Symmetry-Adapted Orbitals

As a last example, we will see that we can create SALC's of atomic orbitals in exactly the same way. Consider the case of ethylene (which is still point-group $D_{2h}$, see Figure 4). The 4 C-H $\sigma$ bonds are symmetry-equivalent, so we can make 4 C-H bonding SALC's.

Figure 4: Ethylene Molecular Orbitals

Table 6: Result of Symmetry Operations on $\sigma _1$

	$E$	$C_2(z)$	$C_2(y)$	$C_2(x)$	$i$	$\sigma(xy)$	$\sigma(xz)$	$\sigma(yz)$
$\sigma _1$	$\sigma _1$	$\sigma_3$	$\sigma_4$	$\sigma_2$	$\sigma_3$	$\sigma _1$	$\sigma_2$	$\sigma_4$

The four non-zero SALC's are:

$${\hat P}_{A_g}(\sigma_1) = \sigma_1 + \sigma_2 + \sigma_3 + \sigma_4$$

$${\hat P}_{B_{1g}}(\sigma_1) = \sigma_1 - \sigma_2 + \sigma_3 - \sigma_4$$

$${\hat P}_{B_{2u}}(\sigma_1) = \sigma_1 - \sigma_2 - \sigma_3 + \sigma_4$$

$${\hat P}_{B_{3u}}(\sigma_1) = \sigma_1 + \sigma_2 - \sigma_3 - \sigma_4$$

These are pictured in Figure 4. Note that four C-H bonds go in to the symmetry-adaptation, and four C-H SALC's come out. We could use similar procedures to construct the remaining four occupied orbitals, which are the symmetric ($a_g$) combination of the two C 1s core orbitals, the antisymmetric ($b_{3u}$) combination of the two C 1s core orbitals, the $a_g$ C-C $\sigma$ bonding orbital, and the $b_{1u}$ C-C $\pi$ bonding orbital (see Fig. 4).

Finally, it is worth commenting that we can often come up with the SALC's by intuition more easily than we can work out the projection operators (especially with a little experience and practice). This is certainly true for the ethylene symmetry-adapted orbitals.

Acknowledgments. The author thanks Dr. Sahan Thanthiriwatte for providing the figures.