Pseudoscalar properties

By forming the maxima or minima respectively of these quantities, the extrema being taken over all r with zf 0, we get chirality numbers which are characteristic properties of the skeleton. It should be emphasized that the condition is zfV 0 and not zr 0 because the latter would only lead to trivial numbers which express the fact that a ligand partition is active for any achiral frame if it contains at least one chiral ligand. The nontrivial numbers we want should present information about the pseudoscalar properties of the particular molecular class in question. We find these chirality numbers from the following maxima and minima ... 

As forcefully stated by Pasteur, whatever the precise arrangement of atoms in the molecule, ce qui ne peut etre l objet d un doute is that the chirality of the atomic arrangement is the necessary and sufficient condition for molecular enantiomorphism and for its manifestation in a pseudoscalar property such as optical activity. It cannot be emphasized too strongly that no recourse to structural theory was needed to arrive at this conclusion, which was based purely on a symmetry argument and which F. M. Jaeger has referred to as la loi de Pasteur ... 

An immediate consequence of Pasteur s law is that the relationship between enantiomers is established by symmetry alone and does not require any knowledge of molecular bonding connectedness (constitution). This is in contrast to diastereomers, the other class of stereoisomers Diastereomers are not related by symmetry, and their relationship can be defined only by first specifying that their constitutions are the same—otherwise, there would be nothing to distinguish them from constitutional isomers. Thus enantiomers, which have identical scalar properties and differ only in pseudoscalar properties, have more in common with homomers than with diastereomers, while diastereomers, which differ in all scalar properties, have more in common with constitutional isomers than with enantiomers.51, 52 It therefore makes more sense, in an isomer classification scheme, to give priority to isometry rather than to constitution.52 In such a scheme there is no need for the concept stereoisomer the concept retains its usefulness only because it normally proves convenient, in chemical reaction schemes, to combine enantiomers and stereoisomers in a common class. 

Chirality in the world of observables is characterized by pseudoscalar properties—properties that remain invariant under proper rotation but change sign under improper rotation. Enantiomers and, in general, enan-tiomorphous molecules, have identical scalar properties, such as melting points or dipole moments, and pseudoscalar properties that are identical in... 

Because enantiomers have oppositely signed pseudoscalar properties, chiral zeroes are unavoidable at some stage in the conversion of a molecule into its enantiomer along a chiral pathway. This is true of chirally connected enantiomeric conformations in chemically achiral molecules, such as (lf )-menthyl (15)-menthyl 2,2, 6,6 -tetranitro-4,4 -diphenate, and of chirally connected enantiomers, such as ( + )- and (- )-isopro-pylmalonamic acids. More generally, as previously noted, any chiral molecule composed of five or more atoms is in principle always capable of conversion into its enantiomer by chiral as well as by achiral pathways, provided that this is energetically feasible. Hence, unless it can be demon-... 

The stereoisomeric relation of the nonplanar D2mirror image. As enantiomers have identical scalar properties (energies, nmr shifts, etc.), but different pseudoscalar properties an experimental differentiation of these chiral molecules is bound to rely on chiral properties, such as optical rotations. The planar E-dimethylbutatriene (5) and Z-dimeth-ylbutatriene (6) exhibit cis-trans isomerism (geometrical isomerism). 5 and 6 differ in their scalar molecular properties. 

The problems of the mathematical structure governing the relationships between arbitrarily multisubstituted cmistitutional isomers have been treated for scalar and pseudoscalar properties in Ref. 7 and Refs. 15b and 15c, respectively. 

All these considerations for scalar molecular properties have counterparts in the discussions of pseudoscalar molecular properties. Pseudoscalar properties are related to the chirality of the molecules under consideration, that is, to effects of enantiomerism. A chiral object cannot be superimposed onto its mirror image by rotation (and/or translation). Therefore, an object is achiral, if only the position in space is altered on reflection or rotation-reflection. It is achiral, if its symmetry group contains no planes of reflection and/or improper rotations (14o, 15). As a consequence, chiral allenes are of the types (b)-(d) in Fig. 5. 

Of the molecules (with an achiral skeleton) under consideration only allenes, pentatetraenes, and ketene imines may become chiral through an appropriate arrangement of (achiral) ligands and thus offer the opportunity to investigate substituent effects on chiral (pseudoscalar) properties. 

It is evident that methods analogous to the ones developed here could be applied to molecular properties which, instead of being pseudoscalar, belong to some other representation of the skeleton point group (vector, tensor, etc. properties). To treat such properties, one needs only to induce from a different representation of than the chiral one. 

Pseudoscalars with the property T T (where the positive sign applies to proper rotations and the negative sign applies to improper rotations) are also called axial tensors of rank 0, 7 (0)ax. A quantity T with three components 7 7 2 7) that transform like the coordinates x x2 x3 of a point P, that is like the components of the position vector r, so that... 

Here A2 symbolizes a pseudo-scalar of A2 symmetry, normalized to unity. The actual form of this pseudoscalar need not bother us. The only property we will have to use later on is that even powers of A2 are equal to +1. Now we can proceed by defining rotation generators f x, y,t 2 in the standard way, as indicated in Table 1 [10]. Note that primed symbols are used here to distinguish the pseudo-operators from their true counterparts in real coordinate space. Evidently the action of the true angular momentum operators t y, (z on the basis functions is ill defined since these functions contain small ligand terms. 

It is clearly necessary to take account of this property of a to establish the parity of the composite system involving a. The particle a is said to be scalar or pseudoscalar, with intrinsic parity of +1 or -1 if, for space inversion its internal wave function does not, or does change sign. 

In equation 1 each term is called a source of optical activity (SOA). Each SO A describes a particular way of generating optical activity and is built from a number of equivalent elements of optical activity (EOAs). The elements of optical activity are combined in such a manner that the corresponding SOA has the correct (pseudoscalar) transformation property to describe optical rotations. 

We have seen that different chirality measures give incommensurable values for the shape and the relative degree of chirality of a given object. In short, there is no such thing as a unique scale by which chirality can be measured. If a choice among the numerous chirality measures that have been discussed in the literature is to have any relevance to the world of observables, it must be predicated on the measure s ability to correlate the calculated degree of chirality with some experimentally determined chirality property. That is, the measure must successfully model some pseudoscalar physical or chemical property. As will be discussed subsequently, this goal may prove to be elusive. 

Symmetry arguments show that parity-odd, time-even molecular properties which have a non-vanishing isotropic part underlie chirality specific experiments in liquids. In linear optics it is the isotropic part of the optical rotation tensor, G, that gives rise to optical rotation and vibrational optical activity. Pseudoscalars can also arise in nonlinear optics. Similar to tlie optical rotation tensor, the odd-order susceptibilities require magnetic-dipole (electric-quadrupole) transitions to be chirally sensitive. 

These remarks are of interest, as the molecular properties which are treated in this contribution are scalar as well as pseudoscalar (chiral) in character. 

If one accepts the preceding premises of the geometrical model any molecular property may be related to intrinsic properties ff(R,) of the ligands R at the various sites i. And thus one may think of a description of the molecular property 5 in terms of a real function F(cr(R ),..., substituent constant and may be treated as an undeHned element that is, it achieves a fixed numerical value for a given experiment, but lacks any information about the physical quality which is represented by molecular property. A function of the ligands that describes a pseudoscalar molecular property of molecules with a common achiral skeleton is called a chirality function x (15). On this level of describing molecular... 

The study of substituent effects on vector molecular quantities usually refer to electric dipole moments. However, dipole moments of allenes are also intimately related to the discussion of the basis of the geometric model underlying the treatments of scalar and pseudoscalar molecular properties (Section II.A). 

As an example for the calculation of pseudoscalar molecular properties of allenes in Table 22 calculated and experimental molar rotations at the wavelength of the sodium D line are given. Concerning the range of the numerical values for the variously substituted allenes there is a striking difference compared with the numerical values for scalar properties. 

In case of scalar molecular properties the variation (in terms of the standard deviations) is of the order of a typical ( average ) substituent constant (molecular property it is related to the square of an average value of the substituent constant (X ). This follows from Equation 51. For the calculation of the linear regression [< ]/>/x Table 22 the compounds 137 and 138 have been omitted, as in these cases solvent and conformational effects influence the experimental rotations (lc,l 3). For 96 and 135 the values in acetonitrile (13) have been used. 

As a summary it may be stated that strict methods of symmetry are powerful for the quantitative treatment of various scalar and pseudoscalar molecular properties of allenes, not only for some model systems, but for chemically interesting and complex molecules. 

A DSP analysis relates the X(R) parameters of those mesomeric groups whose parameters have been shown to give reliable molar rotations of allenes to the ff/ and a k constants (Fig. 40). Only the carbomethoxy group exhibits a larger deviation from the DSP correlation. From Equation 147 R — 0.6827 including COOMe) one can see that also in case of mesomeric groups the X(R) parameters arc more affected by polar than by resonance effects. Irrespectively of the crudeness of the correlation (147) the DSP analysis of the X(R) parameters is the first trial to extend usual substitutuent parameter approaches for scalar molecular properties to pseudoscalar molecular properties. 

The chirality operator is odd under spatial inversion, exhibiting what is called pseudoscalar behavior, that is, it is a scalar with the transformation properties of a vector. For a molecule with two enantiomeric forms, A and B, with respective wave functions and 4 5, we expect space inversion, represented by the operator /, to connect the two forms such that... 

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