Achiral zeroes

For the carbonyl carbon Ij core level ionization, excellent quantitative agreement of the b parameters is found, both between the alternative calculations and between either calculation and experiment (see Section VLB.I). Given the spherical, therefore achiral, nature of the initial orbital in these calculations, any chirality exhibited in the angular distribution must stem from the final-state photoelectron scattering off the chiral molecular ion potential. Successful prediction of any non-zero chiral parameter is clearly then dependent on a reliable potential model describing the final state. At this level, there is nothing significant to choose between the potential models of the two methods. 

It is important to note that the coefficients fp, gp, and hs are always nonvanishing, for both achiral and chiral isotropic films. On the other hand, fs, gs, and hp can only be nonvanishing if the isotropic film is chiral (nonracemic) because they completely depend on the chiral susceptibility components. Note that gs is always equal to zero within the electric dipole approximation. The sign of the chiral expansion coefficients changes between enantiomers, while that of the achiral expansion coefficients stays the same. Experimental determination of all expansion coefficients fully characterizes the nonlinearity and nonlinear optical activity of the sample. Once all expansion coefficients are... 

Figures 9.10a and b show the respective s- and p-polarized components of the second-harmonic intensity recorded from a chiral thin film in transmission. The experimental data points are fitted to Eq. (42) and the values of the expansion coefficients /, g, and h are determined. Note the CD effect observed in both figures. Since fs, gs, and hp are zero for an achiral thin film or surface, (—f+g) and h cannot be simultaneously non vanishing for a particular experimental arrangement and thus no CD effect can be observed. In Figure 9.11, we compare the second-harmonic response as observed experimentally for a chiral and achiral thin film.

From these examples, it becomes clear that molecular symmetry can spontaneously break into chiral domains in the absence of any external force or seed. However, in every known case, the net symmetry remains intact, and the overall chirality sums to zero in the environment. Nevertheless, the spontaneous formation of macroscopic chiral regions in systems of associating achiral molecules is of interest to those who contemplate induction of molecular chirality in the context of prebiotic reaction chemistry. 

In the trans conformation (a), the CH2CH2 moiety is achiral, pi and fi2 colinear and no VCD can be generated, that is, the rotational strength is zero. In the two gauche conformations (b) and (c), the CH2CH2 units have opposite chirality, and generate coupled oscillator VCD intensity of opposite sign. 

Given atomic coordinates for a particular conformation of a molecule and some property value assigned to each atom, one can easily calculate a chirality function that distinguishes enantiomers, is zero for an achiral molecule, and is a continuous function of the coordinates and properties. This is useful as a quantitative measure of chirality for molecular modeling and structure-activity relations. 

The values for two mirror images should have opposite sign but equal magnitude. This implies that the value for achiral molecules should be zero. 

For the optical activity of achiral chromophores with a dissymmetric environment, two types of theoretical treatments have been proposed coupled oscillator treatment and one-electron treatment. The charge distribution of the magnetic dipole transition correlates Coulombically with an electric dipole induced in the substituents, and the colinear component of the induced dipole provides, with the zero-th order magnetic moment, a non-vanishing rotational strength. 

Now, let us consider a system where an achiral molecule (A) and a chiral molecule (C) have a fixed mutual orientation. An electronic transition of the achiral molecule from the ground state z(0> to the excited state Aa, higher in energy by E0a, has a zero-order (non-perturbed) electric dipole moment po0 and an orthogonal magnetic dipole moment ma0. These moments are increased in the molecular pair (A -C) by first-order dynamic coupling as ... 

The zero- and first-order moments give the first-order rotational strength and the second-order rotational strength which are induced for the given transition Aa <- A0 of the achiral molecule ... 

That both phenomena arise as a consequence of macroscopic solvent order and not Intimate solvent-solute Interactions Is clear Saeva and 01In (75) have shown that solute LCICD spectra can be observed In twisted nematic phases only Nakazaki et al. (76) find an excess of one enantiomer of hexahelicene Is produced photochemlcally from achiral precursors In twisted nematic phases no LCICD spectra or optical Induction occurs In untwisted nematic phases and the handedness of the twist can be correlated with the sign of the LCICD and the preferred product enantiomer. Furthermore, Isotropic phases of cholesteric mixtures display no discernible LCICD spectra (12, 67) and the enantiomeric excesses In products of photolablle reactants In Isotropic phases are near zero (51). 

The evolution Eq. 13 of the order parameter has a similar form to the time-dependent Landau equation [17], which is fundamental in nonequihbrium phase transitions. The asymptotic value of the order parameter 4>i,oo is determined as the zero of the velocity 4>i- The main difference from the standard model of phase transitions lies in the time dependence in the coefficients Ait) and B(t) induced by that of the achiral concentration ait) and the total chiral concentration qft). Because the concentrations a and q are nonnegative, A(t) cannot exceed Bit) Ait) < Bit). 

In all the other cases, with either a linear or a nonlinear recycling process or with both, the coefficients A and B are no longer zero at the same time, and a definite value of the order parameter

0, B becomes nonzero since /xqi,oo > 0. If the linear recycling exists as X > 0, not all the achiral substrate transform to chiral products but a finite amount remains asymptotically as a(t = oo) > 0. Therefore, nonzero values of ko, k or k2 k 2 give contributions to the coefficients A or B. 

In support of his proposal, Tauber pointed out that For certain knots Ze = 0. This is exactly as it should be, for precisely these knots are identical with their mirror images. Similarly, Walba asserted that The number of 8s and >cs are summed arithmetically. If there are the same number of 8 and X crossings, then the knot must be topologically achiral. Contrary to these assertions, however, alternating knots whose writhe is zero are not necessarily amphicheiral The simplest example is knot 84. Nineteen of the 32 10-crossing prime knots with writhe zero are topologically chi ral, and 13 of these are alternating.144 Two hundred... 

How can chirality be quantified There are two minimum requirements. First, recall that an object X (no matter whether physical or mathematically abstract) is chiral if and only if it is nonsuperposable on its mirror image X (X X). It follows that any chirality measure % that quantifies this property can equal zero if and only if the object is achiral any function that does not satisfy this conditio sine qua non fails to qualify as a measure of chirality147-148 ... 

In stark contrast to the numerous functions that are available to measure geometrical chirality, no measure has yet been reported for the quantification of topological chirality. In analogy to geometrical chirality measures, topological chirality measures %(K) must satisfy two minimal conditions They can be equal to zero if and only if the knot or link is achiral, and they have to have the same absolute value for two topological enantiomorphs. 

Suppose we had a mixture of equal amounts of ( + )-butan-2-ol and (-)-butan-2-ol. The (+) isomer would rotate polarized light clockwise with a specific rotation of +13.5°, and the (-) isomer would rotate the polarized light counterclockwise by exactly the same amount. We would observe a rotation of zero, just as though butan-2-ol were achiral. A solution of equal amounts of two enantiomers, so that the mixture is optically inactive, is called a racemic mixture. Sometimes a racemic mixture is called a racemate, a ( ) pair, or a (d,l) pair. A racemic mixture is symbolized by placing ( ) or ((1,1) in front of the name of the compound. For example, racemic butan-2-ol would be symbolized by ( )-butan-2-ol or (d,/)-butan-2-ol. ... 

Thus the e.e. of the monoacetate stage will increase due to both steps. When these anticipations apply, the e.e. of the monoacetates should increase throughout the reaction to reach 100% and, moreover, the yield of the two monoacetates R + S) should increase before it eventually declines and reaches zero when the conversion has reached 100% and only the achiral diacetate will comprise the reaction mixture [4d] (Figure 7.8). 

---