Chapter 11, Problem 72FP

As discussed in Are You Wondering 11-1, the sp hybrid orbitals are algebraic combinations of the s and p orbitals. The required combinations of 2s and 2p orbitals are

   $\begin{array}{l}{\psi }_1(sp)=\frac{1}{\sqrt{2}}[\psi (2s)+\psi (2p_z)]\hfill \\ {\psi }_2(sp)=\frac{1}{\sqrt{2}}[\psi (2s)-\psi (2p_z)]\hfill \end{array}$

1. By combining the appropriate functions given in Table 8.2, construct a polar plot in the manner of Figure 8-24 for each of the above functions in the xz plane. In a polar plot the value of ria, set at a fixed value (for example, 1). Describe the shapes and phases of the different portions of the hybrid orbitals, and compare them with those shown in Figure 11-12.

Convince yourself that the combinations employing the 2p, or 2p, orbital also give similar hybrid orbitals but pointing in different directions.

The combinations for the sp²hybrids in the xy plane are

   $\begin{array}{l}{\psi }_1(s{p}^2)=\frac{1}{\sqrt{3}}\psi (2s)+\frac{\sqrt{2}}{\sqrt{3}}\psi (2p_x)\hfill \\ {\psi }_2(s{p}^2)=\frac{1}{\sqrt{3}}\psi (2s)-\frac{1}{\sqrt{6}}\psi (2p_x)+\frac{1}{\sqrt{2}}\psi (2p_y)\hfill \\ {\psi }_3(s{p}^2)=\frac{1}{\sqrt{3}}\psi (2s)-\frac{1}{\sqrt{6}}\psi (2p_x)-\frac{1}{\sqrt{2}}\psi (2p_y)\hfill \end{array}$

By constructing polar plots (in the xy plane), show that these functions correspond to the sp²hybrids depicted in Figure 11-10.