Constructing Molecular Orbital Diagrams for Simple Sigma and Pi Systems

1. Ethene (C₂H₄): Deconstructing the Double Bond

Let's begin with ethene. From a valence bond perspective, we describe each carbon as being sp² hybridized. These sp² orbitals form the $\sigma$ bond framework: one C-C $\sigma$ bond and four C-H $\sigma$ bonds, all lying in a single plane. This leaves one unhybridized 2p orbital on each carbon, perpendicular to the molecular plane. These two parallel p orbitals overlap side-on to form the $\pi$ bond.

While this separation is an approximation, it's an incredibly effective one. The $\sigma$ orbitals are generally much lower in energy and less involved in reactions than the $\pi$ orbitals. We can therefore analyze the crucial $\pi$ system on its own.

Hückel Molecular Orbital (HMO) Theory for the $\pi$ System

The problem of two parallel p orbitals, each contributing one electron, is formally identical to the H₂ problem from our last lesson, but with p orbitals instead of s orbitals. We can use the same LCAO approach and secular determinant framework. However, for $\pi$ systems in organic chemistry, we can make a set of simplifying assumptions known as the Hückel approximations.

Let's summarize the key results from the video. The Hückel method gives us a 2x2 secular determinant for ethene's $\pi$ system:

Solving for the energy $E$ gives two solutions:

• $E_1=\alpha +\beta$ (Bonding MO, ${\psi }_1$)
• $E_2=\alpha -\beta$ (Antibonding MO, ${\psi }_2$)

Since $\alpha$ represents the energy of an electron in an isolated p orbital and $\beta$ is a negative (stabilizing) quantity, $E_1$ is lower in energy and $E_2$ is higher.

The two $\pi$ electrons in ethene occupy the lower-energy bonding orbital, ${\psi }_1$. The total $\pi$ electron energy is $2(\alpha +\beta )=2\alpha +2\beta$. This is $2\beta$ lower in energy than two electrons in isolated p orbitals ($2\alpha$), a quantity called the $\pi$ bond energy.

The wavefunctions are:

• ${\psi }_1=\frac{1}{\sqrt{2}}({\varphi }_A+{\varphi }_B)$ (Bonding, no nodes)
• ${\psi }_2=\frac{1}{\sqrt{2}}({\varphi }_A-{\varphi }_B)$ (Antibonding, one node)

For ethene, the HOMO is the bonding $\pi$ orbital (${\psi }_1$), and the LUMO is the antibonding ${\pi }^{\ast }$ orbital (${\psi }_2$). These frontier orbitals are critical as they are the primary players in chemical reactions. The HOMO is the source of electrons for a nucleophilic molecule, and the LUMO is the destination for electrons for an electrophilic molecule.

2. The Complete MO Diagram of Ethene

Now we can assemble the full picture by combining the $\sigma$ and $\pi$ systems into a single energy level diagram. While we won't derive the $\sigma$ MO energies, we can place them qualitatively based on general principles.

Based on this, we can sketch a more complete, albeit qualitative, MO energy diagram for ethene:

A qualitative MO energy diagram for ethene. The eight $\sigma$ bonding MOs (from C-C and C-H bonds) are at the lowest energies. Above them lie the $\pi$ bonding MO (the HOMO). The ${\pi }^{\ast }$ antibonding MO (the LUMO) is the next lowest in energy, followed by the high-energy ${\sigma }^{\ast }$ antibonding orbitals. The HOMO-LUMO gap is the energy required for the lowest electronic excitation.

3. Ethyne (C₂H₂): Two Orthogonal $\pi$ Systems

Now let's turn to ethyne. With sp hybridization, each carbon has two remaining, unhybridized p orbitals: one $p_y$ and one $p_z$. These two orbitals on each atom are orthogonal (at 90°) to each other.

This orthogonality is key: the $p_y$ orbitals on the two carbons can only interact with each other, and the $p_z$ orbitals can only interact with each other. This means we have two independent $\pi$ systems. Each system is identical to the $\pi$ system of ethene.

Since we have two identical, degenerate $\pi$ systems (${\pi }_y$ and ${\pi }_z$), each will produce a bonding and an antibonding MO with energies $\alpha +\beta$ and $\alpha -\beta$.

• We get two degenerate bonding $\pi$ orbitals.
• We get two degenerate antibonding ${\pi }^{\ast }$ orbitals.

Ethyne has four $\pi$ electrons in total. Following the Aufbau principle, these four electrons fill the two degenerate bonding $\pi$ orbitals.

A qualitative MO energy diagram for ethyne. The $\sigma$ framework gives rise to low-lying $\sigma$ bonding orbitals. The key feature is the pair of degenerate $\pi$ bonding orbitals (the HOMOs) and the pair of degenerate ${\pi }^{\ast }$ antibonding orbitals (the LUMOs). The four pi electrons fill the two bonding orbitals.

Conclusion

Today, we've successfully transitioned from diatomic molecules to the fundamental $\pi$ systems of organic chemistry. By separating the $\sigma$ and $\pi$ frameworks, we can use the powerful, mathematically-grounded Hückel MO theory to gain deep insights into electronic structure.

Key Takeaways:

• Organic molecules can be analyzed by separating their robust $\sigma$ framework from their more reactive $\pi$ system.
• Hückel Molecular Orbital (HMO) theory is a simplified LCAO method that provides an excellent quantitative description of $\pi$ electrons by using the $\alpha$ and $\beta$ parameters.
• Combining N atomic p orbitals in a $\pi$ system results in N molecular $\pi$ orbitals.
• For ethene, two p orbitals combine to form one bonding ($\pi$) and one antibonding (${\pi }^{\ast }$) MO.
• For ethyne, two orthogonal pairs of p orbitals create two independent and degenerate $\pi$ systems.
• The frontier orbitals (HOMO and LUMO) are typically part of the $\pi$ system and are key to understanding a molecule's reactivity.

Preview of the Next Lesson:

We've now laid the quantum mechanical groundwork for hybridization and simple $\pi$ bonds. In our next lesson, we will analyze hybridization itself as a mathematical mixing of atomic orbitals to form sp, sp², and sp³ hybrids, solidifying the connection between this model and molecular geometry.