Homotopical equivalence

In sum, homomorphic ligands (i.e., those which are identical when detached from the molecule) can be classified as homotopic (equivalent) or heterotopic (nonequivalent). The latter can be further distinguished as diastereotopic or enantiotopic. These divisions are shown in Figure A7-5, and the descriptions presented for the different types of ligands are collected in Table A7-1. 

The rules we will present help one assign a simple and unambiguous stereotopicity descriptor - H, H, E, 3, D, D, or F - to each molecular face of an acyclic or monocyclic planar moiety. Paired homotopic (equivalent) faces are designated by the same descriptor H/H or H /H. For enantiotopic faces, the rules will help determine which face is E and which one is 3 for diastereotopic faces they will help tell which face is D, which one is F, which one is D and which one is F. The rules for each molecular system will be detailed below. 

Most of the basic properties of standard triangles involve homotopy, and so are best stated in K( ). For example, the mapping cone C of the identity map A —> A is homotopically equivalent to zero, a homotopy between the identity map of C and the zero map being as indicated ... 

The number k of such circles is precisely equal to the rank of the one-dimensional integer-valued homology group Hi(My2), Moreover, this group is free (in the case of a non-empty boundary) and is isomorphic to Z 0 l(k times) For instance, removing one disk from a two-dimensional torus, we obtain with one hole homotopically equivalent to a bouquet of two circles, that is, in our example k = 2 and x M) = 1 - 2 = -1 (Fig. 77). 

A loop path within a level set F(A) is a path starting and ending at the same nuclear configuration K. Some loop paths of F(A) can be deformed continuously into one another within the level set F(A), while some others cannot. Such continuous deformations are called homotopies. Those loop paths which can be deformed into one another are said to be homotopically equivalent and to belong to the same homotopy class. 

Homotopy theory is the theory of continuous deformations. The topological representation of reaction mechanisms by homotopy equivalence classes of reaction paths is based on the actual chemical equivalence of all those reaction paths that lead from some fixed reactant to some fixed product, and are "not too different" from one another. This "chemical" condition, combined with an energy constraint, corresponds to a precise topological condition two reaction paths are regarded as "not too different" if they can be continuously deformed into each other below some fixed energy value A, that is, within a level set F(A). Such paths are homotopically equivalent at energy bound A. This leads to a classification of all reaction paths a collection of all paths that are deformable into one another within F(A) is a homotopy equivalence class of paths at energy bound A. 

The concepts of homotopy, homotopical equivalence, and homotopy equivalence classes are the main topological tools for the construction of quantum chemical reaction mechanisms within the potential energy hypersurface model. 

Homotopical equivalence within each homotopy class implies that this product, if it exists, is unique and does not depend on the choice of reaction paths Pi, P2 P, representing equivalence classes [pj], [p2] e IT. 

The unit element [Kq] for these homotopy classes is defined as the homotopy equivalence class that contains the constant path p(u) = Kq. Since all these loop paths, when multiplied by their inverse paths, generate a loop path that is homo-topically contractible to Kq, therefore, these path-products are all homotopically equivalent to the constant path p(u) = Ko, so they must all belong to the same, and unique, homotopy equivalence class. 

The inverse of homotopy class [p] is the class [p] = [p ], since pp must be homotopically equivalent to the origin Ko of p, therefore,... 

However, in all instances, the paths p = (pi p2> ps and pB = pi (p2 Ps) are homotopically equivalent. Consequently, for the products of loop homotopy equivalence classes, the associativity condition applies ... 

B The fourth possibility arises in chiral molecules, such as (R)-2-butanol. The two — CH2- hydrogens at C3 are neither homotopic nor enantiotopic. Since replacement of a hydrogen at C3 would form a second chirality center, different diastereomers (Section 9.6) would result depending on whether the pro-R or pro-S hydrogen were replaced. Such hydrogens, whose replacement by X leads to different diastereomers, are said to be diastereotopic. Diastereotopic hydrogens are neither chemically nor electronically equivalent. They are completely different and would likely show different NMR absorptions. 

Both homotopic fluorines such as those in difluoromethane and 2,2-difluoropropane and 1,1-difluoroethene, and enantiotopic fluorines such as those in chlorodifluoromethane and 2,2-difluorobutane (Scheme 2.8) would be chemically equivalent. 

C2-symetric initiators have a pair of equivalent homotopic sites, both of which prefer the same monomer enantioface, that is, both sites prefer the re enantioface or both prefer the si face. Isoselective propagation proceeds with or without migratory insertion since coordination and insertion of monomer at either site give the same stereochemical result. 

The C2-symmetric ansa metallocenes possess a C2 axis of symmetry, are chiral, and their two active sites are both chiral. The two sites are equivalent (homotopic) and enantioselective for the same monomer enantioface. The result is isoselective polymerization. C2 ansa metallocenes are one of two classes of initiators that produce highly isotactic polymer, the other class being the C ansa metallocenes (Sec. 8-5e). C2 ansa metallocenes generally produce the most isoselective polymerizations. 

On the timescale of the NMR experiment, isochrony (chemical shift equivalence) arises from symmetry equivalence of homotopic and enantiotopic nuclei5 17, while anisochrony (chemical shift nonequivalence, JK s <5) arises from symmetry nonequivalence of diastereotopic nuclei. 

Nuclear magnetic resonance chemical shift differences can serve as an indicator of molecular symmetry. If two groups have the same chemical shift, they are isochronous. Isochrony is a property of homotopic groups and of enantiotopic groups under achiral conditions. Diastereotopic or constitutionally heterotopic groups will have different chemical shifts (be anisochronous), except by accidental equivalence and/or lack of sufficient resolution. 

Aliphatic protons which are interconvertible by a rotational axis are termed homotopic and are chemically and magnetically equivalent. For example, the methylene protons of diphenylmethane are homotopic, as are the methylene protons and the methyl protons of propane. 

The term equivalent is overly general and therefore bland and of equivocal meaning. Thus the methylene hydrogen atoms in propionic acid (Fig. 1) are equivalent when detached (i.e. they are homomorphic), but, as already explained, they are not equivalent in the CH3CH2C02H molecules because of their placement — i.e. they are heterotopic. Ligands that are equivalent by the criteria to be described in the sequel are called homotopic from Greek homos = same and topos = place 6>, those that are not are called heterotopic . 

The symmetry planes (a) in molecules 30, 32, 34, 36, 38, Fig. 13 should be readily evident. It is possible to have both homotopic and enantiotopic ligands in the same set, as exemplified by the case of cyclobutanone (34) HA and HD are homotopic as are HB and Hc, HA is enantiotopic with HB and Hc HD is similarly enantiotopic with Hc and HB. The sets HAjB and HC>D may be called equivalent (or homotopic) sets of enantiotopic hydrogen atoms. The unlabeled hydrogens at position 3, constitutionally distinct — see Section 3.4 — from those at C(2, 4), are homotopic with respect to each other. Enantiotopic ligands need not be attached to the same atom — viz. the case of mew-tartaric acid (32) and also the just-mentioned pair Ha, Hc [or Hb, Hd] in cyclobutanone. 

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