4.4 Valence Bond Theory, Hybridization & Sigma/Pi Bonding

Valence Bond Theory, Hybridization & Sigma/Pi Bonding

Quick Summary: Valence Bond (VB) theory explains covalent bonding through localized orbital overlap sharing paired electron spins. To account for observed molecular shapes that pure $s$ and $p$ orbitals cannot explain, atomic wavefunctions mix mathematically into directional hybrid orbitals ($sp, sp^2, sp^3, sp^3d, sp^3d^2$). Direct head-on overlap along the internuclear axis creates cylindrical sigma ($\sigma$) bonds with free rotation, whereas lateral overlap of unhybridized $p$ orbitals creates pi ($\pi$) bonds whose rotational rigidity gives rise to geometric (cis/trans) isomerism.

---

1. Orbital Overlap & Foundations of Valence Bond Theory

Valence Bond (VB) theory models covalent bonding as the spatial overlap of two atomic orbitals, each containing an unpaired electron:

1. Orbital Overlap: Covalent bonds form when an orbital on one atom shares space with an orbital on an adjacent atom.
2. Spin Pairing: Under the Pauli exclusion principle, the two electrons in the overlap region must have opposite (antiparallel) spins ($\uparrow\downarrow$).
3. Maximum Overlap Condition: Bond strength correlates directly with spatial overlap; greater overlap concentrates negative charge between nuclei, lowering potential energy.

The Limitation of Pure Atomic Orbitals

Ground-state carbon has configuration $1s^2 2s^2 2p_x^1 2p_y^1$. With only two unpaired $p$ electrons at $90^\circ$, pure orbital theory predicts carbon would form only two bonds at $90^\circ$. Promoting a $2s$ electron yields $2s^1 2p_x^1 2p_y^1 2p_z^1$, predicting three bonds from $p$ orbitals at $90^\circ$ and one non-directional bond from $s$ overlap. This contradicts experimental reality: methane ($\text{CH}_4$) possesses four identical $\text{C}-\text{H}$ bonds of equal length ($109\text{ pm}$) and strength ($435\text{ kJ/mol}$) oriented at regular tetrahedral angles of $109.5^\circ$.

---

2. Hybridization of Atomic Orbitals

Linus Pauling proposed hybridization: the mathematical mixing of atomic wavefunctions on a central atom to generate directional hybrid orbitals matching VSEPR geometries:

• $sp$ (Steric Number 2): One $s$ + one $p$ orbital yield two equivalent $sp$ hybrid orbitals directed $180^\circ$ apart (linear). Two unhybridized $p$ orbitals remain perpendicular. Examples: $\text{BeCl}_2, \text{CO}_2, \text{C}_2\text{H}_2$ (ethyne).
• $sp^2$ (Steric Number 3): One $s$ + two $p$ orbitals yield three $sp^2$ hybrids directed $120^\circ$ apart in a plane (trigonal planar). One unhybridized $p$ orbital remains perpendicular to the plane. Examples: $\text{BF}_3, \text{C}_2\text{H}_4$ (ethene), $\text{CO}_3^{2-}$.
• $sp^3$ (Steric Number 4): One $s$ + three $p$ orbitals yield four $sp^3$ hybrids directed $109.5^\circ$ apart toward tetrahedral vertices. Zero unhybridized $p$ orbitals remain. Examples: $\text{CH}_4, \text{NH}_3, \text{H}_2\text{O}, \text{C}_2\text{H}_6$.
• $sp^3d$ (Steric Number 5): One $s$ + three $p$ + one $d$ orbital yield five $sp^3d$ hybrids in a trigonal bipyramidal geometry ($90^\circ, 120^\circ, 180^\circ$). Examples: $\text{PCl}_5, \text{SF}_4$.
• $sp^3d^2$ (Steric Number 6): One $s$ + three $p$ + two $d$ orbitals yield six equivalent $sp^3d^2$ hybrids in an octahedral geometry ($90^\circ, 180^\circ$). Examples: $\text{SF}_6, \text{XeF}_4$.

---

3. Sigma ($\sigma$) vs. Pi ($\pi$) Bonding & Multiple Bonds

Covalent bonds are classified by their spatial overlap geometry:

Sigma ($\sigma$) Bonds

• Overlap: Head-on (end-to-end) overlap directly along the internuclear axis ($s-s, s-p, p-p$, or hybrid-hybrid).
• Symmetry & Strength: Exhibiting cylindrical symmetry around the bond axis, $\sigma$ bonds concentrate electron density between nuclei and are stronger than $\pi$ bonds.
• Free Rotation: Rotating atoms around a single $\sigma$ bond maintains constant orbital overlap, permitting free rotation at room temperature.

Pi ($\pi$) Bonds

• Overlap: Lateral (sideways) overlap of parallel unhybridized $p$ orbitals above and below the internuclear axis.
• Nodal Plane: The internuclear axis lies in a nodal plane of zero electron density. Due to less effective lateral overlap, $\pi$ bonds are weaker than $\sigma$ bonds and only form after a primary $\sigma$ bond is established.

Multiple Bond Composition

• Single Bond: $1\,\sigma$ bond.
• Double Bond: $1\,\sigma + 1\,\pi$ bond (e.g., ethene, $\text{H}_2\text{C}=\text{CH}_2$).
• Triple Bond: $1\,\sigma + 2\,\pi$ bonds oriented mutually perpendicular (e.g., ethyne, $\text{HC}\equiv\text{CH}$, and $\text{N}_2$).

---

4. Rotational Rigidity & Geometric Isomerism

Unlike $\sigma$ bonds, double bonds cannot rotate at ambient temperature. Rotating one $sp^2$ carbon twists its unhybridized $2p$ orbital out of parallel coplanarity, destroying the $\pi$ bond. Overcoming this barrier requires $\approx 260\text{ kJ/mol}$, locking substituents into fixed positions and creating geometric (cis/trans) isomerism:

• In cis-1,2-dichloroethene, chlorine atoms reside on the same side of the double bond (net polar).
• In trans-1,2-dichloroethene, chlorines reside on opposite sides across the inversion center (nonpolar).

---

5. Delocalized $\pi$ Systems & Conjugation

When adjacent atoms possess parallel unhybridized $p$ orbitals, $\pi$ electrons delocalize across multiple nuclei:

• Benzene ($\text{C}_6\text{H}_6$): Six $sp^2$ carbons form a planar hexagonal ring. Six perpendicular $2p_z$ orbitals overlap continuously, forming toroidal $\pi$ clouds above and below the ring. This equalizes all $\text{C}-\text{C}$ bonds to $139.7\text{ pm}$ (between single $154\text{ pm}$ and double $134\text{ pm}$) and provides $\approx 150\text{ kJ/mol}$ of aromatic stabilization.
• Carbonate & Nitrate: Continuous $\pi$ overlap across four atoms yields fractional bond orders of $4/3$.

---

6. Hybrid Orbital $s$-Character & Structural Trends

The fraction of $s$-character ($sp = 50\%, sp^2 = 33.3\%, sp^3 = 25\%$) determines orbital proximity to the positive nucleus:

• Bond Length & Strength: Greater $s$-character holds electrons tighter, producing shorter, stronger bonds: $\text{C}-\text{C}$ in ethane ($154\text{ pm}, 347\text{ kJ/mol}$) vs. $\text{C}=\text{C}$ in ethene ($134\text{ pm}, 614\text{ kJ/mol}$) vs. $\text{C}\equiv\text{C}$ in ethyne ($120\text{ pm}, 839\text{ kJ/mol}$).
• Acidity of Hydrocarbons: The $sp$ carbon in ethyne holds the conjugate base lone pair close to the nucleus, stabilizing the acetylide anion ($\text{HC}\equiv\text{C}:^-$) and giving ethyne mild acidity ($pK_a \approx 25$), whereas ethane ($sp^3, pK_a \approx 50$) is non-acidic.

---

7. Hybridization Master Summary Table