Department of Natural Sciences The Hebrew University of Jerusalem The Open University of Israel Shoham - The Center for Technology in Distance Education home page Continuous Symmetry Measures

CoSyM Tutorial

The CoSyM Calculators page provides an online interface for calculations of the following symmetry measures: the continuous chirality measure (CCM), the basic symmetry measure (CSM) and the continuous shape measure (CShM). Click here for theoretical background on the different symmetry measures. In this tutorial, we will demonstrate the use of the CoSyM calculators. We recommend reading the CoSyM calculators technical help and the Jmol help before starting the tutorial. The use of the calculators will be demonstrated with several molecules. To repeat the calculations presented here, download these molecules to your PC.

  1. Example 1: Calculating the CCM
  2. Example 2: Calculating the CSM
  3. Example 3: Calculating the CShM

Example 1: Calculating the CCM
Open the CoSyM Calculators page and upload the molecule dichlorospiroheptane. Note that this molecule is chiral - it does not overlap its mirror image. It therefore has no symmetry operation of improper rotation, Sn. Calculation of the continuous chirality measure for the molecule provides information about the distance of the molecule from an achiral structure. Perform the calculation (click here for technical help). The result is Sch = 2.79. The enantiomer of the molecule gives the same value of the CCM (Why?).

The CCM can be used to compare the chirality content of similar structures. For example, let us explore the chirality content of different conformations of ethane (H3C-CH3). The eclipsed conformation and the staggered conformation are both achiral, therefore Sch = 0 for both molecules.

All other conformations of ethane between the eclipsed and staggered conformation have no symmetry of reflection, nor improper rotation axes of higher order. Therefore, all of these conformations are chiral. The table below shows the chirality content of these conformations as a function of the dihedral angle.

Dihedral angle HCCHSch
   0° (eclipsed)0.00
10° 0.31
50° 0.31

The results show that starting with the eclipsed conformer, the CCM value of ethane increases with the dihedral angle until a maximal value, and then decreases until the staggered conformer. The CCM can be calculated for any value of the dihedral angle and a plot of the CCM as a function of angle can be drawn.

Example 2: Calculating the CSM
Open the CoSyM Calculators page and load the molecule carbonic acid, CO(OH)2. Note that the molecule is planar and that both OH groups point to the same direction in space. In this conformation, there is no symmetry operation of C2, but a reflection operation does exist. Let us check the above claims by means of CSM. Calculate the value of the CSM for C2 and σ (click here for technical help). The results confirm our expectations:
S(C2) = 7.06 and S(σ) = 0.0.

A non-planar conformer of carbonic acid with a dihedral angle OCOH of 120° is shown on the right. Here the OH groups point to opposite directions in space. The molecule has no symmetry of reflection but is expected to have rotational symmetry. Indeed, calculations of the CSM give zero for S(C2) and 3.03 for S(σ). Other conformers of carbonic acid can be characterized by various values of S(C2) and S(σ).

Example 3: Calculating the CShM
On the CoSyM Calculators page, load the molecule CCl4. This is a perfect tetrahedron. Use the Jmol menu to verify that, by examining the bond lengths and bond angles of the molecule (click here for Jmol help). Calculate the distance of the molecule from a tetrahedron (click here for technical help) and note that the result is indeed zero, as expected.

We will now gradually substitute the chlorine atoms with bromine atoms and check how these changes distort the structure. Load the molecules CCl3Br, CCl2Br2, CClBr3. In each one of these molecules, the tetrahedral structure is not perfect since the bond lengths and bond angles are not identical. Use the Jmol menu to verify that and calculate the distance from a tetrahedron for each molecule. The results are shown in the following table.


From the results, we see that the molecule CClBr3 is the less distorted tetrahedron and CCl2Br2 is the most distorted one. The different values of Stetrahedron for CCl3Br and CClBr3 teaches us that C-Cl bonds distort the structure more than C-Br bonds. Can you explain why?

We will now examine the molecule CFClBrI - a distorted tetrahedron with four different atoms. Load this molecule to the CoSyM Calculators page. Use the Jmol menu to determine the bond lengths and bond angles of the molecule (click here for Jmol help). Calculate the distance of this molecule from a perfect tetrahedron (click here for technical help). The value of Stetrahedron is 1.798 - a higher value as compared with the molecules discussed above. As expected, the distortion of the tetrahedron in the molecule CFClBrI is the most significant. For the same molecule, it is also possible to calculate the distance from a square planar structure. Since the structure is closer to a tetrahedron than to a square planar structure, it is expected that the value of Ssquare will be very high. The result of 33.69 confirms this expectation.

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