Natural and magnetic optical activity can be found in quantum states.
To calculate matrix elements of operators between component states of a degenerate level, it is necessary to classify the wavefunctions and operators with respect to the irreducible representations of the symmetry group of the system.
Racah is responsible for the concept of irreducible tensor operators and the development of a formal ism for making practical use of them in spherical systems.
The work was partially based on and developed concurrently with the advances in the theory of angular momentum made by Wigner.
Two authoritative texts summarize this work by Fano and Racah.
The theory was extended to the molecular point groups.
We will not give an account of this work here, but will simply state the formulae required in subsequent chapters and refer the reader to Silver and Piepho for an introduction to most of the aspects required in this book.
The tables ofcoupling coefficients can be used directly if the different versions of the Wigner-Eckart theorem are written down.
The reduced matrix elements can be calculated in some situations, but in many of the applications in this book explicit values are not required because the reduced matrix elements cancel.
An alternative version of the Wigner-Eckart theorem is needed for calculations on systems belonging to finite point groups.
For certain calculations on molecule with odd numbers of electrons, we need an extension of Griffith's methods.
The direct products of some of the irreducible representations contain repeated representations.
Table 5 of Harnung was published in 1973.
An analysis of chirality based on the permutation of ligands among sites on a molecular skeleton is a different aspect of symmetry.
It provides rigorous criteria which can be used to assess any theory of optical activity, as well as giving insight into the phenomenon ofmolecular chirality.
Much of this section is based on reviews by Ruch and Mead, and we refer to these and a later review by King.
A molecule can be seen as a skeleton with sites attached to it.
If the skeleton is achiral, there must be differences between the ligands.
It is well known that chirality is only possible if all four of the ligands are different.
The pseudoscalar optical rotatory parameter is not the only one that can be described with the HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax Ruch and Sch"onhofer gave a definitive form to the systematic group theoretical study of chirality functions.
If we were asked to put our left shoes into one box and our right shoes into another box, we would be able to accomplish the task without mental difficulty, because the right shoes belonging to different people may be quite different in colour, shape and size.
We must capitulate if we are asked to solve the same problem with potatoes.
It is1-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-6556 It is clear that we have to separate them, but for other potatoes we have to make new decisions.
Any classification is not real.
Section 4.5.6 states that the skeleton of any molecule can be assigned to one of two categories.
One of these categories is shoe-like in that it admits a classification into right- and left-handed molecule; the other is potato-like in that it doesn't differentiate between right and left-handed molecule.
A molecule may be completely specified by describing a skeleton and nature at each site.
A particular skeleton can be thought of as defining a class of molecule with individual class members being specified at each site.
For example, ethane can be thought of as the six-site ethane skeleton with six hydrogen atoms as ligands, or as the four-site methane skeleton.
If all of the ligands are of the same kind, then the discussion should be restricted to those that fulfill the condition that the molecules have the symmetry of the bare skeleton.
It also means that the ligand must have a threefold or higher proper rotation axis to make the properties invariant under changes of orientation.
If the skeleton is achiral, a molecule containing only one sort is achiral.
If all the ligands are different, we consider the classes which are specified by skeletons.
skeletons with positions at the corners of regular bodies are examples.
It is assumed that the ligands can be characterized by a physical property associated with a single scalar.
Neither of these functions can be applied without difficulty.
2 is capable of giving a general description of a chiral property in situations where there is no symmetry reason why it should.
We must demand that there is no nonracemic mixture of isomers.
There are more complete accounts in the books.
A 2-cycle is called a transposition.
Every permutation can be written as a product of cycles which operate on mutually exclusive sets of labels.
It is said to be odd.
The product of two even or two odd permutations is even.
Partition and conjugate classes are the subject of discussion.
All elements in a given conjugate class have the same structure.
S4 is an example.
There are five conjugate classes in S4.
Table 4.5 shows the result of applying the theorem to S4.
These ideas can be used to represent dimensions greater than one.
There are two types of permutation, horizontal and vertical, and they interchange only symbols in the same row of a standard tableau.
The identity operation is defined as (1) (2) (3).
The formalism of the permutation group has been used to give a mathematical structure to the concept of qualitative completeness.
G is sometimes isomorphic with the point group of the skeleton.
The behavior of the full permutation group of the ligand sites is related to the transformation properties of the point group of the skeleton.
Depending on whether the isomers are nonracemic or racemic, an ensemble operator is said to be achiral.
M does not disappear.
There is an account of subduced and induced representations included in the proof of this theorem.
The allene skeleton is an explicit example of the meaning of this.
The table shows the character table for S4.
The functions are chosen to correspond with the model on which a particular theory of optical activity is being constructed.
Consider the first procedure, in which the lowest order in one or more parameters is generated.
A simple example can be provided by the two functions.
Our monomial can't be symmetric with respect to any two sites in the same column, that is, it can't have the same power.
The allene skeleton is a simple example of this.
2 l4 is a constant.
The second procedure is used to generate chirality functions that depend on as few ligands as possible.
Any class of skeleton can be applied to the two procedures outlined above.
We will now ask the following question.
The problem is being formulated more generally.
It's possible to associate an assortment of ligands with a Young diagram.
The allene skeleton is dressed with two identical ligands and two different ones.
The active partition is now defined a little more precisely.
We can see from Figure 4.8 that the same result was deduced from the start of this section.
The set of active partition generate a set of numbers which show the properties of the skeleton.
The longest and shortest first lines and columns of all shaded diagrams can be specified with four numbers.
The maximum number of equal ligands and the minimum number of different ligands are defined by the chirality order and the chirality index.
Section 4.5.1 introduces a class of skeleton which allows the use of achiral ligands.
This defines skeletons that are compatible with all of the same types.
Any theory of optical activity must be concerned with the skeleton's idiosyncrasy.
If only one ligand is different, all the others are the same.
The sector rules of the quadrant and octant type can be found in adamantanone derivatives.
In allene derivatives, optical activity is generated by simultaneous perturbations from two ligands, as in this type of skeleton.
This type of skeleton requires three different ligands to support a chiral molecule, as in methane derivatives.
The class of skeleton is achiral since it can't support a chiral molecule if all the ligands are different.
The benzene skeleton is an example.
The property required of satisfactory chirality functions is that they accommodate the concepts of Homochirality and Heterochirality, which were introduced in Section 4.5.1.
Chiral relatedness is based on similarity of the ligands and the molecule.
We can transform any molecule of the class into any other without leaving the class.
The point group of the skeleton causes an achiral molecule to be left invariant.
Section 4.5.2 states that every permutation can be written as a product of cycles which operate on mutually exclusive labels.
The bound ary between R and L can be chosen if this condition is satisfied.
There is an acceptable classification into R and L for shoe-like skeletons, but no direct classification into R and L is possible for potato-like skeletons.
The designation of R and L for the two regions of opposite chirality is arbitrary for shoe-like skeletons.
A molecule originally assigned to R might find itself in L by changing the definition of the ligand parameters.
An example of a shoe-like skeleton can be found in Allene.
The surfaces are determined by the improper rotation.
The one-dimensional space l1 is a subspace of the above because of the improper rotation.
l1 and l2 are three-dimensional hypersurfaces.
The condition developed above can be applied to a given skeleton.
The other skeletons are similar to potatoes.
An example of a potato-like skeleton can be found in the tetragonal bipyramid.
Since an achiral situation is not encountered at any time, we can assign any pair of neighbours encountered on the path to the same subclass.
The same argument applies when we variation with the ligands at positions 3 and 4.
The end result is the enantiomer of the original molecule.
Since there is no privileged point on the path from the original molecule to the enantiomer, the conclusion is that there is no homochirality concept.
We presented Ruch's ideas on permutation symmetry and chirality in some detail because it seems to be of fundamental significance in the theory of optical activity, even though its applications in stereochemistry have been limited.
Section 4.5.5 requires simultaneous contributions from pairs of isotropic ligands, but this could be interpreted as a contribution from a single anisotropic ligand because a pair of interacting atoms is equivalent to an anisotropic group.
Measurement of the optical rotation of various nonracemic mixtures of chiral allene isomers tells us what extent we are justified in taking a chiral allene derivative as having the approximate symmetry of a regular tetrahedron.
Experiments based on optical rotation measure ments have been reviewed by King.
Their success depends on the complexity of the skeleton and the number of sites.
Good approximations of optical rotation data for the derivatives of shoe-like skeletons such as methane and allene are provided by curbless functions.
The approximations get worse for more complicated shoe-like skeletons with several chiral ligand partition or of potato-like skeletons.
The formal development of chirality functions is based on one pseudoscalar observable.
This can be seen immediately in the case of the Raman case.
The isotropic and anisotropic scattering contributions depend on sample density, and the relative amounts of isotropic and anisotropic scattering will be different for different isomers.
The sum of contributions from each molecule will be called L. The entire chirality function formalism will need to be reformulated before any attempt is made to apply it to optical activity.