Molecular Orbital Theory and Energy Diagrams: Comprehensive Study Guide

Principles of Molecular Orbital Overlap and Wave Interference

• Molecular orbital (MO) theory describes the interaction of atomic orbitals (AOs) through the lens of wave interference.
• Bonding Molecular Orbitals: These are formed by constructive interference, described as "overlap and amplify." For example, where "red touches red" (representing the same phase of a wave), the orbital becomes larger. These are typically designated as sigma ($\sigma$) or pi ($\pi$) bonding orbitals.
• Antibonding Molecular Orbitals: These are formed by destructive interference, described as "annihilation." When "opposite colors touch" (e.g., red and blue), they destroy each other, similar to the peak (up part) and trough (down part) of waves meeting. These are designated with an asterisk (e.g., $\sigma^*$).
• Orbital Ownership: In MO theory, orbitals no longer belong to individual atoms but to the entire molecule. Even if certain lobes appear to belong to specific atoms, the resulting MO belongs to "all of us."

Constructing Molecular Orbital Energy Diagrams

• The process of creating an MO diagram follows three primary steps:
    - Step 1: Place Atomic Orbitals on the Wings. Arrange the AOs of the individual atoms on the left and right sides of the energy axis. For $H_2$, this involves placing a $1s$ orbital on each side.
    - Step 2: Place Bonding MOs Lower on the Energy Axis. Bonding result in electrons being attracted to two nuclei simultaneously. This stronger attraction results in lower energy compared to the original AOs.
    - Step 3: Place Antibonding MOs Higher on the Energy Axis. These are placed higher than the starting AOs (e.g., $\sigma_{1s}^*$ is higher than the $1s$ AO).
• Conservation of Orbitals: The number of molecular orbitals in the middle must exactly match the number of atomic orbitals on the wings. For instance, two AOs (one from each Hydrogen) result in two MOs ($\sigma_{1s}$ and $\sigma_{1s}^*$).
• Visual Simplification: While MOs have distinct shapes, diagrams often use boxes or simple lines to represent orbitals for simplicity as molecules become more complex.

Case Study: Diatomic Hydrogen ($H_2$)

• Each Hydrogen ($H$) atom provides one $1s$ atomic orbital containing one electron.
• The total number of electrons to place in the MOs is $1 + 1 = 2$.
• Electrons are filled from the bottom up (lowest energy first).
• Both electrons occupy the $\sigma_{1s}$ bonding orbital.
• This model is consistent with the Lewis structure of $H-H$, representing a single covalent bond.

Case Study: Diatomic Oxygen ($O_2$)

• Valence Orbitals: For Oxygen, the valence AOs are $2s$ and $2p$.
• Electron Count: Oxygen is in the sixth column of the main group elements, meaning it has $6$ valence electrons. For $O_2$, the total valence electron count is $6 + 6 = 12$.
• MO Filling Process for $O_2$:
    - Electrons 1 and 2: $\sigma_{2s}$ (bonding)
    - Electrons 3 and 4: $\sigma_{2s}^<em>$ (antibonding)   - Electrons 5 and 6: $\sigma_{2p}$ (bonding)   - Electrons 7, 8, 9, and 10: $\pi_{2p}$ (bonding - four electrons total across two platforms)   - Electrons 11 and 12: $\pi_{2p}^</em>$ (antibonding - spread across two platforms as unpaired electrons).
• Magnetism: Because the final two electrons in $O_2$ are unpaired in the $\pi_{2p}^*$ orbitals, Oxygen is paramagnetic (attracted to magnets). This explains experimental data that simpler models cannot.

Orbital Ordering and "The Switcheroo"

• The order of energy levels for diatomic molecules is not universal.
• Standard Order ($O_2$ and $F_2$): The $\sigma_{2p}$ orbital is lower in energy than the pair of $\pi_{2p}$ orbitals.
• The Switcheroo ($N_2$ and lighter atoms): For molecules like Nitrogen ($N_2$) and Carbon ($C_2$), the $\pi_{2p}$ orbitals are lower in energy than the $\sigma_{2p}$ orbital.
• Academic Requirement: Users should not worry about memorizing these specific order shifts but must be able to populate and interpret a provided diagram correctly.

Calculating Bond Order

• Bond order ($BO$) can be calculated using the MO diagram with the following formula:
    - $BO = \frac{1}{2} \times (\text{Number of Bonding Electrons} - \text{Number of Antibonding Electrons})$
• Example: Nitrogen ($N_2$):
    - Nitrogen has $5$ valence electrons per atom, totaling $10$.
    - Filling the MOs: $\sigma_{2s}$ (2), $\sigma_{2s}^<em>$ (2), $\pi_{2p}$ (4), $\sigma_{2p}$ (2).   - Bonding electrons: $2 \, (\sigma_{2s}) + 4 \, (\pi_{2p}) + 2 \, (\sigma_{2p}) = 8$.   - Antibonding electrons: $2 \, (\sigma_{2s}^</em>) = 2$.
    - Calculate: $\frac{1}{2} \times (8 - 2) = 3$. This confirms the triple bond in $N_2$.
• Example: Fluorine ($F_2$):
    - Bonding electrons ($8$) minus Antibonding electrons ($6$) results in $\frac{1}{2} \times (2) = 1$. This confirms a single bond.
• Application to Ions ($F_2^{2+}$):
    - Removing two electrons from the highest energy orbitals (the antibonding $\pi_{2p}^*$) changes the calculation.
    - $BO = \frac{1}{2} \times (8 - 4) = 2$. This predicts a double bond for the ion.

Non-bonding Molecular Orbitals

• Definition: Orbitals that do not participate in bonding or antibonding; they are neither constructive nor destructive.
• Occurrence: These form when there are no other orbitals of similar energy on the adjacent atom to combined with.
• Example: Hydrogen Fluoride ($HF$):
    - Fluorine is much more electronegative than Hydrogen; its $2s$ and $2p$ orbitals are significantly lower on the energy axis than Hydrogen's $1s$ orbital.
    - The $H(1s)$ matches closely with one of the $F(2p)$ orbitals to form $\sigma$ and $\sigma^*$.
    - The remaining orbitals from Fluorine (the $2s$ and the other two $2p$ orbitals) "scoot across" to the middle without changing energy. These are labeled as non-bonding.
• Bond Order Calculation: Non-bonding electrons are excluded from the bond order equation. In $HF$, there are $2$ bonding electrons and $0$ antibonding electrons, resulting in a bond order of $1$.

Polyatomic Molecules and Complexity

• Example: Ozone ($O_3$):
    - Ozone consists of three oxygen atoms. Visualizing this requires considering orbitals coming from the left, right, and potentially "behind the screen."
    - Conservation: In Ozone, $12$ atomic orbitals from the three atoms combine to create exactly $12$ molecular orbitals.
    - Energy Patterns: Higher energy MOs are characterized by an increasing number of nodes (color switches).
      - 0 nodes: All lobes are the same color.
      - 1 node: One color switch (shaded to unshaded).
      - 2 nodes: Two color switches (shaded to unshaded to shaded).

Questions & Discussion

• Question on Pole Analogy: A student asked if polarity is the same as being magnetic.
• Response: No, polarity is not the same as being magnetic. The instructor clarified that while a "magnetic pole" analogy is sometimes used, the two concepts are distinct.
• Question on Valence vs. Core Electrons: A student asked if core electrons (like $1s$ in Nitrogen) should be included.
• Response: Typically, we focus on valence electrons. If core electrons are included, they would form their own bonding ($\sigma_{1s}$) and antibonding ($\sigma_{1s}^*$) orbitals. Since both would be full, they would cancel each other out in the bond order calculation, yielding the same final answer.

Historical and Statistical Context

• Professor William Lester Jr.: A renowned scientist famous for using computers to determine molecular orbitals.
• Monte Carlo Method: A statistical and probabilistic method used by Professor Lester for molecular calculations. It is named after the casinos in Monte Carlo because it involves "games of chance" or statistical probability.
• Lester Lecturer: An annual honorary lecture series named after Professor Lester.