Chapter 2: Molecular Structure and Bonding

Lewis Structures

In 1916, the physical chemist G.N. Lewis proposed that a covalent bond is formed when two neighbouring atoms share an electron pair. A single bond (A:B) is denoted A−B; a double bond (A::B) is denoted A═B; and a triple bond (A:::B) is denoted A≡B. An unshared pair of valence electrons on an atom is called a lone pair.

2.1 The Octet Rule

Key Point: Atoms share electron pairs until they have acquired an octet of valence electrons.

Octet Rule:

Each atom shares electrons with neighbouring atoms to achieve a total of eight valence electrons (an 'octet'). Hydrogen is an exception, filling its valence shell with two electrons (a 'duplet').

The octet rule provides a simple way of constructing a Lewis structure, a diagram that shows the pattern of bonds and lone pairs in a molecule:

Decide on the number of electrons by adding valence electrons from all atoms. Each negative charge adds one electron; each positive charge removes one.
Write chemical symbols showing which atoms are bonded together. The less electronegative element is usually central.
Distribute electrons in pairs: one pair between each bonded pair of atoms, then supply remaining pairs as lone pairs or multiple bonds until each atom has an octet.

Example 2.1: Writing a Lewis Structure

Problem: Write a Lewis structure for the BF₄⁻ ion.

Answer: The atoms supply 3 + (4×7) = 31 valence electrons; the negative charge adds one more. We must accommodate 32 electrons (16 pairs) around five atoms. The structure has B at the center bonded to four F atoms, each with three lone pairs.

Write a Lewis structure for the PCl₃ molecule.

2.2 Resonance

Key Point: Resonance between Lewis structures lowers the calculated energy of the molecule and distributes the bonding character of electrons over the molecule.

A single Lewis structure is often inadequate. For ozone (O₃), the Lewis structure suggests one O−O bond differs from the other, but experimentally both bonds are identical (128 pm), intermediate between single (148 pm) and double (121 pm) bonds.

Resonance:

The actual structure is a superposition (average) of all feasible Lewis structures. A resonance hybrid is the blended structure. Resonance averages bond characteristics and lowers energy below any single contributing structure.

The wavefunction is written as a superposition:

ψ = ψ(O═O−O) + ψ(O−O═O)

Important: Resonance occurs only between structures differing in electron allocation, not atomic positions.

2.3 The VSEPR Model

Key Point: In the VSEPR model, regions of enhanced electron density take up positions as far apart as possible, and molecular shape is identified by the locations of atoms.

The Valence Shell Electron Pair Repulsion (VSEPR) model assumes that bonding pairs, lone pairs, and multiple bonds (regions of electron density) position themselves to minimize repulsions.

Basic VSEPR Arrangements

Electron Regions	Arrangement
2	Linear
3	Trigonal planar
4	Tetrahedral
5	Trigonal bipyramidal
6	Octahedral

The molecular shape name refers to atom positions, not electron regions:

Molecular Shapes (VSEPR)

━━━

Linear

HCN, CO₂

∠

Angular (Bent)

H₂O, O₃, NO₂⁻

△

Trigonal Planar

BF₃, SO₃, NO₃⁻

▲

Trigonal Pyramidal

NH₃, SO₃²⁻

◇

Tetrahedral

CH₄, SO₄²⁻

□

Square Planar

XeF₄

⬡

Trigonal Bipyramidal

PCl₅(g), SOF₄⁺

⬢

Octahedral

SF₆, PCl₅⁻

Example 2.2: Using VSEPR to Predict Shapes

(a) BF₃: Three F atoms, no lone pairs → trigonal planar

(b) SO₃²⁻: Three atoms + one lone pair = 4 regions → tetrahedral base → trigonal pyramidal shape

(c) PCl₄⁺: Four Cl atoms, no lone pair → tetrahedral

Lone Pair Effects

Key Point: Lone pairs repel other pairs more strongly than bonding pairs do.

Repulsion order: lone pair/lone pair > lone pair/bonding > bonding/bonding

This ordering explains why:

NH₃ has HNH angle of 107° (less than tetrahedral 109.5°)
H₂O has HOH angle of 104.5°
Lone pairs prefer equatorial positions in trigonal bipyramidal arrangements

Example 2.3: Effect of Lone Pairs on SF₄

SF₄ has 4 F atoms + 1 lone pair = 5 regions → trigonal bipyramidal base. The lone pair occupies an equatorial site, giving a "see-saw" shape. S−F bonds bend away from the lone pair.

Predict the shape of (a) an XeF₂ molecule and (b) an ICl₂⁺ molecule.

Valence Bond Theory

Valence bond (VB) theory was the first quantum mechanical theory of bonding. It considers the interaction of atomic orbitals on separate atoms as they approach to form a molecule.

2.4 The Hydrogen Molecule

Key Point: In VB theory, the wavefunction of an electron pair is formed by superimposing wavefunctions for separated fragments.

For H₂, the VB wavefunction is:

ψ = χ_A(1)χ_B(2) + χ_A(2)χ_B(1)

where χ_A and χ_B are H1s orbitals on atoms A and B. The bond forms due to high probability of finding both electrons between the nuclei. Only electrons with paired spins can be described this way—hence the importance of spin pairing.

σ Bond:

A bond with cylindrical symmetry around the internuclear axis, formed by head-on orbital overlap. Electrons have zero orbital angular momentum about the axis.

2.5 Homonuclear Diatomic Molecules

For N₂, with the z-axis along the bond:

One σ bond forms from 2p_z-2p_z overlap (head-on)
Two π bonds form from 2p_x-2p_x and 2p_y-2p_y overlap (side-by-side)

π Bond:

A bond with a nodal plane containing the internuclear axis, formed by side-by-side p orbital overlap. Electrons have one unit of orbital angular momentum about the axis.

σ Bond (Single)

σ + π (Double)

σ + 2π (Triple)

2.6 Polyatomic Molecules

VB theory for H₂O predicts two σ bonds from O2p orbitals overlapping with H1s orbitals. Since 2p_x and 2p_y are perpendicular, VB predicts a 90° bond angle—but the actual angle is 104.5°.

(a) Promotion

Key Point: Promotion of electrons may occur if it allows more or stronger bonds.

Promotion is the excitation of an electron to a higher energy orbital during bond formation. For carbon, promotion from 2s²2p² to 2s¹2p³ creates four unpaired electrons for four bonds—the energy invested is recovered in stronger bonding.

(b) Hypervalence

Elements beyond Period 2 can exceed the octet. PCl₅ (10 electrons around P) and SF₆ (12 electrons around S) are hypervalent. This may involve d orbital participation or simply geometric packing considerations.

(c) Hybridization

Key Point: Hybrid orbitals form when atomic orbitals on the same atom interfere; specific schemes correspond to each geometry.

Hybrid orbitals are mixtures of atomic orbitals on one atom that have enhanced directional character and form equivalent bonds.

sp

Linear

180°

sp²

Trigonal Planar

120°

sp³

Tetrahedral

109.5°

sp³d

Trigonal Bipyramidal

90°, 120°

sp³d²

Octahedral

90°

The sp³ hybrid orbitals of carbon:

h₁ = s + p_x + p_y + p_z h₂ = s − p_x − p_y + p_z
h₃ = s − p_x + p_y − p_z h₄ = s + p_x − p_y − p_z

Molecular Orbital Theory

Molecular orbital (MO) theory provides a more sophisticated model where electrons spread over all atoms, binding them together. Almost all modern calculations use MO theory.

2.7 Introduction to the Theory

(a) Approximations

Key Point: Molecular orbitals are constructed as linear combinations of atomic orbitals (LCAO); each MO can hold up to two electrons.

The orbital approximation assumes the wavefunction is a product of one-electron functions:

ψ = ψ(1)ψ(2)...ψ(N_e)

The LCAO approximation builds molecular orbitals from atomic orbitals:

ψ = c_Aχ_A + c_Bχ_B

The basis set is the collection of atomic orbitals used. The coefficients c_A² and c_B² give probabilities of finding electrons near each atom.

Key principle: From N atomic orbitals, we construct N molecular orbitals.

(b) Bonding and Antibonding Orbitals

Key Point: Bonding orbitals arise from constructive interference; antibonding orbitals arise from destructive interference with a node between atoms.

Bonding Orbital (ψ₊):

Electrons have enhanced probability between nuclei. The molecule has lower energy than separated atoms. Formed by same-sign overlap of atomic orbitals.

Antibonding Orbital (ψ₋):

A nodal plane between nuclei excludes electrons from the bonding region. Denoted with asterisk (*). The molecule has higher energy than separated atoms.

Nonbonding Orbital:

Same energy as original atomic orbitals. Often a lone orbital on one atom with no matching symmetry partner.

Important: An antibonding orbital is slightly more antibonding than its partner is bonding—this asymmetry explains weak bonds between electron-rich 2p elements.

2.8 Homonuclear Diatomic Molecules

(a) The Orbitals

Key Point: Molecular orbitals are classified as σ, π, or δ by rotational symmetry, and as g or u by inversion symmetry.

For Period 2 diatomics, the minimal basis set is: one 2s and three 2p orbitals per atom = 8 atomic orbitals → 8 molecular orbitals.

σ orbitals: From 2s-2s and 2p_z-2p_z overlap (cylindrical symmetry)
π orbitals: From 2p_x-2p_x and 2p_y-2p_y overlap (nodal plane through axis)
g (gerade): Unchanged under inversion through center
u (ungerade): Changes sign under inversion

Molecular Orbital Diagram for O₂ and F₂

2p

2s

Atom A

2σ_u*

1π_g* ↑↑ (O₂)

2σ_g ↑↓

1π_u ↑↓ ↑↓

1σ_u* ↑↓

1σ_g ↑↓

Molecular Orbitals

2p

2s

Atom B

Note: For Li₂ to N₂, the order of 2σ_g and 1π_u is reversed due to s-p mixing when the 2s-2p energy gap is small.

(b) The Building-Up Principle

Key Point: Fill molecular orbitals in order of increasing energy, following the Pauli principle and Hund's rule.

Example 2.4: Electron Configurations

O₂ (12 valence e⁻): 1σ_g²1σ_u²2σ_g²1π_u⁴1π_g²

The 1π_g orbitals each get one electron with parallel spins → O₂ is paramagnetic!

O₂⁻ (13 e⁻): ...1π_g³ O₂²⁻ (14 e⁻): ...1π_g⁴

(a) Determine unpaired electrons in O₂, O₂⁻, O₂²⁻. (b) Write configurations for S₂²⁻ and Cl₂²⁻.

Frontier Orbitals:

HOMO (Highest Occupied MO) and LUMO (Lowest Unoccupied MO) play special roles in structure and reactivity. SOMO denotes a singly occupied MO in radicals.

2.9 Heteronuclear Diatomic Molecules

Key Point: Heteronuclear diatomics are polar; bonding electrons concentrate on the more electronegative atom.

For heteronuclear molecules like HF and CO:

ψ = c_Aχ_A + c_Bχ_B where c_A² ≠ c_B²

More electronegative atom contributes more to bonding orbitals
Less electronegative atom contributes more to antibonding orbitals
Energy mismatch reduces orbital mixing strength

Carbon Monoxide (CO)

Configuration: 1σ²2σ²1π⁴3σ²

HOMO (3σ): Largely C2p character, almost nonbonding, localized on C atom—this is the "lone pair" that bonds to metals

LUMO (2π*): Antibonding π orbitals with mainly C2p character—accepts electrons in π-backbonding

Bond order = 3, consistent with C≡O. Despite electronegativity difference, the dipole moment is small (0.1 D) with negative end on C!

2.10 Bond Properties

(a) Bond Order

Key Point: Bond order = ½(n − n*), where n = bonding electrons and n* = antibonding electrons.

b = ½(n − n*)

Species	Configuration	Bond Order
H₂	σ_g²	1
He₂	σ_g²σ_u*²	0
N₂	1σ_g²1σ_u²1π_u⁴2σ_g²	3
O₂	...1π_u⁴2σ_g²1π_g*²	2
O₂⁻	...1π_g*³	1.5
O₂²⁻	...1π_g*⁴	1
F₂	...1π_g*⁴	1

(b) Bond Correlations

Key Point: For a given pair of elements: bond strength ↑ as bond order ↑, and bond length ↓ as bond order ↑.

Important comparison:

C═C is less than 2× as strong as C−C → organic polymerization favored
N═N is more than 2× as strong as N−N → N₂ is very stable
P═P is less than 2× as strong as P−P → P prefers single-bonded structures (P₄)

Example 2.7: Predicting Bond Properties

For O₂, O₂⁻, O₂²⁻ with bond orders 2, 1.5, 1:

Bond enthalpy: O₂²⁻ < O₂⁻ < O₂

Bond length: O₂²⁻ > O₂⁻ > O₂

Experimental: O−O 146 kJ/mol, 132 pm; O═O 496 kJ/mol, 121 pm

2.11 Polyatomic Molecules

MO theory extends naturally to polyatomic molecules. From N atomic orbitals, construct N molecular orbitals.

Key Point: MOs are formed from linear combinations of atomic orbitals of the same symmetry.

ψ = Σ c_iχ_i

Guiding principles:

More nodes → higher energy
Lower-energy AOs produce lower-energy MOs
Same-sign lobes on non-neighbors are weakly bonding; opposite signs are weakly antibonding

Symmetry labels:

a, b: nondegenerate orbital
e: doubly degenerate (two orbitals of same energy)
t: triply degenerate (three orbitals of same energy)

NH₃ Molecular Orbitals

Basis: N2s, N2p_x, N2p_y, N2p_z + 3 H1s = 7 AOs → 7 MOs

Configuration: 1a₁²1e⁴2a₁²

The 2a₁ HOMO is largely N2p_z character—the "lone pair" that determines basicity. Photoelectron spectrum shows peaks at 11 eV (2a₁) and 16 eV (1e).

(b) Hypervalence in MO Context

Key Point: MO delocalization allows electron pairs to bond more than two atoms—no d orbitals required for hypervalence.

For SF₆: 10 basis orbitals (S 3s, 3p + 6 F 2p) → 10 MOs. Four bonding + four antibonding + two nonbonding. The 12 electrons fill 1a₁²1t₁⁶1e⁴ with no antibonding occupation.

(c) Localization

Key Point: Localized and delocalized descriptions are mathematically equivalent but suit different properties.

Localized Description	Delocalized Description
Bond strengths	Electronic spectra
Force constants	Photoionization
Bond lengths	Electron attachment
Brønsted acidity	Magnetism
VSEPR description	Standard potentials

(e) Electron Deficiency

Key Point: Electron-deficient compounds exist because MO delocalization spreads bonding over many atoms.

Diborane (B₂H₆): Only 12 valence electrons but 8 atoms! Lewis theory requires 16 electrons for 8 bonds.

MO explanation: 14 basis orbitals → ~7 bonding/nonbonding MOs accommodate 12 electrons. The BHB "banana bonds" involve 3-center-2-electron bonding.

Structure and Bond Properties

2.13 Bond Length

Key Point: Equilibrium bond length is the separation of bonded atom centers; covalent radii are additive.

The covalent radius is an atom's contribution to a bond length. The P−N bond length ≈ 110 pm + 74 pm = 184 pm (experimental: ~180 pm).

The van der Waals radius is the closest nonbonded approach—important for molecular packing and conformations.

Element	Covalent Radius (pm)
H	37
C	77 (single), 67 (double), 60 (triple)
N	74 (single), 65 (double), 54 (triple)
O	66 (single), 57 (double)
F	64
Cl	99

2.14 Bond Strength

The bond dissociation enthalpy ΔH°(A−B) is the enthalpy for AB(g) → A(g) + B(g), always positive.

The mean bond enthalpy B is an average over various molecules—useful for estimates when specific data unavailable.

Example 2.9: Using Mean Bond Enthalpies

Reaction: SF₄(g) + F₂(g) → SF₆(g)

Bonds broken: 1 F−F (158 kJ) + 4 S−F (4×343 kJ) = +1530 kJ

Bonds formed: 6 S−F (6×327 kJ) = −1962 kJ

ΔH° = +1530 − 1962 = −432 kJ (exothermic)

Experimental value: −434 kJ (excellent agreement)

2.15 Electronegativity and Bond Enthalpy

Key Point: Pauling electronegativity is useful for estimating bond enthalpies and assessing bond polarities.

Pauling defined electronegativity difference from excess bond energy:

|χ_P(A) − χ_P(B)| = 0.102(ΔE/kJ mol⁻¹)^1/2

ΔE = B(A−B) − ½[B(A−A) + B(B−B)]

Compounds with electronegativity difference > 1.7 are generally ionic.

The Ketelaar Triangle

The van Arkel–Ketelaar triangle classifies bonding based on electronegativity difference (Δχ) and average electronegativity (χ_mean):

IONIC (CsF) METALLIC (Cs) COVALENT (F₂)

Ionic: Large Δχ, intermediate χ_mean (e.g., CsF: Δχ = 3.19, χ_mean = 2.38)
Covalent: Small Δχ, high χ_mean (e.g., F₂: Δχ = 0, χ_mean = 3.98)
Metallic: Small Δχ, low χ_mean (e.g., Cs: Δχ = 0, χ_mean = 0.79)

2.16 Oxidation States

Key Point: Oxidation numbers are assigned by exaggerating ionic character—the more electronegative atom "takes" the bonding electrons.

Rules for assigning oxidation number:

Sum of oxidation numbers = total charge
Elemental form: 0
Group 1: +1; Group 2: +2
H: +1 with nonmetals, −1 with metals
F: always −1
O: usually −2 (except with F, or in peroxides −1, superoxides −½)
Halogens: usually −1

Example 2.10: Assigning Oxidation Numbers

(a) H₂S: 2(+1) + N_ox(S) = 0 → N_ox(S) = −2

(b) [MnO₄]⁻: N_ox(Mn) + 4(−2) = −1 → N_ox(Mn) = +7

Find oxidation numbers: (a) O in O₂⁺, (b) P in PO₂³⁻, (c) Mn in [MnO₄]²⁻, (d) Cr in [Cr(H₂O)₆]Cl₃

2.12 Computational Methods

Key Point: Computational methods range from rigorous ab initio calculations to parametrized semi-empirical methods.

Ab Initio Methods

Calculate structures from first principles using only atomic numbers and geometry. Methods include:

Hartree-Fock (HF): Primary approximation for electron-electron repulsion
Møller-Plesset (MPn): Perturbation theory corrections
Configuration Interaction (CI): Correlation corrections
Coupled Cluster (CC): Advanced correlation treatment

Density Functional Theory (DFT)

Expresses energy in terms of electron density ρ = |ψ|² rather than wavefunction. Solves Kohn-Sham equations iteratively. Less computationally demanding than ab initio; often gives better agreement for d-metal complexes.

Semi-Empirical Methods

Use experimental parameters for certain integrals. Much faster than ab initio but quality depends on parameter transferability. Successful for organic chemistry with limited element types.

Molecular Mechanics

"Ball and spring" model treating atoms as particles and bonds as springs. Uses classical mechanics. Applicable to large systems including proteins.

Output visualization: Isodensity surfaces show constant electron density. Electrostatic potential surfaces (elpot) show charge distribution with color-coded positive/negative regions.