8.3 Multiple Bonds and 8.4 Molecular Orbital Theory

Multiple Covalent Bonding and Hybrid Orbitals

• Conceptual Overview: Multiple bonds consist of a combination of $\sigma$ (sigma) and $\pi$ (pi) bonds. The hybrid orbital model, which explains the geometry of single covalent bonds, also accounts for molecules with double and triple bonds.

• Double Bonds (Example: Ethene, $C_2H_4$):

  • Lewis Structure and Geometry: Each carbon atom in ethene is bonded to one other carbon and two hydrogen atoms. These three bonding regions result in a trigonal planar electron-pair geometry.

  • Hybridization: To achieve this geometry, the carbon atoms use a set of $sp^2$ hybrid orbitals. These result from the hybridization of two $2p$ orbitals and the $2s$ orbital.

  • Bond Formation:

    • The $sp^2$ hybrid orbitals form the $C-H$ single bonds and the $\sigma$ bond component of the $C=C$ double bond.

    • The $\pi$ bond results from the side-by-side overlap of the third, unhybridized $2p$ orbital on each carbon atom.

  • Orientation: The unhybridized $p$ orbitals are perpendicular to the plane of the $sp^2$ hybrid orbitals. The overlap occurs in two lobes, located above and below the internuclear axis.

Rotation in Sigma and Pi Bonds

• Sigma ($\sigma$) Bond Rotation: Rotation around single bonds occurs easily. This is because the end-to-end orbital overlap is symmetric about the internuclear axis, meaning the extent of overlap does not change with the relative orientation of the atoms.

• Multiple ($\pi$) Bond Rotation: Rotation around the internuclear axis is significantly more difficult for multiple bonds. Rotating the atoms would drastically decrease the off-axis side-by-side overlap of the $\pi$ bonding orbitals, effectively breaking the $\pi$ bond.

• Molecular Stability: In ethene, all four hydrogen atoms and both carbon atoms lie in the same plane. This planar configuration is the most stable arrangement because it allows for the most efficient overlap of the $p$ orbitals to create the $\pi$ bond.

Triple Bonds and sp Hybridization in Acetylene

• Acetylene ($C_2H_2$): This is a linear molecule ($H-C \equiv C-H$).

• Hybridization: Each carbon atom is $sp$ hybridized, leaving two unhybridized $2p$ orbitals per carbon.

• Bonding Components:

  • Sigma Bonds: The $sp$ hybrid orbitals overlap end-to-end to form one $C-C$ $\sigma$ bond. The remaining $sp$ orbitals form $\sigma$ bonds with hydrogen atoms.

  • Pi Bonds: The two sets of unhybridized $p$ orbitals are positioned to overlap side-by-side. This results in the formation of two $\pi$ bonds.

  • Total Bond: The carbon atoms are held together by one $\sigma$ bond and two $\pi$ bonds, totaling a triple bond.

Resonance and Hybridization

• Hybridization Assignment: Hybridization involves only $\sigma$ bonds, lone pairs, and radicals (single unpaired electrons). Because resonance forms involve different arrangements of $\pi$ bonds (which only use unhybridized orbitals), resonance does not influence the assignment of hybridization.

• Benzene ($C_6H_6$):

  • Each carbon atom is bonded to three other atoms with no lone pairs, meaning every carbon in benzene is $sp^2$ hybridized regardless of the resonance structure considered.

  • Delocalization: The electrons in the $\pi$ bonds are not fixed in one position but are delocalized throughout the ring. While Valence Bond Theory requires multiple structures to describe this, Molecular Orbital Theory handles delocalization more naturally.

Molecular Orbital Theory and the Oxygen Discrepancy

• Limitations of Lewis Theory: The Lewis structure for oxygen ($:O=O:$) predicts that all electrons are paired. However, experimental evidence shows that liquid oxygen is attracted to magnetic fields (paramagnetism), implying it has unpaired electrons.

• Magnetic Susceptibility:

  • Paramagnetism: Occurs in molecules with unpaired electrons. These substances are attracted to a magnetic field.

  • Diamagnetism: Occurs in molecules where all electrons are paired. These substances weakly repel magnetic fields, and their apparent weight decreases slightly in an inhomogeneous magnetic field.

  • Measurement: A Gouy balance is used to determine the number of unpaired electrons by comparing the weight of a sample with the electromagnet turned on versus off. Experiments confirm that each $O_2$ molecule has two unpaired electrons.

• Molecular Orbital (MO) Theory defined: Unlike Valence Bond Theory, which assigns hybrid orbitals to specific atoms (localization), MO theory describes molecular orbitals that are delocalized over the entire molecule. It explains the energies and probable locations of electrons and accounts for violations of the octet rule.

Comparison of Bonding Theories: Valence Bond vs. Molecular Orbital

Linear Combination of Atomic Orbitals (LCAO)

• Mathematical Approach: MO theory uses quantum mechanics to describe electron behavior via wave functions ($\psi$). The process of combining atomic orbitals to generate molecular orbitals is called the Linear Combination of Atomic Orbitals (LCAO).

• Interference of Waves:

  • Constructive Interference: Occurs when in-phase waves combine (peaks line up with peaks), producing regions with higher electron density probability. This results in a bonding orbital.

  • Destructive Interference: Occurs when out-of-phase waves combine (peaks line up with troughs), producing nodes (regions of zero electron density). This results in an antibonding orbital.

Types of Molecular Orbitals

• Sigma ($\sigma$) and Sigma-star ($\sigma^*$) from s orbitals:

  • $\sigma_s$ (Bonding): Lower energy; formed by in-phase combination. Electron density is concentrated directly between the nuclei, creating an attractive force that holds them together.

  • $\sigma_s^*$ (Antibonding): Higher energy; formed by out-of-phase addition (subtraction). Contains a node between the nuclei. Electrons here pull the nuclei apart.

• Orbitals from p orbitals:

  • End-to-end overlap: Forms $\sigma_{px}$ (bonding) and $\sigma_{px}^*$ (antibonding). The asterisk ($*$) denotes the higher-energy orbital with a node between the nuclei.

  • Side-by-side overlap: Forms $\pi$ (bonding) and $\pi^*$ (antibonding) orbitals. A $\pi$ bond exists when electrons occupy the $\pi$ orbital. Out-of-phase combinations create two nodal planes.

Molecular Orbital Energy Diagrams and Bond Order

• MO Diagrams: These diagrams show the relative energy levels of atomic orbitals (on the sides) and molecular orbitals (in the center). Combining six $2p$ atomic orbitals results in three bonding orbitals (one $\sigma$, two $\pi$) and three antibonding orbitals (one $\sigma^*$, two $\pi^*$).

• Filling Rules: Follows the Aufbau principle (lowest energy first), Hund's rule (spread electrons across degenerate orbitals before pairing), and Pauli exclusion principle (maximum two electrons per orbital with opposite spins).

• Bond Order Calculation: Bond order is a guide to bond strength; higher bond order translates to stronger bonds. The formula is:     $\text{bond order} = \frac{(\text{number of bonding electrons}) - (\text{number of antibonding electrons})}{2}$

• Examples:

  • Dihydrogen ($H_2$): Two electrons in $\sigma_{1s}$. Electron configuration: $(\sigma_{1s})^2$. Bond order = $\frac{(2-0)}{2} = 1$ (Single bond).

  • Dihelium ($He_2$): Hypothetical configuration $(\sigma_{1s})^2(\sigma_{1s}^*)^2$. Bond order = $\frac{(2-2)}{2} = 0$. A bond order of zero means no stable bond forms.

Homonuclear Diatomic Molecules of the Second Period

• Molecular Series: Includes $Li_2, Be_2, B_2, C_2, N_2, O_2, F_2, Ne_2$. $Be_2$ and $Ne_2$ are unstable due to bond orders of zero.

• Orbital Ordering and s-p Mixing:

  • Typically, $\sigma$ bonds are more stable than $\pi$ bonds, meaning $\sigma_{2p}$ should be lower in energy than $\pi_{2p}$.

  • s-p Mixing: For atoms with three or fewer $p$ electrons ($Li$ through $N$), the $\sigma_{2s}$ and $\sigma_{2p}$ wavefunctions interact. This makes $\sigma_{2s}$ more stable and pushes $\sigma_{2p}$ higher in energy, often above the $\pi_{2p}$ set.

  • Across the Period: Orbital energies decrease across the period as effective nuclear charge increases and atomic radius decreases. The orbital order switch happens between $N_2$ and $O_2$.

General Principles of Hybridization

1. Existence: Hybrid orbitals do not exist in isolated atoms; they form only in covalently bonded atoms.

2. Geometry: They have shapes and orientations different from atomic orbitals.

3. Conservation of Orbitals: The number of hybrid orbitals in a set equals the number of atomic orbitals combined.

4. Equivalence: All orbitals in a hybrid set are equivalent in shape and energy.

5. VSEPR Connection: The hybridization type is determined by the electron-pair geometry predicted by VSEPR theory.

6. Bond Function: Hybrid orbitals form $\sigma$ bonds; unhybridized orbitals form $\pi$ bonds.

• sp Hybridization Example (Beryllium Chloride, $BeCl_2$):

  • In gaseous $BeCl_2$, Be is the central atom with two regions of valence electron density (linear arrangement).

  • Mixing one valence $s$ orbital and one valence $p$ orbital yields two equivalent $sp$ hybrid orbitals oriented $180^{\circ}$ apart.

  • This leaves two unhybridized $2p$ orbitals.

Questions & Discussion

• Example 8.4: Hybridization in Sulfur Dioxide ($SO_2$): Sulfur dioxide is a major component of volcanic gases and a contributor to acid rain. Determining its hybridization requires looking at its resonance structures to identify the regions of electron density around the Sulfur (S) atom.

• Portrait of a Chemist (Walter Kohn): Walter Kohn was a theoretical physicist who, along with mathematician John Pople, won the 1998 Nobel Prize in Chemistry. Kohn developed density functional theory, which allowed for the computation of molecular orbital shapes and energies. He was also a survivor of the Kindertransport program during WWII.

• Practical Application (Computational Chemistry): MO theory and computational methods are used in drug design, such as targeting the HIV-1 protease to inhibit the progress of the disease.