl 2-methoxycarbonyl-3-oxocycloheptancarboxylate in the presence of 1.2 equivalents of potassium tertiary butoxide in dimethylsulfoxide (41% yield). 

Cyclopropylcycloalkanones (38) undergo three-atom photochemical ring expansion into a mixture of trans and cis cyclonon-4-enones 39 and 40 (equation 29) . 

Table 4. Mean values of the expansion coefficients for the CH2, NH, and C—C groups in three-atom rings...

An alternative would retain the number of centers at three, but extend the expansion around each center beyond monopole, to include dipoles, quadrupoles, or perhaps even higher moments. One may, for example, place dipole moment vectors on each of the three atoms, as illustrated in Fig. 2g, to supplement the atomic charges in 2e. This scheme would not, however, provide for any charge distribution above or below the molecular plane. On the other hand, one can account for this aspect by adding a perpendicular component of a quadrupole moment to each atomic center, as indicated in Fig. 2h. This addition could add an important, and qualitatively correct, out-of-plane component to this molecule s interaction with another and help to model the O lone pairs. 

The angle 6 is the bond angle defined by three atoms ky, and k are constants. The torsional term torsion four-atom term for units of the form A—B—C—D, is usually written as a Fourier-series expansion, where the periodicity takes the values 1, 2 and 3, and to is the dihedral angle between the ABC and BCD planes ... 

MCI were introduced in the study of aromaticity by Giambiagi et al. [17,18] and extended by some of us for the quantification of electron delocalization in aromatic compounds [19], They are the simple extension of the SEDI described above to larger numbers of atoms. At the single determinant level of theory, the formulae become especially simple because of the Lowdin expansion and for MCI over three atoms, one has for example [53] ... 

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