Collinear central configuration

Also notice that the classical Euler condition (24), for a collinear central configuration with m2 between mi and m3, gives the position of the center of mass in terms of the positions of the three bodies (and independently of the three masses themselves). Indeed if rj, with j = 1, 2, 3 are the three radius vectors from the center of mass, that Euler condition (24) is equivalent to the following beautiful symmetrical condition ... 

Here we consider an optical transition between Aj and E electronic states of a center of a trigonal symmetry. To describe the vibrations of the center we use the collinear-configurational approximation [27] in which only the central forces are taken into account in the optical center (taking account of deviations from this approximation, see later). If one restricts oneself to the linear vibronic coupling in the e state, then in this approximation the potential energy operators in the Ai and E electronic states can be presented in the form ... 

The e.d. data for this halide are consistent with a model consisting of two -SF5 groups joined in the staggered configuration so that the central S—S bond is collinear with two S-F bonds, with all interbond angles 90° (or 180°), S-F, 1-56 A, and S-S, 2-21 The length of the S-S bond may be due to repulsion of the F atoms. 

Inversion of configuration always accompanies SN2 reactions the transition state is shown schematically in Figure 7.1. An everyday analogy of this inversion is shown by an umbrella that blows inside out on a windy day of course, the umbrella has more spokes than a carbon atom has valencies. This analogy has some further merit in that the nucleophile, the central carbon and the leaving group are collinear, and remain so in the transition state for the reaction. 

The attacking nucleophile approaches the H C-X molecule perpendicular to the H -plane at the central carbon atom and under inversion of the CH -umbrella (Walden inversion [69]) the system reacts via a necessarily collinear arranged Y. .. C. .. X transition complex (due to the electronic sp -hybridization at the central C atom) to the product configuration Y-CH +X. Hence in this reaction the bond-breaking and the bond-forming distance are fixed to a collinear configuration. In a zero-order approximation this bimolecular reaction can be treated as... 

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