Molecular Orbital Theory (MOT) for Diatomic and Simple Triatomic Molecules

Molecular Orbital Theory (MOT) is a powerful model used to explain the bonding, magnetic properties, and spectral characteristics of molecules. Unlike Valence Bond Theory, which localizes electrons between two atoms, MOT considers molecular orbitals (MOs) that are delocalized over the entire molecule. This theory is particularly useful for understanding diatomic molecules and simple polyatomic molecules where simple Lewis structures or hybridization models might be insufficient.

Basic Principles of MOT

The fundamental idea behind MOT is that atomic orbitals (AOs) of individual atoms combine to form molecular orbitals (MOs) in the molecule. This combination occurs through the Linear Combination of Atomic Orbitals (LCAO) method. For a stable molecule to form, the AOs must have similar energies and appropriate symmetry.

When atomic orbitals combine, they form an equal number of molecular orbitals. These MOs can be of two types:

• Bonding Molecular Orbitals (BMOs): These are formed by the constructive interference of atomic orbitals. They have lower energy than the parent AOs and result in increased electron density between the nuclei, leading to a stable bond.
• Antibonding Molecular Orbitals (ABMOs): These are formed by the destructive interference of atomic orbitals. They have higher energy than the parent AOs and have a node (a region of zero electron density) between the nuclei, weakening the bond.

Formation of Molecular Orbitals from Atomic Orbitals

The combination of two atomic orbitals, $\psi_A$ and $\psi_B$, can be represented mathematically as:

Bonding MO: $\psi_{MO, bonding} = \psi_A + \psi_B$

Antibonding MO: $\psi_{MO, antibonding} = \psi_A - \psi_B$

The energy of the MOs follows the order: $E_{MO, antibonding} > E_{AO} > E_{MO, bonding}$.

Filling of Molecular Orbitals

Electrons are filled into the molecular orbitals following the same rules as for atomic orbitals:

• Aufbau Principle: Electrons are filled starting from the lowest energy MO.
• Pauli Exclusion Principle: Each MO can hold a maximum of two electrons with opposite spins.
• Hund's Rule: Degenerate MOs (orbitals of the same energy) are filled singly with electrons of parallel spin before pairing up.

Bond Order

Bond order is a key concept in MOT that indicates the strength and stability of a chemical bond. It is calculated as half the difference between the number of electrons in bonding and antibonding MOs.

Bond Order (BO) = $\frac{1}{2} \times (\text{Number of bonding electrons} - \text{Number of antibonding electrons})$

A higher bond order indicates a stronger and shorter bond. A bond order of zero suggests that no stable bond is formed.

• BO = 1: Single bond
• BO = 2: Double bond
• BO = 3: Triple bond

Magnetic Properties

The magnetic properties of a molecule are determined by the presence of unpaired electrons.

• Paramagnetic: Molecules with one or more unpaired electrons are attracted to a magnetic field.
• Diamagnetic: Molecules with all electrons paired are weakly repelled by a magnetic field.

MOT for Homonuclear Diatomic Molecules

Homonuclear diatomic molecules are composed of atoms of the same element (e.g., H₂, N₂, O₂). The formation of MOs in these molecules depends on the overlap of s and p atomic orbitals.

Formation of Sigma ($\sigma$) and Pi ($\pi$) Molecular Orbitals

Atomic orbitals overlap along or perpendicular to the internuclear axis to form sigma ($\sigma$) and pi ($\pi$) molecular orbitals, respectively.

• Sigma ($\sigma$) MOs: Formed by the head-on overlap of atomic orbitals (e.g., s-s, s-p, p-p along the internuclear axis). They are symmetrical around the internuclear axis.
• Pi ($\pi$) MOs: Formed by the sideways overlap of atomic orbitals (e.g., p-p perpendicular to the internuclear axis). They have a nodal plane containing the internuclear axis.

MO Energy Level Diagrams

The arrangement of MOs in order of increasing energy is crucial for constructing MO diagrams. For diatomic molecules of second-period elements, the order of MOs can vary slightly due to s-p mixing.

Case 1: Li₂ to N₂ (s-p Mixing occurs)

In molecules like Li₂, Be₂, B₂, C₂, and N₂, the energy difference between the 2s and 2p atomic orbitals is small enough that the $\sigma_{2p}$ MO formed from the head-on overlap of 2p orbitals interacts with and is pushed to higher energy than the $\pi_{2p}$ MOs formed from the sideways overlap of 2p orbitals. This phenomenon is called s-p mixing.

The energy order for these molecules is: $\sigma_{1s} < \sigma^*_{1s} < \sigma_{2s} < \sigma^*_{2s} < \pi_{2px} = \pi_{2py} < \sigma_{2p} < \pi^*_{2px} = \pi^*_{2py} < \sigma^*_{2p}$

Case 2: O₂, F₂, Ne₂ (No significant s-p Mixing)

For O₂, F₂, and Ne₂, the energy difference between 2s and 2p AOs is larger. The $\sigma_{2p}$ MO is lower in energy than the $\pi_{2p}$ MOs.

The energy order for these molecules is: $\sigma_{1s} < \sigma^*_{1s} < \sigma_{2s} < \sigma^*_{2s} < \sigma_{2p} < \pi_{2px} = \pi_{2py} < \pi^*_{2px} = \pi^*_{2py} < \sigma^*_{2p}$

Examples of Homonuclear Diatomic Molecules

1. Hydrogen Molecule (H₂)

Electronic configuration of H atom: $1s^1$. Total electrons = 2. Atomic orbitals involved: $1s$ from each H atom.

MO configuration: $(\sigma_{1s})^2$

Bond Order = $\frac{1}{2} \times (2 - 0) = 1$. This indicates a single bond between the two H atoms.

Magnetic property: Diamagnetic (all electrons are paired).

2. Helium Molecule (He₂)

Electronic configuration of He atom: $1s^2$. Total electrons = 4. Atomic orbitals involved: $1s$ from each He atom.

MO configuration: $(\sigma_{1s})^2 (\sigma^*_{1s})^2$

Bond Order = $\frac{1}{2} \times (2 - 2) = 0$. No stable He₂ molecule exists.

Magnetic property: Diamagnetic.

3. Lithium Molecule (Li₂)

Electronic configuration of Li atom: $1s^2 2s^1$. Total valence electrons = 2 (from 2s orbitals). Atomic orbitals involved: $2s$ from each Li atom. (Core $1s$ orbitals also form MOs, but they are filled and do not contribute to bonding).

Valence MO configuration: $(\sigma_{2s})^2$

Bond Order = $\frac{1}{2} \times (2 - 0) = 1$. Li₂ exists with a single bond.

Magnetic property: Diamagnetic.

4. Boron Molecule (B₂)

Electronic configuration of B atom: $1s^2 2s^2 2p^1$. Total valence electrons = 2 (from 2s) + 2 (from 2p) = 4. Atomic orbitals involved: $2s$ and $2p$ from each B atom. Order of MOs (s-p mixing): $\sigma_{2s} < \sigma^*_{2s} < \pi_{2p} < \sigma_{2p} < \dots$

Valence MO configuration: $(\sigma_{2s})^2 (\sigma^*_{2s})^2 (\pi_{2p})^2$

Bond Order = $\frac{1}{2} \times (4 - 2) = 1$.

Magnetic property: Paramagnetic. The two electrons in the degenerate $\pi_{2p}$ orbitals are filled singly according to Hund's rule, resulting in unpaired electrons. This explains why B₂ is paramagnetic, which simple VB theory might miss.

5. Nitrogen Molecule (N₂)

Electronic configuration of N atom: $1s^2 2s^2 2p^3$. Total valence electrons = 2 (from 2s) + 6 (from 2p) = 8. Atomic orbitals involved: $2s$ and $2p$ from each N atom. Order of MOs (s-p mixing): $\sigma_{2s} < \sigma^*_{2s} < \pi_{2p} < \sigma_{2p} < \dots$

Valence MO configuration: $(\sigma_{2s})^2 (\sigma^*_{2s})^2 (\pi_{2p})^4 (\sigma_{2p})^2$

Bond Order = $\frac{1}{2} \times (8 - 2) = 3$. This corresponds to the triple bond in N₂.

Magnetic property: Diamagnetic. All electrons are paired.

6. Oxygen Molecule (O₂)

Electronic configuration of O atom: $1s^2 2s^2 2p^4$. Total valence electrons = 2 (from 2s) + 8 (from 2p) = 6. Atomic orbitals involved: $2s$ and $2p$ from each O atom. Order of MOs (no s-p mixing): $\sigma_{2s} < \sigma^*_{2s} < \sigma_{2p} < \pi_{2p} < \pi^*_{2p} < \sigma^*_{2p}$

Valence MO configuration: $(\sigma_{2s})^2 (\sigma^*_{2s})^2 (\sigma_{2p})^2 (\pi_{2p})^4 (\pi^*_{2p})^2$

Bond Order = $\frac{1}{2} \times (6 - 2) = 2$. This corresponds to the double bond in O₂.

Magnetic property: Paramagnetic. The two electrons in the degenerate $\pi^*_{2p}$ orbitals are filled singly according to Hund's rule, resulting in two unpaired electrons. This is a key success of MOT, as O₂ is experimentally found to be paramagnetic.

7. Fluorine Molecule (F₂)

Electronic configuration of F atom: $1s^2 2s^2 2p^5$. Total valence electrons = 2 (from 2s) + 10 (from 2p) = 7. Atomic orbitals involved: $2s$ and $2p$ from each F atom. Order of MOs (no s-p mixing): $\sigma_{2s} < \sigma^*_{2s} < \sigma_{2p} < \pi_{2p} < \pi^*_{2p} < \sigma^*_{2p}$

Valence MO configuration: $(\sigma_{2s})^2 (\sigma^*_{2s})^2 (\sigma_{2p})^2 (\pi_{2p})^4 (\pi^*_{2p})^4$

Bond Order = $\frac{1}{2} \times (7 - 3) = 2$. This corresponds to the double bond in F₂. (Note: F₂ has a single bond in reality, and MOT predicts a bond order of 1 for F₂ if considering all valence electrons. Let's re-calculate with correct valence electron count for F₂. Each F has 7 valence electrons, so F₂ has 14 valence electrons. However, for diatomic molecules, we typically consider only the valence shell electrons involved in bonding. For F₂, it should be $2s^2 2p^5$. So, 2 electrons from 2s orbitals and 10 electrons from 2p orbitals. Total valence electrons from 2s and 2p are 2+5+2+5 = 14. Let's consider the valence shell MOs for F₂. Valence MO configuration for F₂ (14 valence electrons): $(\sigma_{2s})^2 (\sigma^*_{2s})^2 (\sigma_{2p})^2 (\pi_{2p})^4 (\pi^*_{2p})^4 (\sigma^*_{2p})^2$ Bond Order = $\frac{1}{2} \times (2+2+4 - (2+4+2)) = \frac{1}{2} \times (8 - 8) = 0$. This is incorrect. Let's re-examine the electron filling for F₂. Each F atom has 7 valence electrons. So F₂ has 14 valence electrons. The MO diagram for O₂, F₂, Ne₂ is: $\sigma_{2s} < \sigma^*_{2s} < \sigma_{2p} < \pi_{2p} < \pi^*_{2p} < \sigma^*_{2p}$. Filling for F₂ (14 valence electrons): $(\sigma_{2s})^2 (\sigma^*_{2s})^2 (\sigma_{2p})^2 (\pi_{2p})^4 (\pi^*_{2p})^4$. Number of bonding electrons = 2 (from $\sigma_{2s}$) + 2 (from $\sigma_{2p}$) + 4 (from $\pi_{2p}$) = 8. Number of antibonding electrons = 2 (from $\sigma^*_{2s}$) + 4 (from $\pi^*_{2p}$) = 6. Bond Order = $\frac{1}{2} \times (8 - 6) = \frac{1}{2} \times 2 = 1$. This matches the experimental observation of a single bond in F₂.

Magnetic property: Diamagnetic. All electrons are paired.

MOT for Heteronuclear Diatomic Molecules

Heteronuclear diatomic molecules are composed of atoms of different elements (e.g., CO, NO, HF). The atomic orbitals of the two different atoms have different energies. This asymmetry affects the energies and shapes of the resulting molecular orbitals.

When combining AOs from atoms with different electronegativities:

• The more electronegative atom contributes AOs of lower energy.
• The bonding MOs will have more character (contribution) from the AO of the more electronegative atom.
• The antibonding MOs will have more character from the AO of the less electronegative atom.
• The energy difference between the bonding and antibonding MOs is generally smaller than in homonuclear diatomics, leading to weaker bonds for a given bond order.

The general order of MOs is often similar to that of homonuclear diatomics, but the energy levels are not degenerate ($\pi_{2px} \neq \pi_{2py}$ if the atoms are different and oriented differently). However, for simplicity, we often assume similar energy levels or slight perturbations.

Examples of Heteronuclear Diatomic Molecules

1. Carbon Monoxide (CO)

C: $2s^2 2p^2$ (Electrons = 6) O: $2s^2 2p^4$ (Electrons = 8) Total valence electrons = 2 + 2 + 2 + 4 = 10. Electronegativity: O > C. So, O's AOs are lower in energy.

The MO diagram is similar to N₂ (due to 10 valence electrons and similar atomic orbital energies after considering electronegativity differences). s-p mixing is expected. Order: $\sigma_{2s} < \sigma^*_{2s} < \pi_{2p} < \sigma_{2p} < \pi^*_{2p} < \sigma^*_{2p}$

Valence MO configuration: $(\sigma_{2s})^2 (\sigma^*_{2s})^2 (\pi_{2p})^4 (\sigma_{2p})^2$

Bond Order = $\frac{1}{2} \times (8 - 2) = 3$. This explains the triple bond character in CO.

Magnetic property: Diamagnetic.

Note: The triple bond in CO is often described as having one $\sigma$ bond and two $\pi$ bonds. However, MOT suggests that the $\sigma_{2p}$ MO has more contribution from O, and the $\pi_{2p}$ MOs are more evenly distributed, contributing to the polar nature of the bond.

2. Nitric Oxide (NO)

N: $2s^2 2p^3$ (Electrons = 7) O: $2s^2 2p^4$ (Electrons = 8) Total valence electrons = 2 + 3 + 2 + 4 = 11. Electronegativity: O > N. So, O's AOs are lower in energy.

The MO diagram is similar to O₂ (11 valence electrons, O is more electronegative). Order: $\sigma_{2s} < \sigma^*_{2s} < \sigma_{2p} < \pi_{2p} < \pi^*_{2p} < \sigma^*_{2p}$

Valence MO configuration: $(\sigma_{2s})^2 (\sigma^*_{2s})^2 (\sigma_{2p})^2 (\pi_{2p})^4 (\pi^*_{2p})^1$

Bond Order = $\frac{1}{2} \times (8 - 3) = 2.5$. This fractional bond order is characteristic of molecules with an odd number of electrons.

Magnetic property: Paramagnetic. The single electron in the $\pi^*_{2p}$ orbital is unpaired.

3. Hydrogen Fluoride (HF)

H: $1s^1$ (Electrons = 1) F: $1s^2 2s^2 2p^5$ (Valence: $2s^2 2p^5$) (Electrons = 9) Total valence electrons = 1 (from 1s) + 7 (from 2s, 2p) = 8. Electronegativity: F > H. So, F's AOs ($2s, 2p$) are lower in energy than H's $1s$.

The $1s$ AO of H combines with the $2p_z$ AO of F (assuming z-axis is internuclear axis) to form $\sigma$ MOs. The $2s$ AO of F is too low in energy to participate significantly. The $2p_x$ and $2p_y$ AOs of F are perpendicular to the internuclear axis and do not overlap with H's $1s$ AO, forming non-bonding MOs.

The MOs formed are approximately: $\sigma_{1s-2pz}$ (bonding) $\sigma^*_{1s-2pz}$ (antibonding) $2p_x$ (non-bonding) $2p_y$ (non-bonding)

The $2s$ AO of F forms $\sigma_{2s}$ and $\sigma^*_{2s}$ MOs which are very low in energy and filled by F's $2s$ electrons, essentially acting as core orbitals.

Considering valence electrons: The $2s$ electrons of F form a stable $\sigma_{2s}$ MO (low energy). The $2p$ electrons of F and $1s$ electron of H form MOs. Order (simplified, focusing on interaction): $\sigma_{2s}$ (F core) < $\sigma_{1s-2pz}$ < $\pi_{2px}$ (non-bonding) = $\pi_{2py}$ (non-bonding) < $\sigma^*_{1s-2pz}$

Valence MO configuration: $(\sigma_{2s})^2 (\sigma_{1s-2pz})^2 (\pi_{2px})^2 (\pi_{2py})^2$

Bond Order = $\frac{1}{2} \times (\text{bonding } e^- - \text{antibonding } e^-)$. Bonding electrons: 2 from $\sigma_{1s-2pz}$. Antibonding electrons: 0. Total electrons involved in bonding MOs = 2. Number of non-bonding electrons = 4 (from $\pi_{2px}, \pi_{2py}$). Bond Order = $\frac{1}{2} \times (2 - 0) = 1$. This corresponds to the single bond in HF.

Magnetic property: Diamagnetic.

The $\pi_{2px}$ and $\pi_{2py}$ orbitals are essentially the non-bonding $2p$ orbitals of Fluorine, as they do not participate in bonding with Hydrogen's $1s$ orbital.

MOT for Simple Triatomic Molecules

Applying MOT to triatomic molecules like H₂O, BeCl₂, or B₃ (hypothetical) becomes significantly more complex. It requires considering the symmetry of the molecule and the AOs involved. The process involves constructing a symmetry-adapted linear combination of atomic orbitals (SALC) and then combining these SALCs with atomic orbitals of matching symmetry to form MOs.

For simple triatomic molecules, the symmetry considerations can be managed using Group Theory. However, for introductory purposes, we often rely on simplified MO diagrams derived from symmetry principles or by considering localized bonding first.

Example: Beryllium Chloride (BeCl₂)

BeCl₂ is a linear molecule. Be: $1s^2 2s^2$ (Valence: $2s^2$) Cl: $3s^2 3p^5$ (Valence: $3s^2 3p^5$)

Be contributes its $2s$ and $2p$ orbitals. Cl atoms contribute their $3s$ and $3p$ orbitals.

Due to the linear geometry (D_∞h symmetry), the AOs combine to form MOs with specific symmetry labels (e.g., $\sigma_g, \sigma_u, \pi_g, \pi_u$).

The central Be atom's $2s$ orbital is of $\sigma_g$ symmetry. Its $2p_z$ orbital (along the axis) is also $\sigma_g$. Its $2p_x, 2p_y$ orbitals are $\pi_u$. The Cl atoms' $3s$ orbitals are $\sigma_g$. Their $3p_z$ orbitals are $\sigma_g$. Their $3p_x, 3p_y$ orbitals are $\pi_u$.

Combinations: - $\sigma_g$ AOs combine to form $\sigma_g$ MOs. - $\pi_u$ AOs combine to form $\pi_u$ MOs.

This leads to MOs like: - Bonding and antibonding $\sigma_g$ MOs from Be($2s, 2p_z$) and Cl($3s, 3p_z$). - Bonding and antibonding $\pi_u$ MOs from Be($2p_x, 2p_y$) and Cl($3p_x, 3p_y$). - Non-bonding MOs derived primarily from Cl's orbitals.

The resulting MO diagram shows that BeCl₂ has a bond order close to 1 for each Be-Cl bond, consistent with a single covalent bond. The molecule is diamagnetic.

Example: Water (H₂O)

H₂O is a bent molecule (C_2v symmetry). O: $2s^2 2p^4$ (Valence: $2s^2 2p^4$) H: $1s^1$ (x2)

Oxygen's $2s$ orbital forms a low-energy $\sigma$ MO. Oxygen's $2p_z$ orbital (along the C2 axis) forms a $\sigma$ MO. Oxygen's $2p_x$ orbital (perpendicular to axis, in the molecular plane) forms a $\pi$ MO. Oxygen's $2p_y$ orbital (perpendicular to axis and plane) forms a $\pi$ MO.

The two Hydrogen $1s$ orbitals combine to form SALCs of $\sigma$ and $\sigma'$ symmetry.

Interactions: - O($2s$) interacts with H SALC($\sigma'$) to form bonding/antibonding MOs. - O($2p_z$) interacts with H SALC($\sigma$) to form bonding/antibonding MOs. - O($2p_x$) forms a $\pi$ MO (non-bonding character). - O($2p_y$) forms a $\pi$ MO (antibonding character with H SALC).

The MO diagram for water shows: - A low-lying bonding MO from O($2s$). - Two bonding MOs derived from O($2p_z$) and O($2p_x$) interacting with H SALCs. One is mainly $\sigma$ and the other has $\pi$ character. - A non-bonding MO primarily localized on oxygen (corresponding to the lone pair). - Antibonding MOs.

The highest occupied molecular orbital (HOMO) is often the non-bonding one, and the next HOMO is the $\pi$ bonding MO. The molecule is diamagnetic.

Limitations of MOT

While powerful, MOT has limitations:

• Constructing MO diagrams for polyatomic molecules becomes very complex and often requires computational methods.
• It can be difficult to visualize the exact shapes and locations of electrons in complex MOs.
• Predicting bond strengths and lengths accurately for larger molecules can be challenging without advanced calculations.

Despite these limitations, MOT provides invaluable insights into the electronic structure, stability, and properties of molecules that are not easily explained by simpler theories.