D2 Unit 8 Assignment

The identity operation, E, leaves the molecule unchanged. The C2 axis lies along the z-axis. The C2 operation transforms the dichloromethane molecule as so. Carrying out two consecutive C2 operations is equivalent to the identity transformation. There are two reflection planes in the molecule; both contain the rotation axis. One plane is the plane of the page containing the ClCCl plane. We will label this plane σ′(yz). The second plane is perpendicular to the plane of the page; we will label it σ(xz). The action of σ′(yz) is to give the arrangement of atoms shown, where the two hydrogen atoms have been interchanged, while the two chlorine atoms and carbon are unchanged. σ(xz) permutes the chlorine atoms, but leaves carbon and the two hydrogen atoms fixed.

This means that σ′(yz) is its own inverse (see (d) above). Similarly, we find that (σ(xz))2 = σ(xz)σ(xz) = E, and σ(xz) is its own inverse. Now, if we carry out a σ(xz) reflection first and follow it by a σ′(yz) reflection, we get the following. Comparing this diagram to that of a C2 rotation we see that the result is identical. Therefore, we say that σ′ (yz)σ(xz) = C2. You can show that performing the reflections in reverse order yields the same result. Note that the symmetry elements remain fixed and are not transformed to new positions when the atoms in the molecule move to new positions. What about carrying out a C2 rotation followed by the reflection σ′(yz)? Performing these symmetry operations yields the following. 8 This is equivalent to a σ(xz) operation. You can show that carrying out these operations in reverse order affords the same result. Next, we compute the product