Part 2, Topic 1 | Lectures 9–11

Chemical Bonding & Lewis Structures

From electron dots to three-dimensional molecular geometry

3.1 Lewis Dot Structures

Gilbert N. Lewis proposed a simple yet powerful model for chemical bonding based on electron pairs. In Lewis structures, valence electrons are represented as dots around atomic symbols. A covalent bond consists of a shared pair of electrons between two atoms.

Drawing Lewis Structures: Step-by-Step

Count total valence electrons (adjust for charge)
Draw single bonds between all bonded atoms
Distribute remaining electrons as lone pairs, starting with outer atoms
If the central atom lacks an octet, form multiple bonds
Calculate formal charges to identify the best structure

Formal Charge

Formal charge helps determine the most stable Lewis structure among resonance forms:

$$FC = V - L - \frac{B}{2}$$

$V$ = valence electrons, $L$ = lone pair electrons, $B$ = bonding electrons

The best Lewis structure minimizes formal charges, places negative formal charges on more electronegative atoms, and avoids like charges on adjacent atoms.

3.2 The Octet Rule and Exceptions

The octet rule states that atoms tend to form bonds until they are surrounded by eight valence electrons. While remarkably useful, several important exceptions exist:

Electron-Deficient Species

Some elements in Group 2 and 13 (Be, B, Al) form stable compounds with fewer than 8 electrons. Example: BF$_3$ has only 6 electrons around boron. These species are strong Lewis acids.

Odd-Electron Species

Molecules with an odd number of total valence electrons (free radicals) cannot satisfy the octet rule for all atoms. Examples: NO, NO$_2$, ClO$_2$. These are paramagnetic.

Expanded Octet

Elements in Period 3 and beyond can use d orbitals to accommodate more than 8 electrons. Examples: PCl$_5$ (10 e$^-$), SF$_6$ (12 e$^-$), XeF$_4$ (12 e$^-$).

3.3 VSEPR Theory

Valence Shell Electron Pair Repulsion (VSEPR) theory predicts molecular geometry by assuming that electron pairs (both bonding and lone) around a central atom arrange themselves to minimize mutual repulsion.

Repulsion Ordering

Lone pair – Lone pair > Lone pair – Bond pair > Bond pair – Bond pair

This ordering explains why lone pairs compress bond angles. For example, in the tetrahedral family: CH$_4$ has bond angles of 109.5$^\circ$, NH$_3$ has 107$^\circ$, and H$_2$O has 104.5$^\circ$.

Common Molecular Geometries

2 electron pairs

Linear (180$^\circ$): CO$_2$, BeCl$_2$

3 electron pairs

Trigonal planar (120$^\circ$): BF$_3$, or bent with 1 LP: SO$_2$

4 electron pairs

Tetrahedral (109.5$^\circ$): CH$_4$, pyramidal: NH$_3$, bent: H$_2$O

5 electron pairs

Trigonal bipyramidal (90$^\circ$/120$^\circ$): PCl$_5$, seesaw: SF$_4$

6 electron pairs

Octahedral (90$^\circ$): SF$_6$, square planar: XeF$_4$

3.4 Bond Order, Energy, and Length

Bond order, bond energy, and bond length are closely correlated properties that reflect the strength and character of chemical bonds:

Bond Order Increases

Single → Double → Triple

1 → 2 → 3

Bond Energy Increases

C–C → C=C → C$\equiv$C

346 → 614 → 839 kJ/mol

Bond Length Decreases

C–C → C=C → C$\equiv$C

154 → 134 → 120 pm

3.5 Electronegativity and Polarity

Electronegativity ($\chi$) measures an atom's ability to attract shared electrons in a bond. Pauling developed the most widely used scale, where fluorine (the most electronegative element) is assigned $\chi = 3.98$.

The electronegativity difference between bonded atoms determines the bond's polar character:

Nonpolar Covalent

$\Delta\chi < 0.5$

C–H, N–H

Polar Covalent

$0.5 < \Delta\chi < 1.7$

O–H, N–O

Ionic

$\Delta\chi > 1.7$

Na–Cl, K–F

The dipole moment of a polar bond is the product of the partial charge and the bond length:

$$\mu = q \cdot d$$

Measured in Debye (D); 1 D = 3.336 $\times$ 10$^{-30}$ C$\cdot$m

Pauling estimated the percent ionic character of a bond from the electronegativity difference:

$$\% \text{ ionic} = 100 \times \left(1 - e^{-0.25(\Delta\chi)^2}\right)$$

Interactive Simulation: VSEPR Geometry

This simulation visualizes how bond angles vary with the number of electron pairs and lone pairs, demonstrating VSEPR predictions for common molecular geometries.

VSEPR Geometry Visualizer

Python

Plots bond angles for different electron pair geometries and shows how lone pairs compress bond angles in the tetrahedral family.

script.py82 lines

import numpy as np
import matplotlib.pyplot as plt
# VSEPR geometry data
# Format: (electron_pairs, lone_pairs, name, bond_angle, shape_description)
geometries = [
    (2, 0, 'Linear', 180.0, 'BeCl\u2082, CO\u2082'),
    (3, 0, 'Trigonal Planar', 120.0, 'BF\u2083, SO\u2083'),
    (3, 1, 'Bent (from TP)', 117.0, 'SO\u2082, O\u2083'),
    (4, 0, 'Tetrahedral', 109.5, 'CH\u2084, SiH\u2084'),
    (4, 1, 'Trigonal Pyramidal', 107.0, 'NH\u2083, PH\u2083'),
    (4, 2, 'Bent (from Td)', 104.5, 'H\u2082O, H\u2082S'),
    (5, 0, 'Trigonal Bipyramidal', 90.0, 'PCl\u2085, AsF\u2085'),
    (5, 1, 'Seesaw', 117.0, 'SF\u2084, TeCl\u2084'),
    (5, 2, 'T-shaped', 90.0, 'ClF\u2083, BrF\u2083'),
    (5, 3, 'Linear (from TBP)', 180.0, 'XeF\u2082, I\u2083\u207b'),
    (6, 0, 'Octahedral', 90.0, 'SF\u2086, XeF\u2086'),
    (6, 1, 'Square Pyramidal', 85.0, 'BrF\u2085, IF\u2085'),
    (6, 2, 'Square Planar', 90.0, 'XeF\u2084, ICl\u2084\u207b'),
]

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(15, 8))
fig.patch.set_facecolor('#0f172a')
for ax in (ax1, ax2):
    ax.set_facecolor('#1e293b')
    ax.tick_params(colors='#cbd5e1')
    ax.xaxis.label.set_color('#cbd5e1')
    ax.yaxis.label.set_color('#cbd5e1')
    ax.title.set_color('#cbd5e1')
    for sp in ax.spines.values():
        sp.set_color('#334155')
    ax.grid(True, color='#334155', alpha=0.4)

# Left: Bond angles for different electron pair counts
colors_map = {0: '#4ade80', 1: '#facc15', 2: '#fb923c', 3: '#f472b6'}
for ep, lp, name, angle, examples in geometries:
    color = colors_map.get(lp, '#94a3b8')
    ax1.scatter(ep, angle, c=color, s=120, zorder=5, edgecolors='white', linewidths=0.5)
    offset_y = 3 if lp == 0 else -5
    ax1.annotate(name, (ep, angle), textcoords="offset points",
                xytext=(10, offset_y), fontsize=7, color=color)

ax1.set_xlabel('Total Electron Pairs', fontsize=12)
ax1.set_ylabel('Bond Angle (degrees)', fontsize=12)
ax1.set_title('VSEPR Bond Angles vs Electron Pairs', fontsize=14, fontweight='bold')

# Custom legend
from matplotlib.lines import Line2D
legend_elements = [
    Line2D([0], [0], marker='o', color='w', markerfacecolor='#4ade80', markersize=10, label='0 lone pairs'),
    Line2D([0], [0], marker='o', color='w', markerfacecolor='#facc15', markersize=10, label='1 lone pair'),
    Line2D([0], [0], marker='o', color='w', markerfacecolor='#fb923c', markersize=10, label='2 lone pairs'),
    Line2D([0], [0], marker='o', color='w', markerfacecolor='#f472b6', markersize=10, label='3 lone pairs'),
]
ax1.legend(handles=legend_elements, loc='lower right', facecolor='#1e293b',
           edgecolor='#334155', labelcolor='#cbd5e1', fontsize=9)

# Right: Effect of lone pairs on bond angle (tetrahedral family)
categories = ['CH\u2084\n(0 LP)', 'NH\u2083\n(1 LP)', 'H\u2082O\n(2 LP)']
angles = [109.5, 107.0, 104.5]
bar_colors = ['#4ade80', '#facc15', '#fb923c']

bars = ax2.bar(categories, angles, color=bar_colors, edgecolor='white', linewidth=0.5, width=0.5)
ax2.set_ylabel('Bond Angle (degrees)', fontsize=12)
ax2.set_title('Lone Pair Effect on Bond Angle\n(Tetrahedral Family)', fontsize=14, fontweight='bold')
ax2.set_ylim(100, 115)

for bar, angle in zip(bars, angles):
    ax2.text(bar.get_x() + bar.get_width()/2, bar.get_height() + 0.3,
             f'{angle}\u00b0', ha='center', va='bottom', color='#cbd5e1', fontsize=12, fontweight='bold')

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0f172a')
plt.close()

print("VSEPR Geometry Analysis")
print("=" * 50)
print(f"{'Pairs':>5} {'LP':>3} {'Geometry':<25} {'Angle':>7} {'Examples':<20}")
print("-" * 60)
for ep, lp, name, angle, examples in geometries:
    print(f"{ep:>5} {lp:>3} {name:<25} {angle:>6.1f}\u00b0 {examples:<20}")
print("\nKey: Lone pairs compress bond angles due to greater repulsion")
print("LP-LP > LP-BP > BP-BP (repulsion ordering)")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Fortran: Bond Energy Calculator

This Fortran program calculates percent ionic character and estimates heteronuclear bond energies using Pauling's electronegativity-based formulas.

Bond Energy & Ionic Character Calculator

Fortran

Computes percent ionic character from electronegativity differences and estimates bond energies using Pauling's geometric mean formula.

bond_energy.f9071 lines

program bond_energy_calculator
  implicit none
  integer :: i, j, n_elements
  double precision :: chi_diff, percent_ionic, bond_energy
  double precision :: chi(10), cov_energy_AA(10), cov_energy_BB(10)
  character(len=2) :: symbols(10)
  double precision :: delta, extra_ionic

! Pauling electronegativities for selected elements
  n_elements = 8
  symbols = (/ 'H ', 'C ', 'N ', 'O ', 'F ', 'Cl', 'Br', 'I ', '  ', '  ' /)
  chi     = (/ 2.20d0, 2.55d0, 3.04d0, 3.44d0, 3.98d0, 3.16d0, 2.96d0, 2.66d0, 0d0, 0d0 /)

! Homonuclear bond energies (kJ/mol)
  cov_energy_AA = (/ 436d0, 346d0, 946d0, 498d0, 159d0, 242d0, 193d0, 151d0, 0d0, 0d0 /)

write(*,'(A)') '======================================================'
  write(*,'(A)') '  Bond Energy & Ionic Character Calculator'
  write(*,'(A)') '======================================================'
  write(*,*)

! Part 1: Percent ionic character from electronegativity difference
  write(*,'(A)') '--- Percent Ionic Character ---'
  write(*,'(A)') '  Pauling formula: % ionic = 100 * (1 - exp(-0.25 * dx^2))'
  write(*,*)
  write(*,'(A5, A5, A10, A12)') 'A', 'B', 'dx(chi)', '% Ionic'
  write(*,'(A)') '--------------------------------------'

do i = 1, n_elements
    do j = i+1, n_elements
      chi_diff = abs(chi(i) - chi(j))
      ! Pauling's equation for percent ionic character
      percent_ionic = 100.0d0 * (1.0d0 - exp(-0.25d0 * chi_diff**2))
      if (chi_diff > 0.3d0) then
        write(*,'(A5, A5, F10.2, F12.1)') symbols(i), symbols(j), chi_diff, percent_ionic
      end if
    end do
  end do

write(*,*)
  write(*,'(A)') '--- Bond Type Classification ---'
  write(*,'(A)') '  dx < 0.5:  Nonpolar covalent'
  write(*,'(A)') '  0.5 < dx < 1.7:  Polar covalent'
  write(*,'(A)') '  dx > 1.7:  Ionic'
  write(*,*)

! Part 2: Pauling's bond energy estimation
  write(*,'(A)') '--- Bond Energy Estimation (Pauling) ---'
  write(*,'(A)') '  D(A-B) = sqrt(D(A-A) * D(B-B)) + 96.5 * dx^2  (kJ/mol)'
  write(*,*)
  write(*,'(A5,A5,A12,A12,A12,A12)') 'A','B','D(A-A)','D(B-B)','D(A-B)est','dx'
  write(*,'(A)') '--------------------------------------------------------------'

do i = 1, 5
    do j = i+1, min(i+3, n_elements)
      chi_diff = abs(chi(i) - chi(j))
      extra_ionic = 96.5d0 * chi_diff**2
      bond_energy = sqrt(cov_energy_AA(i) * cov_energy_AA(j)) + extra_ionic
      write(*,'(A5,A5,F12.0,F12.0,F12.0,F12.2)') &
        symbols(i), symbols(j), cov_energy_AA(i), cov_energy_AA(j), &
        bond_energy, chi_diff
    end do
  end do

write(*,*)
  write(*,'(A)') '--- Key Relationships ---'
  write(*,'(A)') '  Greater electronegativity difference => stronger, more ionic bond'
  write(*,'(A)') '  Bond energy increases with bond order (single < double < triple)'
  write(*,'(A)') '  Bond length decreases with bond order'

end program bond_energy_calculator

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server

Video Lectures

Lecture 9: Lewis Structures I

Lecture 10: Lewis Structures II

Lecture 11: Shapes of Molecules and VSEPR

Goodie Bag 4: VSEPR

Key Takeaways

●Lewis structures represent bonding and lone pair electrons; formal charge = $V - L - B/2$
●The octet rule has important exceptions: electron-deficient, odd-electron, and expanded octet species
●VSEPR theory predicts 3D geometry from electron pair repulsion (LP-LP > LP-BP > BP-BP)
●Bond order correlates with energy (stronger) and length (shorter)
●Electronegativity difference determines bond polarity: nonpolar, polar covalent, or ionic

← Part 2 Overview Molecular Orbital Theory →