Distance between mirror images

In this Appendix we will demonstrate that the distance P P2 between a pair of points Pi = rj — (21,22) and P2 = (x2 2A2) remains unaltered upon 

We wish to demonstrate that the length of the line P P2 remains unaltered if the points P and P2 are reflected at the line y = —x as indicated in Fig. C.l. In general the points Pi and P[ an. related through 

In this Appendix we will demonstrate that the distance Pi P2 between a pair of points P = rj = (21,22) and P2 = I J ( 2 W2) remains unaltered upon reflection at some straight line in the x-y plane. Consider the situation depicted in Fig. C.l where the two points are separated by distances 

---

Now let us examine the relationships between handedness of Xs, Ca—Ca distance, and X2 values. Among the disulfides for which coordinates were available at 2 A resolution or better (Deisenhofer and Steigemann, 1975 Imoto et al., 1972 Wyckoff et al., 1970 Quiocho and Lipscomb, 1971 Saul et al., 1978 Epp et al., 1975 Huber et al., 1974 Chambers and Stroud, 1979 Hendrickson and Teeter, 1981 Brookhaven Data Bank, 1980 Feldmann, 1977), there are equal numbers with right-handed and left-handed xs- The average Ca—Ca distance across the left-handed ones is 6.1 A, exactly what was seen in the small-molecule structures, but for the right-handed ones the average Ca—Ca distance is 5.2 A. Clearly the two sets of disulfides as they occur in proteins cannot simply be mirror images of one another. 

The r-delocalisation in the parent phospholide anion I (Fig. 3, R =R =H) can be expressed in the valence bond picture by resonance between the canonical structures lA-IC (and their mirror images). Phosphonio-sub-stituents (R =R =PH3 ) increase the weight of the 1,2-dipolaric canonical structure IB and induce thus, in essence, a partial r-bond localisation and a shift of r-electron density from the phosphorus to the adjacent carbon atoms [16]. Consequences of this effect are the decrease in delocalisation energy for reaction (1) depicted in Fig. 4, and lower C2-C3/C4-C5 and higher C3-C4 bond orders which are reproduced in concomitant variations of computed bond distances [16]. 

The distance between peaks in a fluorescence spectrum is a measure of the energy differences between vibrational levels in the ground state likewise, the peaks in the lowest-energy band of the absorption spectrum give the energy differences between vibrational levels in Sv If the spacings are similar, the absorption and emission spectra are mirror images of one another. 

Identical molecules have the same formula, the same atomic connectivity, and the same distances between corresponding atoms. In addition, they are superimposable (homomers). Enantiomers have the same formula, the same atomic connectivity, and the same distances between corresponding atoms, but they are not superimposable, instead, they are mirror images of each other (cf. Section 2.7 on chirality). 

The Hausdorff distance was applied in chemistry in various chirality studies. The Hausdorff distance was used to measure the deviation of a chiral nuclear arrangement from some arbitrary reference arrangement, as proposed by Rassat. Mislow and co-workers used the Hausdorff distance between the object and its optimally overlapping mirror image to provide a chirality measure of the second type. - Using this Hausdorff distance criterion, Buda and Mislow determined the most chiral constrained and unconstrained simplexes in two and three dimensions, that is, the most chiral triangles and tetrahedra." ... 

By applying the SNSM similarity measure to mirror images, the quantity is a measure of achirality, whereas the dissimilarity measure d A,A ), denoted as Xs J A), is a measure of chirality, where the interrelation (137) between Xs,J A) and implies that this measure can take values from the unit interval. The measure Xs A), first proposed as an example of dissimilarity measures of the second kind, is zero for achiral objects and takes positive values for all chiral objects. Objects perceived as having prominent chirality tend to have large Xs A) values. The SNSM measures have also been applied to more general molecular shape problems. More recently, Klein showed that by a logarithmic transformation of the scaling factors s g, a metric can be constructed to provide a proper distance-like measure of dissimilarity of shapes. 

The projection of the cubic (10, 3) net on a face of the cubic unit cell, the full circles and lines of Fig. 3.28(b), shows that the net is built of 4-fold helices which are all anticlockwise upwards. The figures indicate the heights of the points in terms of c/8, where c is the length of the cell edge. A second net can be accommodated in the same volume, and if the second net is a mirror-image of the first in no case is the distance between points of different nets as short as the distance between (connected) points within a given net. In the second net of Fig. 3.28(b) (dotted circles and lines) the helices are clockwise. This type of structure, which would be a 3D racemate, is not yet known, but in view of its similarity to the /3-quinol structure described later, there is no reason why it should not be adopted by some suitable compound. 

This means that the Hausdorff distance between two sets of points, Q and Q, representing geometric obj ects, can be zero only if these two obj ects are identical, that is, achiral mirror images. 

The value of the Hausdorff distance between a geometric object Q and its mirror image Q depends not only on the shape of these objects but also on their size and their relative... 

---