PHY.K02UF Molecular and Solid State Physics

sp, sp², and sp³ hybrid orbitals

The 2s and 2p atomic orbitals are,

$$\phi_{2s}=\frac{1}{4} \sqrt{\frac{Z^3}{2\pi a_0^3}}\left(2-\frac{Zr}{a_0}\right)\exp\left(-\frac{Zr}{2a_0}\right),$$ $$\phi_{2p_x}=\frac{1}{4} \sqrt{\frac{Z^5}{2\pi a_0^5}}x \exp\left(-\frac{Zr}{2a_0}\right),$$ $$\phi_{2p_y}=\frac{1}{4} \sqrt{\frac{Z^5}{2\pi a_0^5}}y \exp\left(-\frac{Zr}{2a_0}\right),$$ $$\phi_{2p_z}=\frac{1}{4} \sqrt{\frac{Z^5}{2\pi a_0^5}}z \exp\left(-\frac{Zr}{2a_0}\right).$$

Here $Z$ is the effective nuclear charge. These atomic orbitals can be combined to form sp, sp² and sp³ hybrid orbitals.

Sp hybridization

$$\psi_1 = \frac{1}{\sqrt{2}}\left( \phi_{2s}+\phi_{2p_x}\right)$$ $$\psi_2 = \frac{1}{\sqrt{2}}\left( \phi_{2s}-\phi_{2p_x}\right)$$ $$\psi_3 = \phi_{2p_y}$$ $$\psi_4 = \phi_{2p_z}$$

Sp² hybridization

$$\psi_1 = \frac{1}{\sqrt{3}}\phi_{2s}+\frac{2}{\sqrt{3}}\phi_{2p_x}$$ $$\psi_2 = \frac{1}{\sqrt{3}} \phi_{2s}-\frac{1}{\sqrt{6}}\phi_{2p_x}+\frac{1}{\sqrt{2}}\phi_{2p_y}$$ $$\psi_3 = \frac{1}{\sqrt{3}} \phi_{2s}-\frac{1}{\sqrt{6}}\phi_{2p_x}-\frac{1}{\sqrt{2}}\phi_{2p_y}$$ $$\psi_4 = \phi_{2p_z}$$

Sp³ hybridization

$$\psi_1 = \frac{1}{2}\left(\phi_{2s}+\phi_{2p_x}+\phi_{2p_y}+\phi_{2p_z}\right)$$ $$\psi_2 = \frac{1}{2}\left(\phi_{2s}+\phi_{2p_x}-\phi_{2p_y}-\phi_{2p_z}\right)$$ $$\psi_3 = \frac{1}{2}\left(\phi_{2s}-\phi_{2p_x}+\phi_{2p_y}-\phi_{2p_z}\right)$$ $$\psi_4 = \frac{1}{2}\left(\phi_{2s}-\phi_{2p_x}-\phi_{2p_y}+\phi_{2p_z}\right)$$