CHEM 548: Materials Chemistry

Lecture 12: Atomic Orbitals and Molecular Orbital Theory

Wed 10/14/2026 · Meeting 14

Reading: Woodward, Karen, Evans, and Vogt, Solid State Materials Chemistry
Ch. 5, §§5.2–5.3

Learning Objectives: Atomic Orbitals

Atomic Orbitals

Recognize the wave nature of electrons and the structure of atomic orbitals.
Identify and interpret the four quantum numbers (n, l, m_l, m_s).
Apply Pauli’s exclusion principle and Hund’s rule to electron configurations.
Distinguish between radial and angular nodes.
Relate orbital type (s, p, d, f) to shape, orientation, and nodal structure.
Explain how orbital energies vary with n, l, and atomic number.

Orbital Wavefunctions

$\Psi_{n,l,m_l} = R_{n,l}(r)\,Y_{l,m_l}(\theta,\varphi)$

Spherical polar axes X, Y, and Z with the radius r to a point, the polar angle theta measured from the z-axis, and the azimuthal angle phi measured in the xy-plane.

Radial part	Angular part
Defines the size of the orbital	Defines the shape of the orbital

Quantum Numbers

Principle quantum # (n = 1,2,3,…)
- Largely defines the size and energy of the orbital
Orbital angular momentum quantum # (ℓ = 0, 1, … n−1)
- Defines the shape of the orbital
Magnetic quantum # (m_ℓ = −ℓ to ℓ)
- Defines the orientation of the orbital
Spin quantum # (m_s = +½, −½)
- Defines the spin of the electron

Quantum Mechanics and Orbitals

Pauli Exclusion Principle
- No two electrons can have the same set of quantum numbers, this limits each orbital to holding two electrons
Hund’s 1^st Rule
- For degenerate orbitals the lowest energy configuration maximizes the electron spin
Nodes
- A node is a place where the wavefunction changes sign
- The value of the wavefunction is 0 at a node
- The probability density is 0 at a node
- Radial nodes, R = 0
- Nodal planes, Y = 0

Principal Quantum Number and Radial Nodes

Radial wavefunctions and radial probability functions for the 1s, 2s, and 3s orbitals, with radial nodes marked; the number of radial nodes equals n minus l minus 1.

S orbitals are spherical, so their angular wave function is constant, and all structure comes from the radial wave function.

1s, 2s, 3s all peak at the nucleus and decay outward: 1s always positive, no nodes; 2s one node (~1 Å), like a ripple crossing zero; 3s two nodes, positive–negative–positive
Radial nodes = n − l − 1: 0, 1, 2 for 1s, 2s, 3s
Radial probability function: squared wave function × 4πr² — probability at distance r
1s peaks at ~0.54 Å; 2s: inner and outer peak split by a node; 3s: outer maximum around 7 Å
As n increases: orbitals grow larger, density shifts farther from the nucleus
ψ(r) peaks at r = 0 (wave amplitude); 4πr²|ψ|² peaks ~0.5 Å — the r² shell-area term grows while |ψ|² falls

Angular nodes come next: they give the p, d, and f orbitals their shapes.

Angular Momentum and Nodal Planes

s, p, d, and f orbital shapes showing that the number of nodal planes equals the orbital angular momentum quantum number l.

A Closer Look at the Wavefunction

The 2p x wavefunction plotted along the x-axis, rising to a peak, crossing zero at the nodal plane, and turning negative on the other side.

Orbital Shapes: p Orbitals

The three 2p orbitals with nodal planes perpendicular to the x, y, and z axes.

Orbital Shapes: d Orbitals

The five d orbitals, four with two nodal planes and four lobes, and the d z squared orbital with nodal cones.

Orbitals and the Periodic Table

Periodic table with the s, p, d, and f blocks reflecting which orbitals fill across each row.

Azimuthal Quantum Number and Orbital Energy

Radial probability functions of the 3s, 3p, and 3d orbitals showing the inner peaks that let 3s and 3p penetrate close to the nucleus.

To understand why orbital energies differ, examine the radial probability functions for the 3s, 3p, and 3d orbitals in a hydrogen-like atom.

Maximum probability: farthest for 3s, closer for 3p, closest for 3d — as l increases the orbital shrinks
You might expect closer = lower energy; but near the nucleus, differences appear due to radial nodes
3s: two nodes, strong inner peak ~0.2 Å — substantial density near the nucleus
3p: one node, smaller inner peak ~0.7–0.8 Å — moderate penetration
3d: no radial nodes, very little density close to the nucleus
Lacking that penetrating density, 3d feels less nuclear attraction → higher energy than 3s and 3p, despite its smaller overall extent

With penetration in mind, let's look at calculated orbital energies across the whole periodic table.

Orbital Energies

Most of what we’ve covered so far should be review.

Dirac–Fock calculations for isolated atoms; up and to the right (toward neon), energies more negative — increasing effective nuclear charge
K, Ca: only 4s occupied, so it must be lower in energy than 3d; by Sc, Ti the 3d have dropped below 4s
End of the transition series (Zn, Ga): 3d much deeper and more stable than 4s
Zinc forms Zn²⁺ by losing its 4s electrons only; the 3d act as core levels
Gap: ~20 eV by gallium, ~70 eV by krypton → 3d become deep core states, key to elemental trends and bonding

[advance: highlight the second-period 2s and 2p energies]

s–p gap, second period: boron 2s ~5 eV below 2p, widening to ~23 eV in fluorine as effective nuclear charge increases
Fluorine 2s: a core orbital, largely uninvolved in bonding; boron and carbon use both 2s and 2p
Heavier p-block: valence s below p by ~6 eV (Ga), 8 eV (Ge), 5–7 eV (In, Sn), growing to ~9 eV in lead, similar in bismuth
These s orbitals sit 2–3 eV lower than lighter-element trends predict: relativistic effects — inner electrons near light speed, mass up, s orbitals (no nodes at the nucleus) contract
Group 4A: oxidation states shift from +4 (Si, Ge, Sn) to +2 dominance in Pb — stabilized 6s less available for bonding, the origin of the inert-pair effect in Tl, Pb, Hg, Bi

Energy is one half of the story; orbital size is the other.

Orbital Sizes

Orbital size is hard to define because it extends infinitely, but the maximum of the radial distribution function gives a practical measure.

First filling of a subshell: electrons drawn unusually close — stronger-than-expected attraction, orbital smaller than predicted
Reflects reduced shielding and greater effective nuclear charge for those new electrons

[advance: highlight the second-period 2s and 2p sizes]

So what do I mean? Look at the sizes of the 2s and the 2p orbitals across the second period
By and large, they're almost the same

[advance: clear the highlight]

Third, fourth, fifth periods: s smaller than p; the exception is the second period, where 2p is anomalously small
Why: no p orbitals in the core to shield — only s closer in, so 2p feels unusually strong attraction and contracts
Result: 2s and 2p nearly the same size, and this similarity allows strong overlap in π bonding — dominates carbon chemistry, far weaker for silicon, whose larger 3p orbitals overlap poorly
The ubiquity of π bonds in carbon and their near absence in silicon: rooted in this size anomaly; it underpins the chemistry of life’s carbon-based molecules

[advance: highlight the 3d orbital sizes]

3d unusually small vs the s orbitals; d always smaller than same-n s, but the difference is greatest for 3d
Between valence and core, so they can easily lose different numbers of electrons, leading to multiple oxidation states; partially filled d shells allow unpaired electrons → magnetic behavior
4d, 5d: similar effects, most pronounced for 3d
Lanthanides and actinides: 4f even smaller, core-like → +3 dominates the whole lanthanide series
Small, shielded 4f → sharp f-shell transitions → the vivid luminescence behind lanthanide phosphors and optical materials

Let's collect the atomic-orbital ideas in a summary.

Summary: Atomic Orbitals

Summary

Electrons behave as standing waves described by wavefunctions (ψ).
Wavefunctions separate into radial (distance-dependent) and angular (direction-dependent) parts.
Each electron is defined by four quantum numbers (n, l, m_l, m_s).
Nodes occur where ψ = 0:
- Radial nodes = n − l − 1
- Angular nodes = l
Orbital types:
- s (spherical), p (two lobes), d (four lobes), f (complex, six lobes).
Pauli Exclusion and Hund’s Rule determine electron configurations.
Orbital energy depends on n, l, and electron shielding.
Relativistic effects stabilize 6s orbitals in heavy elements, shaping their oxidation states.
Orbital sizes and energies explain periodic trends and bonding behavior.

Check Your Understanding: Nodes and Quantum Numbers

Question asking how the number of radial and angular nodes relates to the quantum numbers and why that matters for bonding in solids, then its worked answer.

Check Your Understanding: 4s versus 3d Energies

Question asking why 3d orbitals are higher in energy than 4s in potassium but lower by zinc and what that means for transition-metal chemistry, then its worked answer.

Check Your Understanding: The Inert-Pair Effect

Question asking what causes the inert pair effect in heavy p-block elements and how it arises from relativistic effects, then its worked answer.

Check Your Understanding: π Bonds in the Second Period

Question asking why second-period elements form strong pi bonds while heavier congeners rarely do, then its worked answer.

Learning Objectives: Molecular Orbital Theory

Molecular Orbital Theory

Explain how atomic orbitals combine to form molecular orbitals.
Distinguish between bonding, antibonding, and nonbonding orbitals based on constructive and destructive interference.
Interpret simple MO diagrams (H₂, HeH⁺, O₂) and relate orbital overlap to bond order and stability.
Describe how orbital energy differences influence covalent, polar covalent, and ionic bonding character.
Understand σ vs. π interactions and how orbital symmetry governs overlap strength.
Recognize s–p mixing and its effect on molecular orbital energy ordering.
Apply the concept of symmetry-adapted linear combinations (SALCs) to polyatomic molecules (e.g., BeH₂, CH₄).
Define HOMO and LUMO and explain their importance in determining chemical reactivity.
Identify how conjugated π systems (e.g., benzene) generate delocalized bonding and characteristic energy levels.

Molecular Orbitals: LCAO

Molecular Orbitals

The electrons in an atom reside in atomic orbitals, each of which has a characteristic wavefunction.

When atoms combine to form molecules we need new orbitals, molecular orbitals, that have their own wavefunctions.

We will choose to describe molecular orbital wavefunctions as linear combination of atomic orbital wavefunctions (LCAO):

$\psi_{MO} = \sum c_1\psi_{AO(1)} + c_2\psi_{AO(2)} + \ldots c_i\psi_{AO(i)}$

MO Diagram: H₂

Molecular orbital diagram of H2 with a bonding orbital from constructive interference and an antibonding orbital from destructive interference; clicks circle the antibonding orbital and redraw the diagram in symbolic style with coefficients and energies.

The simplest and best-known molecular orbital diagram is for H₂.

Two MOs: a lower-energy bonding orbital and a higher-energy antibonding orbital
Bonding: constructive interference of the two 1s wave functions → more density between the nuclei
That density pulls on both nuclei → stabilization: the bonding MO lies below the isolated atomic orbitals

[advance: circle the antibonding orbital]

Antibonding: destructive interference; color = sign of the wave function, red positive, blue negative
Node between the atoms: zero density, electrons pushed out of the internuclear region → energy above the separate atoms

[advance: redraw the diagram in symbolic style]

Symbolic style from here on: same sign → bonding, opposite signs → antibonding
Extended Hückel: bonding coefficients smaller than antibonding — the antibonding positive and negative regions partly cancel in the normalization integral, so the remaining portions scale up
Bonding stabilized ~4 eV; antibonding destabilized ~17 eV
Stabilization always less than destabilization → filling both orbitals gives no net stabilization, bond formation unfavorable

Plotting the wavefunctions along the bond axis makes those interference patterns explicit.

Molecular Orbital Wavefunctions

MO Diagram: HHe

Molecular orbital diagram of HHe with the helium 1s orbital lower in energy than the hydrogen 1s; a click marks the reduced stabilization of the bonding orbital.

Now let’s look at a more complex case with two different atoms.

Helium’s higher nuclear charge → He 1s lower than hydrogen’s
The energy difference creates partial ionicity: a polar covalent bond
Bonding MO: helium coefficient larger — resembles helium’s 1s more strongly; the antibonding is weighted toward hydrogen
Bigger atomic-orbital gap → bonding resembles the more electronegative atom, antibonding the more electropositive; in the limit, purely ionic

[advance: mark the reduced stabilization of the bonding orbital]

Stabilization of the bonding MO decreases when the atomic orbitals differ: H₂ ~4 eV, H–He only ~1.8 eV
Larger energy mismatch → less overlap and mixing → weaker bonding interaction

Next, we will move to O₂, where both the s and the p orbitals come into play.

O₂ and s-p Mixing

Molecular orbital diagram of O2 with sigma and pi levels in a 1-2-2-1 pattern; clicks bring up s-p mixing of the sigma orbitals and return to the O2 diagram with mixing included.

Let’s examine O₂, starting with a simplified view and then moving toward the full MO diagram.

The two 2s overlap → bonding and antibonding MOs, like H₂; the 2p interactions split into a 1–2–2–1 pattern, not three equal pairs
p_z head-on along z → σ bonding and antibonding; p_x, p_y side-on → π, with less overlap
Greater spatial overlap → larger bonding–antibonding gap → σ stronger than π
The true diagram also needs σ–σ mixing of 2s and 2p, which slightly alters energies and ordering

[advance: bring up s-p mixing of the sigma orbitals]

Adding the wave functions constructively → density between the nuclei → stabilizes; subtracting → density to the periphery → raises the energy
Mixing stabilizes the orbital with more 2s character, destabilizes the one with more 2p; same pattern for the σ antibonding pair

[advance: return to the O₂ diagram with mixing included]

σ2s gains p character → stabilized; σ2p gains s character → destabilized; the antibonding pair shifts the same way
Net effect: smaller σ2p–π2p gap; in lighter diatomics like N₂ or C₂, mixing lifts σ2p above the π2p

So far everything has been diatomic; next we move to molecules built around a central atom.

Beyond Diatomics: SALCs

Central-atom s and p valence orbitals; clicks add the ligand symmetry-adapted linear combinations for linear BeH2, tetrahedral CH4, and octahedral SH6.

To move beyond diatomic molecules, we decompose the system into a central atom and its surrounding atoms, called ligands.

Group the ligand orbitals by symmetry into symmetry-adapted linear combinations (SALCs): how the ligand orbitals combine and interact with the central atom’s orbitals, consistent with the molecule’s overall symmetry

[advance: add the linear BeH₂ SALCs]

Symmetric SALC can overlap with the Be 2s; antisymmetric SALC aligns with the Be 2p_x
The remaining 2p_y, 2p_z: no net overlap, no interaction — formally group theory, here evident by visual inspection

[advance: add the tetrahedral CH₄ SALCs]

Methane: the four hydrogens form four SALCs, each matching one of carbon’s valence orbitals
Totally symmetric SALC → carbon 2s; the other three → the three 2p → four equivalent σ bonds, tetrahedral

[advance: add the octahedral SH₆ SALCs]

Octahedral S–H₆: six ligands, one orbital each → six SALCs (shapes require group theory)
First matches the sulfur s, next three the sulfur p; the remaining two have no matching symmetry → nonbonding

Let's put the BeH₂ SALCs into an energy-level diagram.

BeH₂: MO Diagram and Frontier Orbitals

BeH2 molecular orbital diagram built from the ligand SALCs and the Be 2s and 2p orbitals; a click labels the HOMO and LUMO.

In BeH₂, the ligand SALCs are shown on the left and the Be 2s and 2p orbitals on the right.

Electrons: Be contributes two in its 2s, each hydrogen one
Symmetric (σ_g) SALC + Be 2s → bonding and antibonding MOs; antisymmetric (σ_u) SALC + the axial Be 2p → another bonding/antibonding pair
2p above 2s → its MOs sit higher in energy
The two perpendicular Be 2p: no matching SALCs → nonbonding, effectively atomic orbitals

[advance: label the HOMO and LUMO]

Frontier orbitals dominate a molecule’s behavior: HOMO = highest occupied, LUMO = lowest unoccupied
BeH₂: HOMO = the Be 2p–SALC bonding interaction; LUMO = the degenerate nonbonding Be 2p

Methane gets the same treatment in three dimensions.

CH₄ MO Diagram

Methane molecular orbital diagram with the A1 and T2 bonding levels and their antibonding counterparts built from carbon valence orbitals and hydrogen SALCs.

Delocalized π Bonding in Benzene

The six delocalized pi molecular orbitals of benzene ordered by energy, from all-bonding a2u to all-antibonding b2g.

Summary: Molecular Orbital Theory

Summary — Molecular Orbital Theory

Molecular orbitals (MOs) form from linear combinations of atomic orbitals (LCAO); electrons are described by delocalized wave functions over the entire molecule.
Bonding MOs arise from constructive interference (increased electron density between nuclei → stabilization).
Antibonding MOs arise from destructive interference (nodal plane between nuclei → destabilization).
Energy splitting between bonding and antibonding orbitals increases with orbital overlap and decreases with energy mismatch.
Polarity arises when orbitals of unequal energy mix, giving partial ionic character.
σ and π bonds differ by overlap geometry: σ bonds form along the internuclear axis, π bonds form side-on.
s–p mixing alters MO energy ordering, notably in lighter diatomics (C₂, N₂).
For polyatomic molecules, ligand orbitals combine into SALCs that mix with central-atom orbitals of matching symmetry.
HOMO and LUMO define the frontier orbitals controlling reactivity and optical transitions.
In conjugated π systems, delocalization spreads bonding over many atoms, producing evenly spaced energy levels (e.g., benzene).

Let's read back through the molecular-orbital summary.

LCAO: delocalized wave functions over the entire molecule
Bonding = constructive interference, density between the nuclei, stabilization; antibonding = destructive, a nodal plane, destabilization
Splitting grows with overlap, shrinks with energy mismatch; unequal-energy mixing → polarity, partial ionic character
σ along the internuclear axis, π side-on; s–p mixing reorders the levels, most notably in light diatomics like C₂ and N₂
Polyatomics: ligand orbitals combine into SALCs that mix with matching central-atom orbitals
HOMO and LUMO: frontier orbitals controlling reactivity and optical transitions; conjugated π systems spread bonding over many atoms → evenly spaced levels, as in benzene

That leaves the Lecture 12 practice questions.

Practice questions

Work the Lecture 12 practice questions before the next class. They cover orbital wavefunctions and the four quantum numbers, radial and angular nodes, orbital energies and sizes across the periodic table, and molecular orbital theory from LCAO and bonding versus antibonding through s–p mixing, SALCs, and frontier orbitals.

Open the Lecture 12 practice questions

Every question carries a worked explanation, so you can check your reasoning as you go.