# Theoretical Background

## Types of Calculations

**
(1) Measuring the degree of symmetry point group
**

As described above, the general principle can be translated into a computational
method for measuring the degree of any symmetry point group (the CSM).
Some are already available, others will be added soon, and in the meantime can be accessed
by writing directly to us. From that general principle, additional more specific important measures are derived:

- Measuring the degree of the basic symmetry elements (CSM)
- The Continuous Shape Measure (CShM)
- The Continuous Chirality Measure (CCM)

**
(2) Measuring the degree of symmetry elements
**

The basic symmetry elements are: the rotation, *C _{n}* (e.g.

*C*); the reflection, σ; and the rotation-reflection combination (the improper rotation),

_{2}*S*(e.g. S

_{n}_{2}, namely, inversion,

*i*). While Eq.(1) is applicable to any symmetry point group, composed of as many symmetry elements as are needed for their characterization, it is often the symmetry content of a single element, say, four-fold rotation,

*C*which is of interest. Evaluation of

_{4}*S(C*is done by the same algorithms, namely the distance is measured to the nearest structure that has the symmetry point group

_{4})*C*(namely, a group composed of identity,

_{4}*E*, and

*C*). For relevant literature see Zabrodsky & Avnir (1995).

_{4}
**
Example: The near C_{2}-symmetry of inhibitor/HIV protease complexes
**

When inhibitors bind to the active site of the enzyme protease of the human immunodeficiency virus (HIV), they cause a deviation from the perfect

*C*-symmetry of that protein (Keinan & Avnir, 2000). Various parts of the protein deviate differently from

_{2}*C*-symmetry, and one can draw a quantitative map of these deviations (Fig. 8). In the original study, a correlation was found between

_{2}*C*-symmetry deviations and free energy of binding.

_{2}**Figure 8:**The symmetry sensitivity map of 1HVI. Red represents the highest symmetry deviation (Ile50

*S(C*), green represents symmetry deviations in the range

_{2}) = 1.31*0.1 < S(C*, yellow are deviations in the

_{2}) < 1*0.05 < S(C*range, and light gray represents very small deviations (below 0.05). The catalytic triad (Asp25, Thr26, Gly27) is shown in purple, and the inhibitor is seen above it in dark gray.

_{2}) < 0.1Reprinted with permission from Keinan, S. and Avnir, D. Copyright (2000), "Quantitative Symmetry in Structure-Activity Correlations: The Near

*C*Symmetry of Inhibitor/HIV Protease Complexes",

_{2}*Journal of the American Chemical Society*,

**122**(18), 4378-4384.

**
(3) The Continuous Shape Measure (CShM)
**

The Continuous Shape Measure (CShM) evaluates the distance of a given structure to any predetermined shape.
It is therefore a special case of Eq. (1)
(for which the nearest ideal structure is usually searched).
For instance, if one asks, what is the degree of tetrahedricity, *S(T _{d})*, of an AB

_{4}structure, then it is known a-priori that the nearest symmetric structure must have the shape of an exact tetrahedron. Although the CShM is general in its definition and covers all shapes, symmetric or not, we emphasize in this site the distance to polyhedral shapes. For instance, for a distorted hexacoordinated complex, we can evaluate its degree of being an equilateral triangular prism (etp) - which must be of

*D*symmetry - namely its

_{3h}*S(D*. Note that

_{3h}-etp)*S(D*is not the same as

_{3h}-etp)*S(D*: The former is a measure of the distance to a specific shape - the prism - while the latter looks for the distance to the nearest hexacoordinated shape which has a

_{3h})*D*symmetry (it will be some triangular prism, but not necessarily equilateral).

_{3h}
In some cases, such as the determination of the tetrahedricity of AB_{4},
the measure of symmetry and the measure of shape, coincide. For relevant
literature see Pinsky & Avnir (1998).

**
Example: The degree of icosahedricity of distorted C _{60}-fullerene anions. **

The tool of CShM is capable in principle of dealing with

*any*number of vertexes, even the icosahedral C

_{60}fullerene (Fig.9). It undergoes symmetry distortion upon substitution, ionization, and intracage entrapment.

** Figure 9:** Icosahedral polyhedrons

The example we have chosen is the study of Green et al. (1996) on the electronic structures of a series of C

_{60}anions. Green et al. found that the perfect icosahedricity of the fullerene is distorted upon extra charging of this molecule and that the degree of this JT-induced distortion changes with amount of charge. An excellent correlation between the degree of icosahedricity

*S(I*and charge was obtained (Fig.10) (Pinsky & Avnir, 1998).

_{h})**Figure 10:**C

_{60}-fullerene anions undergo Jahn-Teller distortion of the original prefect icosahedral structure of the neutral molecule. Shown is the correlation between the degree of icosahedricity of C

_{60}-fullerene anions and their charge. (For the distorted dianion: upper point, singlet; lower, triplet.) Reprinted with permission from Pinsky, M.; Avnir, D. Copyright (1998), "Continuous Symmetry Measures. 5.The Classical Polyhedra",

*Inorganic Chemistry,*

**37**(21), 5575-5582.

**
(4) The Continous Chirality Measure (CCM)
**

In evaluating the degree of chirality we ask, how far is a given structure from achirality? In terms of
Eq.(1)
we therefore evaluate the distance to the nearest object which has an improper symmetry: reflection plane,
inversion, or any *S _{n}*. Thus, the chirality measure is the searching of the minimal

*S(S*. In many cases, the degree of chirality is the distance to the nearest object which has at least one mirror plane; in others it is the distance to objects of higher order of achirality, e.g.,

_{n})*S*. For relevant literature see Salomon & Avnir (1999).

_{4}
**
Example: Catalysis and enantioselectivity
**

The degree of chirality of a catalyst determines its degree of enantioselectivity. Intuitively,
the enantiomeric excess (ee) of a chiral product, obtained with a chiral catalyst,
will be dictated by the degree of chirality of that catalyst. The following is an example that this works
(Lipkowitz & Schefzick, 2001).
The chiral spirocyclic bisoxazoline copper complexes (Fig.11a, n=1-4),
have been reported to catalyze a Diels-Alder reaction between acrylimide and cyclopentadiene (Fig.11b), resulting
in a chiral addition product. It was found that the ee of the product changes with the degree of chirality
of the catalyst, as shown in Fig.11c.

**
Figure 11:
**
a. The catalysts - spirocyclic bisoxazoline copper complexes.
b. A Diels-Alder reaction between acrylimide with cyclopentadiene.
c. Plot of the computed chirality content (CCM) for catalysts **1-4** versus experimental enantiomeric
excess of a cycloaddition. The structures used for the CCM calculation correspond to the
lowest energy conformation of the spirocycle.
Reprinted with permission from Lipkowitz K. B.; Schefzick S. Copyright (2001),
"Enhancement of Enantiomeric Excess by Ligand Distortion", *Journal of the American Chemical
Society,*

**123**, 6710-6711.

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