Constructing Molecular Orbitals for Diatomic Molecules

1. From Atomic to Molecular Orbitals: The LCAO Approximation

The central idea of MO theory is that when atoms combine to form a molecule, their atomic orbitals (AOs) cease to exist and are replaced by a new set of molecular orbitals (MOs) that belong to the molecule as a whole.

While the Schrödinger equation for a molecule is too complex to solve exactly, we can construct excellent approximations for the MO wavefunctions. The most common method is the Linear Combination of Atomic Orbitals (LCAO). We express the unknown molecular orbital wavefunction (${\psi }_{\text{MO}}$) as a weighted sum of the constituent atomic orbital wavefunctions (${\varphi }_{\text{AO}}$).

For the simplest molecule, H₂, formed from two hydrogen atoms (A and B) with their respective 1s orbitals (${\varphi }_{1sA}$ and ${\varphi }_{1sB}$), the LCAO is:
${\psi }_{\text{MO}}=c_A{\varphi }_{1sA}\pm c_B{\varphi }_{1sB}$
Where $c_A$ and $c_B$ are weighting coefficients.

This combination of wavefunctions leads to two possible outcomes, which you may recall from wave physics.

As the video explained, the two combinations are:

1. Bonding MO (Constructive Interference): The in-phase combination (${\psi }_{+}=c_A{\varphi }_{1sA}+c_B{\varphi }_{1sB}$) increases the electron probability density in the region between the two nuclei. This increased density attracts both nuclei, holding them together. This orbital is lower in energy than the original AOs.
2. Antibonding MO (Destructive Interference): The out-of-phase combination (${\psi }_{-}=c_A{\varphi }_{1sA}-c_B{\varphi }_{1sB}$) results in a node—a region of zero electron density—between the nuclei. Without this "electronic glue," the nuclei repel each other. This orbital is higher in energy than the original AOs.

2. A Deeper Dive: The Energy of Molecular Orbitals

How do we determine the precise energies of these orbitals and the coefficients in the LCAO? For this, we turn to the Variational Principle, a cornerstone of quantum mechanics. It states that the energy calculated from any approximate wavefunction will always be greater than or equal to the true ground-state energy. Our goal is to find the coefficients that minimize the energy, giving us the best possible approximation.

The energy expectation value is given by:
$E=\frac{\int \psi \hat{H}\psi d\tau }{\int {\psi }^{2}d\tau }$

For a homonuclear diatomic molecule like H₂, where $\psi =c_A{\varphi }_A+c_B{\varphi }_B$, substituting and minimizing the energy with respect to the coefficients leads to a set of equations that can be solved. This process is mathematically elegant and reveals the deep structure of chemical bonding.

The video's matrix approach is powerful. To see the algebraic solution step-by-step, we can solve the secular determinant derived from the variational principle.

A Key Insight from the Mathematics

Look closely at the energy expressions. Since the overlap integral $S$ is a small positive number (typically ~0.2-0.3), the denominator $(1-S)$ is smaller than $(1+S)$. This means the energy of the antibonding orbital is raised by more than the energy of the bonding orbital is lowered.

$\Delta E_{\text{destabilization}}=E_{\text{antibonding}}-\alpha >\alpha -E_{\text{bonding}}=\Delta E_{\text{stabilization}}$

This is a profound result: antibonding orbitals are more antibonding than bonding orbitals are bonding. This explains why Helium does not form a stable He₂ molecule. The two electrons in the bonding orbital provide stabilization, but the two electrons in the more-destabilizing antibonding orbital lead to a net increase in energy.

3. Constructing MO Diagrams for Diatomic Molecules

We can now use these principles to build MO diagrams for any diatomic molecule.

Rules for Construction:

1. The total number of MOs equals the number of AOs combined.
2. Orbitals are filled with electrons starting from the lowest energy level (Aufbau Principle).
3. Each MO can hold a maximum of two electrons with opposite spins (Pauli Exclusion Principle).
4. Electrons enter degenerate orbitals singly before pairing up (Hund's Rule).

We classify MOs by their symmetry:

• σ (sigma) orbitals: Formed from head-on overlap of s-s, s-p, or p-p orbitals. They are cylindrically symmetric about the internuclear axis.
• π (pi) orbitals: Formed from side-on overlap of p-p orbitals. They have a nodal plane that includes the internuclear axis. Pi orbitals always come in degenerate pairs (e.g., ${\pi }_{2px}$ and ${\pi }_{2py}$).

A useful metric is Bond Order, which predicts the net number of bonds:
$\text{Bond Order}=\frac{1}{2}\left(({\text{number of bonding e}}^{-})-({\text{number of antibonding e}}^{-})\right)$

Diatomics of the 2nd Period

For second-period elements (Li to Ne), we consider the combination of 2s and 2p atomic orbitals. The 2s AOs combine to form ${\sigma }_{2s}$ and ${\sigma }_{2s}^{\ast }$ MOs. The 2p AOs combine to form one ${\sigma }_{2p}$ and two degenerate ${\pi }_{2p}$ bonding MOs, along with their antibonding counterparts (${\sigma }_{2p}^{\ast }$ and ${\pi }_{2p}^{\ast }$).

A subtle but critical point is s-p mixing. The ${\sigma }_{2s}$ and ${\sigma }_{2p}$ orbitals have the same symmetry and can interact. This interaction pushes the ${\sigma }_{2s}$ down in energy and the ${\sigma }_{2p}$ up. The strength of this mixing depends on the energy gap between the 2s and 2p AOs.

• For O₂, F₂, Ne₂: The s-p gap is large, s-p mixing is weak. The ${\sigma }_{2p}$ orbital is lower in energy than the ${\pi }_{2p}$ orbitals.
• For B₂, C₂, N₂: The s-p gap is small, s-p mixing is strong. The effect is large enough to push the ${\sigma }_{2p}$ orbital above the ${\pi }_{2p}$ orbitals in energy.

Conclusion

Today, we have built a quantum mechanical model of the chemical bond from first principles. We have seen that bonding is not just about "sharing electrons," but about the constructive interference of wavefunctions, leading to a new set of molecular orbitals with specific, calculable energies.

Key Takeaways:

• Molecular orbitals are formed by the Linear Combination of Atomic Orbitals (LCAO).
• In-phase combination gives a low-energy bonding MO, while out-of-phase combination gives a high-energy antibonding MO.
• The Variational Principle allows us to find the best approximation for the MO energies, revealing that antibonding orbitals are more destabilizing than bonding orbitals are stabilizing.
• MO theory successfully explains the bonding, stability (or lack thereof), and magnetic properties of diatomic molecules like N₂ and O₂.
• s-p mixing is a crucial effect that inverts the ${\sigma }_{2p}$ and ${\pi }_{2p}$ energy levels for diatomics up to and including N₂.

Preview of the Next Lesson:

We now have a powerful and versatile tool for describing chemical bonds. In the next lesson, we will apply these MO concepts to simple organic molecules, constructing the molecular orbital energy diagrams for the $\sigma$ and $\pi$ systems in ethene and ethyne. This will lay the groundwork for understanding the structure and reactivity of alkenes and alkynes.