Symmetry duplication

Plane of symmetry (Section 7.3) A plane that bisects an object, such as a molecule, into two mirror-image halves also called a mirror plane. When a line is drawn from any element in the object perpendicular to such a plane and extended an equal distance in the opposite direction, a duplicate of the element is encountered. 

At metaphase, mammalian chromosomes possess a twofold symmetry, with the identical duplicated sister chromatids connected at a centromere, the relative po-... 

Now consider Structure 6.15. In this case, there is no such symmetry and so all the signals of the spectrum of this compound would be expected to be broadened or duplicated Always consider the symmetry of the molecule in anticipation of the extent of rotameric complexity. 

The three-dimensional structure of OxlT, an oxalate/formate antiporter from Oxalobacter formigenes, most recently solved [124], is explicitly different from that of both antiporter NhaA and symporter MelB (see above). There is an obvious twofold symmetry in the organisation of the 12 transmembrane helices. This supports the previous idea that the MFS proteins evolved from a duplication of a 6-TM protein to form a 12-TM protein. Moreover, it is possible that the 6-TM predecessor was created from a duplication of an ancestral 3-TM protein. 

Since there are no symmetry considerations embedded in the isomer generation approaches described above, duplicate isomers may be generated for symmetric parts of a molecule. 

Table 5.63 lists the structural characters of the various feldspar polymorphs, according to Smyth and Bish (1988). Note duplication of cell edge c in anorthite and celsian and the progressive increase in volume that accompanies the lowering of symmetry in KAlSi30g. 

We briefly mention the case where the number of points m is less than n, i.e., less than the number of elements in the symmetry group G with respect to which we measure symmetry. In this case, m should be a factor of n such that there exists a subgroup H of G with n/m elements. In this case, we duplicate each point trim times and fold/unfold the points with elements of a left coset of G with respect to H. Following the folding/unfolding method, the relocated points will align on symmetry elements of G (on a reflection plane or on a rotation axis for example). Further details of this case and proof can be found in Ref. 2. 

For any open-chain single bond that connects two sp carbon atoms, an inhnite number of conformations are possible, each of which has a certain energy associated with it. As a practical matter, the number of conformations is much less. If one ignores duplications due to symmetry, the number of conformations can be estimated as being greater than 3", where = the number of internal C—C bonds. -Pentane, for example, has 11, -hexane 35, w-heptane 109, -octane 347, n-non-ane 1101, and n-decane 3263. For ethane there are two extremes, a conformation of highest and one of lowest potential energy, depicted in two ways as ... 

If one wants to estimate the ee via/lF g 9 (cf Regioselectivity), it is necessary to consider all possible ligand coordination modes (for example, all possible coordination modes (axial-equatorial ae/equatorial-equatorial ee) are shown in transition states 8,9 for a C2-symmetric bidentate phosphine ligand (e.g., LIO). In the case of Cl symmetry, each coordination mode is duplicated). Each coordination mode contributes with its asymmetric induction to the total stereoselectivity. To achieve high ee values, at least one of the following requirements should be met [7, 37]. 

The NH4 ion and the dimeric ZHZ of the first set are sited on crystallographic centres of symmetry. (A centre of symmetry is, of course, impossible for the tetrahedral NH4 ion, which must be disordered in this case.) In the second set, the NH4, X and HZ are in general positions, duplicated by operation of crystallographic centres of inversion. Thus the crystal contains both Type A and Type B structure NH4HZ2 and 2 (NH4Z. HZ) respectively. The structural plan, as it affects the acetate residues, is represented schematically by Fig. 10. 

The bilateral symmetry of the worm implies duplication of lineages on the two sides curiously, though, functionally equivalent cells on each side can arise from a pattern of division that is different on the two sides. Some of the embryonic cells function as stem cells, dividing repeatedly to form more of themselves or another type of stem cell, while... 

In some circumstances it may be impractical to achieve symmetry between the sample and tare sides of a two-sided balance, e.g. if the sample is in a cryogenic environment that caimot readily be duplicated on the tare side. Equation (7.14) applies to this scenario. While measurements of Avh(To) and Avb(To) may be made as described in the p eceding sub-section, separated values for Vbs(Ts), Vbt(Tt), Vhs(Ts) and Vht(Tt) are required. This can be accomplished by measurements of the resultant force at roughly constant high pressure in the full range of sample temperatures to which the second and third lines of Eq. (7.14) are separately fitted subject to the constraints AVh(To) = Vhs(To) - Vht(To) and AVb(To) = Vbs(To) - Vbt(To). The same procedure is then applied to individually determining values for v iTs) and... 

It should be realized modern ab initio programs do not rigidly follow the basic SCF procedure described in this section. To speed up calculation they employ a variety of mathematical tricks. Among these are the use of symmetry to avoid duplicate calculation of identical integrals testing two-electron integrals quickly to see if they are... 

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