Part 2, Topic 2 | Lectures 12–14

Molecular Orbital Theory & Intermolecular Forces

Quantum mechanical bonding, hybridization, and the forces between molecules

4.1 LCAO-MO Theory

Molecular Orbital (MO) theory provides a quantum mechanical description of bonding that goes beyond the localized bond model of Lewis structures. In the Linear Combination of Atomic Orbitals (LCAO) approach, molecular orbitals are formed by combining atomic orbitals:

$$\psi_{MO} = c_1\phi_A \pm c_2\phi_B$$

The + combination gives a bonding orbital (lower energy); the − gives an antibonding orbital (higher energy)

Bonding vs. Antibonding Orbitals

Bonding Orbital ($\sigma$, $\pi$)

Constructive interference: electron density accumulates between nuclei

• Lower energy than parent atomic orbitals
• Stabilizes the molecule
• Node-free between nuclei (for $\sigma$)

Antibonding Orbital ($\sigma^$, $\pi^$)

Destructive interference: node between nuclei depletes electron density

• Higher energy than parent atomic orbitals
• Destabilizes the molecule
• Nodal plane between nuclei

Bond Order

The bond order in MO theory is calculated from the number of electrons in bonding ($n_b$) and antibonding ($n_a$) orbitals:

$$BO = \frac{n_b - n_a}{2}$$

BO = 0 means no bond (molecule doesn't exist); higher BO = stronger, shorter bond

4.2 MO Diagrams for Homonuclear Diatomics

For homonuclear diatomic molecules of the second period, the MO ordering depends on whether the 2s and 2p atomic orbitals interact (s-p mixing). This leads to two different orderings:

With s-p mixing (Li$_2$ to N$_2$)

$\sigma_{2s}$ < $\sigma^*_{2s}$ < $\pi_{2p}$ <$\sigma_{2p}$ < $\pi^*_{2p}$ < $\sigma^*_{2p}$

The $\pi_{2p}$ orbitals are lower than $\sigma_{2p}$

Without s-p mixing (O$_2$ to Ne$_2$)

$\sigma_{2s}$ < $\sigma^*_{2s}$ < $\sigma_{2p}$ <$\pi_{2p}$ < $\pi^*_{2p}$ < $\sigma^*_{2p}$

The $\sigma_{2p}$ orbital is lower than $\pi_{2p}$

Key Examples

Molecule	Valence e$^-$	Bond Order	Magnetic	Bond Energy (kJ/mol)
N$_2$	10	3 (triple bond)	Diamagnetic	945
O$_2$	12	2 (double bond)	Paramagnetic	498
F$_2$	14	1 (single bond)	Diamagnetic	159

A triumph of MO theory: it correctly predicts that O$_2$ is paramagnetic (has unpaired electrons), which Lewis structures cannot explain. The two unpaired electrons in the$\pi^*_{2p}$ orbitals make liquid oxygen attracted to a magnet.

4.3 Hybridization

Hybridization is a model that mixes atomic orbitals on the same atom to form equivalent hybrid orbitals that better describe bonding geometry. The type of hybridization is determined by the number of electron groups (bonding + lone pairs) around the central atom:

sp Hybridization (2 groups)

One s + one p orbital $\rightarrow$ two sp hybrids at 180$^\circ$ (linear)

Examples: BeCl$_2$, CO$_2$, C$_2$H$_2$ (acetylene)

sp$^2$ Hybridization (3 groups)

One s + two p orbitals $\rightarrow$ three sp$^2$ hybrids at 120$^\circ$ (trigonal planar)

Examples: BF$_3$, C$_2$H$_4$ (ethylene), graphite

sp$^3$ Hybridization (4 groups)

One s + three p orbitals $\rightarrow$ four sp$^3$ hybrids at 109.5$^\circ$ (tetrahedral)

Examples: CH$_4$, NH$_3$, H$_2$O, diamond

Hybridization explains why carbon forms four equivalent bonds in methane at 109.5$^\circ$, despite having 2s and 2p orbitals at different energies. The sp$^3$ hybrid orbitals are all equivalent and point toward the vertices of a tetrahedron.

4.4 Intermolecular Forces

While intramolecular forces (chemical bonds) hold atoms together within molecules, intermolecular forces (IMFs) operate between molecules and determine bulk properties such as boiling point, viscosity, and surface tension. IMFs are much weaker than covalent bonds but crucial for understanding material behavior.

London Dispersion Forces (all molecules)

Instantaneous dipole-induced dipole interactions arising from electron cloud fluctuations. Present in all molecules; the only IMF in nonpolar species.

Strength increases with molecular size (more electrons = more polarizable). Typical energy: 0.05–40 kJ/mol.

Dipole-Dipole Forces (polar molecules)

Attractive interactions between the positive end of one polar molecule and the negative end of another. Strength depends on the magnitude of the permanent dipole moment.

Typical energy: 5–25 kJ/mol. Examples: HCl, acetone, SO$_2$.

Hydrogen Bonding (N–H, O–H, F–H)

An exceptionally strong type of dipole-dipole interaction where H bonded to N, O, or F interacts with a lone pair on another N, O, or F atom. Crucial for water's properties and biological structures (DNA, proteins).

Typical energy: 10–40 kJ/mol. Water forms up to 4 hydrogen bonds per molecule.

Ion-Dipole Forces (ions + polar molecules)

Strongest IMF type. Electrostatic attraction between an ion and the partial charge of a polar molecule. Responsible for dissolving ionic compounds in polar solvents.

Typical energy: 50–200 kJ/mol. Key to solvation of Na$^+$ and Cl$^-$ in water.

4.5 Phase Transitions & Clausius-Clapeyron Equation

Phase transitions occur when a material changes between solid, liquid, and gas states. The Clausius-Clapeyron equation relates vapor pressure to temperature for a liquid-gas phase transition:

$$\ln\frac{P_2}{P_1} = -\frac{\Delta H_{vap}}{R}\left(\frac{1}{T_2} - \frac{1}{T_1}\right)$$

$\Delta H_{vap}$ = enthalpy of vaporization, $R = 8.314$ J/(mol$\cdot$K)

Key implications:

●A plot of ln(P) vs 1/T gives a straight line with slope $= -\Delta H_{vap}/R$
●Higher $\Delta H_{vap}$ (stronger IMFs) means lower vapor pressure at a given temperature
●The boiling point is the temperature where vapor pressure equals atmospheric pressure

The equation also derives from thermodynamics: along a coexistence curve on a phase diagram, the Gibbs free energies of the two phases are equal ($G_l = G_g$), and the Clausius-Clapeyron equation describes how the equilibrium pressure changes with temperature.

Interactive Simulation: MO Diagrams

This simulation constructs molecular orbital energy level diagrams for N$_2$, O$_2$, and F$_2$, showing how electrons fill bonding and antibonding orbitals and computing bond orders.

MO Energy Level Diagrams & Bond Order Calculator

Python

Constructs molecular orbital diagrams for N2, O2, and F2 with electron filling, bond order calculation, and magnetic property prediction.

script.py139 lines

import numpy as np
import matplotlib.pyplot as plt
# MO energy level diagram for homonuclear diatomics
fig, axes = plt.subplots(1, 3, figsize=(16, 9))
fig.patch.set_facecolor('#0f172a')

molecules = [
    {
        'name': 'N\u2082',
        'Z': 7,
        'electrons': 10,  # total valence electrons in MOs
        'levels': [
            ('\u03c3(2s)', -2.0, 2, '#f472b6'),
            ('\u03c3*(2s)', -1.2, 2, '#fb923c'),
            ('\u03c0(2p)', 0.0, 4, '#4ade80'),
            ('\u03c3(2p)', 0.5, 2, '#38bdf8'),
            ('\u03c0*(2p)', 1.5, 0, '#facc15'),
            ('\u03c3*(2p)', 2.5, 0, '#a78bfa'),
        ],
        'bond_order': 3,
        'magnetic': 'Diamagnetic',
    },
    {
        'name': 'O\u2082',
        'Z': 8,
        'electrons': 12,
        'levels': [
            ('\u03c3(2s)', -2.0, 2, '#f472b6'),
            ('\u03c3*(2s)', -1.2, 2, '#fb923c'),
            ('\u03c3(2p)', -0.2, 2, '#38bdf8'),
            ('\u03c0(2p)', 0.5, 4, '#4ade80'),
            ('\u03c0*(2p)', 1.5, 2, '#facc15'),
            ('\u03c3*(2p)', 2.5, 0, '#a78bfa'),
        ],
        'bond_order': 2,
        'magnetic': 'Paramagnetic',
    },
    {
        'name': 'F\u2082',
        'Z': 9,
        'electrons': 14,
        'levels': [
            ('\u03c3(2s)', -2.0, 2, '#f472b6'),
            ('\u03c3*(2s)', -1.2, 2, '#fb923c'),
            ('\u03c3(2p)', -0.2, 2, '#38bdf8'),
            ('\u03c0(2p)', 0.5, 4, '#4ade80'),
            ('\u03c0*(2p)', 1.5, 4, '#facc15'),
            ('\u03c3*(2p)', 2.5, 0, '#a78bfa'),
        ],
        'bond_order': 1,
        'magnetic': 'Diamagnetic',
    },
]

for idx, mol in enumerate(molecules):
    ax = axes[idx]
    ax.set_facecolor('#1e293b')
    ax.tick_params(colors='#cbd5e1', left=True, bottom=False, labelbottom=False)
    ax.yaxis.label.set_color('#cbd5e1')
    ax.set_title(f"{mol['name']}  (BO = {mol['bond_order']})", fontsize=14,
                 fontweight='bold', color='#cbd5e1')
    for sp in ax.spines.values():
        sp.set_color('#334155')

for label, energy, n_electrons, color in mol['levels']:
        # Draw the level line
        ax.hlines(energy, 0.3, 0.7, colors=color, linewidth=3)
        ax.text(0.75, energy, label, color=color, fontsize=9, va='center')

# Draw electrons as arrows
        is_pi = '\u03c0' in label and '*' not in label
        if is_pi:
            # Pi orbitals are doubly degenerate
            positions = [0.4, 0.6]
            e_per_orbital = n_electrons // 2 if n_electrons > 0 else 0
            remainder = n_electrons % 2
            for p_idx, pos in enumerate(positions):
                n_here = min(e_per_orbital + (1 if p_idx < remainder else 0), 2)
                if n_here >= 1:
                    ax.annotate('', xy=(pos - 0.02, energy + 0.15),
                               xytext=(pos - 0.02, energy - 0.05),
                               arrowprops=dict(arrowstyle='->', color='white', lw=1.5))
                if n_here >= 2:
                    ax.annotate('', xy=(pos + 0.02, energy - 0.05),
                               xytext=(pos + 0.02, energy + 0.15),
                               arrowprops=dict(arrowstyle='->', color='white', lw=1.5))
        elif '\u03c0*' in label:
            # Antibonding pi - also doubly degenerate
            positions = [0.4, 0.6]
            e_per_orbital = n_electrons // 2 if n_electrons > 0 else 0
            remainder = n_electrons % 2
            for p_idx, pos in enumerate(positions):
                n_here = min(e_per_orbital + (1 if p_idx < remainder else 0), 2)
                if n_here >= 1:
                    ax.annotate('', xy=(pos - 0.02, energy + 0.15),
                               xytext=(pos - 0.02, energy - 0.05),
                               arrowprops=dict(arrowstyle='->', color='white', lw=1.5))
                if n_here >= 2:
                    ax.annotate('', xy=(pos + 0.02, energy - 0.05),
                               xytext=(pos + 0.02, energy + 0.15),
                               arrowprops=dict(arrowstyle='->', color='white', lw=1.5))
        else:
            # Sigma orbitals
            pos = 0.5
            if n_electrons >= 1:
                ax.annotate('', xy=(pos - 0.02, energy + 0.15),
                           xytext=(pos - 0.02, energy - 0.05),
                           arrowprops=dict(arrowstyle='->', color='white', lw=1.5))
            if n_electrons >= 2:
                ax.annotate('', xy=(pos + 0.02, energy - 0.05),
                           xytext=(pos + 0.02, energy + 0.15),
                           arrowprops=dict(arrowstyle='->', color='white', lw=1.5))

# Add bond order and magnetic info
    ax.text(0.5, -2.7, f"Bond Order: {mol['bond_order']}", ha='center',
            color='#4ade80', fontsize=11, fontweight='bold')
    ax.text(0.5, -3.1, mol['magnetic'], ha='center',
            color='#facc15' if mol['magnetic'] == 'Paramagnetic' else '#94a3b8', fontsize=10)

ax.set_xlim(0, 1)
    ax.set_ylim(-3.5, 3.2)
    ax.set_ylabel('Energy (relative)', fontsize=10)

plt.suptitle('Molecular Orbital Diagrams for Homonuclear Diatomics',
             fontsize=16, fontweight='bold', color='#cbd5e1', y=0.98)
plt.tight_layout(rect=[0, 0, 1, 0.95])
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0f172a')
plt.close()

print("MO Diagrams for N2, O2, F2")
print("=" * 50)
print(f"{'Molecule':<10} {'Bond Order':<12} {'Magnetic':<15} {'Bond E (kJ/mol)'}")
print("-" * 50)
print(f"{'N2':<10} {'3':<12} {'Diamagnetic':<15} {'945'}")
print(f"{'O2':<10} {'2':<12} {'Paramagnetic':<15} {'498'}")
print(f"{'F2':<10} {'1':<12} {'Diamagnetic':<15} {'159'}")
print(f"\nBond Order = (bonding e- - antibonding e-) / 2")
print(f"O2 is paramagnetic because pi*(2p) has 2 unpaired electrons")
print(f"Note: N2 has sigma-pi ordering swap (pi below sigma for 2p)")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Fortran: Clausius-Clapeyron Vapor Pressure

This Fortran program computes vapor pressures as a function of temperature for water, ethanol, and diethyl ether using the Clausius-Clapeyron equation.

Clausius-Clapeyron Vapor Pressure Calculator

Fortran

Computes vapor pressure vs temperature for water, ethanol, and diethyl ether using the Clausius-Clapeyron equation.

clausius_clapeyron.f9080 lines

program clausius_clapeyron
  implicit none
  double precision :: R, T1, T2, P1, P2, dH_vap, ln_ratio
  double precision :: T_range(20), P_range(20)
  integer :: i, n_points
  character(len=10) :: substance

R = 8.314d0  ! J/(mol*K)

write(*,'(A)') '======================================================'
  write(*,'(A)') '  Clausius-Clapeyron Vapor Pressure Calculator'
  write(*,'(A)') '======================================================'
  write(*,*)

! Water
  write(*,'(A)') '--- Water (H2O) ---'
  write(*,'(A)') '  dH_vap = 40.7 kJ/mol, P(100 C) = 101.325 kPa'
  dH_vap = 40700.0d0  ! J/mol
  T1 = 373.15d0  ! K (100 C)
  P1 = 101.325d0  ! kPa
  write(*,*)
  write(*,'(A10, A15, A15)') 'T (C)', 'T (K)', 'P (kPa)'
  write(*,'(A)') '----------------------------------------'
  n_points = 11
  do i = 1, n_points
    T2 = 273.15d0 + dble(i-1) * 10.0d0  ! 0 to 100 C
    ln_ratio = -dH_vap / R * (1.0d0/T2 - 1.0d0/T1)
    P2 = P1 * exp(ln_ratio)
    write(*,'(F10.1, F15.2, F15.3)') T2 - 273.15d0, T2, P2
  end do

write(*,*)

! Ethanol
  write(*,'(A)') '--- Ethanol (C2H5OH) ---'
  write(*,'(A)') '  dH_vap = 38.6 kJ/mol, P(78.4 C) = 101.325 kPa'
  dH_vap = 38600.0d0
  T1 = 351.55d0  ! K (78.4 C)
  P1 = 101.325d0
  write(*,*)
  write(*,'(A10, A15, A15)') 'T (C)', 'T (K)', 'P (kPa)'
  write(*,'(A)') '----------------------------------------'
  do i = 1, 9
    T2 = 273.15d0 + dble(i-1) * 10.0d0
    ln_ratio = -dH_vap / R * (1.0d0/T2 - 1.0d0/T1)
    P2 = P1 * exp(ln_ratio)
    write(*,'(F10.1, F15.2, F15.3)') T2 - 273.15d0, T2, P2
  end do

write(*,*)

! Comparison at 25 C
  write(*,'(A)') '--- Vapor Pressures at 25 C (298.15 K) ---'
  T2 = 298.15d0

! Water
  dH_vap = 40700.0d0; T1 = 373.15d0; P1 = 101.325d0
  ln_ratio = -dH_vap / R * (1.0d0/T2 - 1.0d0/T1)
  P2 = P1 * exp(ln_ratio)
  write(*,'(A20, F10.3, A5)') 'Water:', P2, ' kPa'

! Ethanol
  dH_vap = 38600.0d0; T1 = 351.55d0; P1 = 101.325d0
  ln_ratio = -dH_vap / R * (1.0d0/T2 - 1.0d0/T1)
  P2 = P1 * exp(ln_ratio)
  write(*,'(A20, F10.3, A5)') 'Ethanol:', P2, ' kPa'

! Diethyl ether
  dH_vap = 26520.0d0; T1 = 307.75d0; P1 = 101.325d0
  ln_ratio = -dH_vap / R * (1.0d0/T2 - 1.0d0/T1)
  P2 = P1 * exp(ln_ratio)
  write(*,'(A20, F10.3, A5)') 'Diethyl ether:', P2, ' kPa'

write(*,*)
  write(*,'(A)') '--- Key Points ---'
  write(*,'(A)') '  Higher dH_vap => stronger IMFs => lower vapor pressure'
  write(*,'(A)') '  ln(P2/P1) = -(dH_vap/R)(1/T2 - 1/T1)'
  write(*,'(A)') '  Plot of ln(P) vs 1/T gives slope = -dH_vap/R'

end program clausius_clapeyron

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server

Video Lectures

Lecture 12: Molecular Orbitals

Lecture 13: Hybridization

Lecture 14: Intermolecular Forces

Additional Lecture: Phases

Key Takeaways

●MO theory: atomic orbitals combine to form bonding (lower energy) and antibonding (higher energy) MOs
●Bond order = ($n_b - n_a$)/2 determines bond strength; BO = 0 means no stable molecule
●MO theory correctly predicts O$_2$ is paramagnetic (2 unpaired electrons in $\pi^*$)
●Hybridization (sp, sp$^2$, sp$^3$) explains molecular geometry from atomic orbital mixing
●IMF strength hierarchy: ion-dipole > H-bonding > dipole-dipole > London dispersion
●Clausius-Clapeyron connects vapor pressure to temperature via $\Delta H_{vap}$

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