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

Wave natureRecognize the wave nature of electrons and the structure of atomic orbitals.

Quantum numbersIdentify and interpret the four quantum numbers (n, l, m_l, m_s).

Pauli and HundApply Pauli’s exclusion principle and Hund’s rule to electron configurations.

NodesDistinguish between radial and angular nodes.

Orbital typeRelate orbital type (s, p, d, f) to shape, orientation, and nodal structure.

Orbital energyExplain how orbital energies vary with n, l, and atomic number.

Orbital Wavefunctions

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

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 numbers per Solid State Materials Chemistry, §5.2.

Quantum Mechanics and Orbitals

Two rules for how electrons fill orbitals, and the places where the wavefunction goes to zero.

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

The number of radial nodes is equal to n − ℓ − 1

These impact the energy of the orbital

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

Now let's square the wave function and ask where the electron is most likely to be found.

[advance: show the radial probability functions]

The radial probability function multiplies the squared wave function by 4πr² to give the probability of finding the electron 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

The number of nodal planes is equal to ℓ

These impact the shape of the orbital

A Closer Look at the Wavefunction

Orbital Shapes: p Orbitals

Nodal planes1 nodal plane

OrientationsThree different values of m_ℓ (1, 0, −1) each leading to a different orientation (p_x, p_y, p_z)

Orbital Shapes: d Orbitals

Nodal planes2 nodal planes

OrientationsFive different values of m_ℓ (2, 1, 0, −1, −2)

Orbitals and the Periodic Table

	1	2		3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
1	1H																		2He
2	3Li	4Be												5B	6C	7N	8O	9F	10Ne
3	11Na	12Mg												13Al	14Si	15P	16S	17Cl	18Ar
4	19K	20Ca		21Sc	22Ti	23V	24Cr	25Mn	26Fe	27Co	28Ni	29Cu	30Zn	31Ga	32Ge	33As	34Se	35Br	36Kr
5	37Rb	38Sr		39Y	40Zr	41Nb	42Mo	43Tc	44Ru	45Rh	46Pd	47Ag	48Cd	49In	50Sn	51Sb	52Te	53I	54Xe
6	55Cs	56Ba	57–70	71Lu	72Hf	73Ta	74W	75Re	76Os	77Ir	78Pt	79Au	80Hg	81Tl	82Pb	83Bi	84Po	85At	86Rn
7	87Fr	88Ra	89–102	103Lr	104Rf	105Db	106Sg	107Bh	108Hs	109Mt	110Ds	111Rg	112Cn	113Nh	114Fl	115Mc	116Lv	117Ts	118Og

				57La	58Ce	59Pr	60Nd	61Pm	62Sm	63Eu	64Gd	65Tb	66Dy	67Ho	68Er	69Tm	70Yb
				89Ac	90Th	91Pa	92U	93Np	94Pu	95Am	96Cm	97Bk	98Cf	99Es	100Fm	101Md	102No

s blockp blockd blockf block

Azimuthal Quantum Number and Orbital Energy

In a many-electron atom, 3s lies lowest and 3d highest: 3s < 3p < 3d.

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
Hydrogen, one electron: 3s, 3p, 3d same energy; the split needs core electrons to shield

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

Orbital Energies

Valence-orbital energies in eV (relativistic Dirac–Fock).

Across period 4 the 3d falls below the 4s and becomes a core level.

Across period 2 the 2s–2p gap grows from 5.0 eV at B to 29.3 eV at Ne.

Down group 14 the p levels off, but the s drops back down at germanium and again at lead.

Per Solid State Materials Chemistry, Table 5.2.

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

Calculated orbital energies: Dirac–Fock, isolated atoms
Fourth period, K to Kr: red 4s, green 3d
K, Ca: only 4s occupied, so 4s is lower
By Sc, Ti the 3d have dropped below 4s, then fall much faster
Gap ~20 eV at gallium, ~70 eV at krypton → filled 3d is a core level
Zinc: Zn²⁺ loses 4s only; the 3d stays put

Now look at how far the 2s sits below the 2p across the second period.

[advance: show the second-period 2s–2p gap]

The blue line is the 2p, the red line is the 2s, and the shaded band between them is the s–p gap.

Boron ~5 eV → fluorine ~23 eV → neon ~29 eV
Rising effective nuclear charge pulls 2s down faster than 2p
B, C: 2s and 2p both bond; F: 2s is core, largely out of bonding

Now follow the s and the p down group 14.

[advance: show the s and p energies down group 14]

This is group 14, carbon to lead: red is the s, blue is the p, plotted against period.

p levels off after Si, about −7 eV
s rises C → Si, then Ge 4s below Si 3s: 3d just filled, shields 4s poorly
Pb 6s ~1.5 eV deeper than Sn 5s: relativistic — inner electrons near light speed, mass up, s orbitals contract and drop
Tl and Bi, either side of Pb: same drop
Deep 6s → inert-pair effect
Group 14: +4 in Si, Ge, Sn → +2 in Pb; same for Tl, Bi

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

Orbital Sizes

Distances in Å where the radial distribution function is a maximum.

The first subshell of each shape, 1s, 2p, 3d and 4f, is compact.

Across period 2 the 2s and 2p are the same size.

From period 3 on, the p orbital is 11–34% larger than the s.

The 3d is the smallest d next to its s, and the 4f is smaller still.

Per Solid State Materials Chemistry, Table 5.3.

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

First subshell of each shape, 1s, 2p, 3d, 4f: unusually compact
No inner orbital of the same shape to shield it → stronger attraction, sits closer in

The second period shows this most clearly, so let's start there.

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

Each panel is one period, groups 13 to 18 across; the red solid line is the s, the blue dashed line is the p.

Second period: the lines lie on top of each other
2s and 2p almost the same size

Now compare that with the third, fourth, and fifth periods.

[advance: add periods 3 to 5]

From the third period on, the p line sits clearly above the s: the p orbital is 11 to 34 percent larger.

Period 2 is the exception: 2p anomalously small
No core p to shield, only s → strong attraction, 2p contracts
2s ≈ 2p → strong π overlap in carbon; Si 3p too large, overlaps poorly
π bonds everywhere in carbon, rare in silicon: rooted in this size anomaly → carbon-based life

Now the d and f orbitals, which have a size story of their own.

[advance: show the d and f sizes against s]

Each bar is the size of a d orbital divided by the size of the s orbital in the same atom: 3d against 4s, 4d against 5s, 5d against 6s.

Red 3d bar shortest in every group: a quarter to a third of 4s
3d small → overlaps neighbors weakly → stays partly filled in compounds
Partly filled 3d: Fe²⁺ / Fe³⁺, unpaired electrons → magnetism
4d, 5d: same effects, weaker
Gold bar: Ce 4f/6s ~1/6 → 4f almost core
Ln³⁺ = lose both 6s + one more: the 5d if present (La, Ce, Gd), else one 4f
Remaining 4f deep → a 4th electron costs more than bonding repays → +3
Exceptions Ce⁴⁺, Eu²⁺, Yb²⁺: empty, half, full 4f
4f barely touches neighbors → levels undisturbed → sharp absorption/emission colors
Eu red, Tb green phosphors

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

Summary: Atomic Orbitals

Electrons behave as standing waves described by wavefunctions (ψ).

Radial and angularWavefunctions separate into radial (distance-dependent) and angular (direction-dependent) parts.

Quantum numbersEach electron is defined by four quantum numbers (n, l, m_l, m_s).

Nodesoccur where ψ = 0: radial nodes = n − l − 1, angular nodes = l.

Orbital typess (spherical), p (two lobes), d (four lobes), f (complex, six lobes).

Pauli and HundPauli Exclusion and Hund’s Rule determine electron configurations.

Orbital energydepends on n, l, and electron shielding.

Relativistic effectsstabilize 6s orbitals in heavy elements, shaping their oxidation states.

Sizes and energiesOrbital sizes and energies explain periodic trends and bonding behavior.

Check Your Understanding: Nodes and Quantum Numbers

An orbital has two radial nodes and one angular node. Which orbital is it?

A3p

B3d

C4pCorrect

D4d

ENone of the above

Angular nodes = l, so one angular node gives l = 1, a p orbital. Radial nodes = n − l − 1, so 2 = n − 2 and n = 4.

Learning Objectives: Molecular Orbital Theory

MO formationExplain how atomic orbitals combine to form molecular orbitals.

Bonding classesDistinguish between bonding, antibonding, and nonbonding orbitals based on constructive and destructive interference.

MO diagramsInterpret simple MO diagrams (H₂, HeH⁺, O₂) and relate orbital overlap to bond order and stability.

Energy differencesDescribe how orbital energy differences influence covalent, polar covalent, and ionic bonding character.

σ and π overlapUnderstand σ vs. π interactions and how orbital symmetry governs overlap strength.

s–p mixingRecognize s–p mixing and its effect on molecular orbital energy ordering.

SALCsApply the concept of symmetry-adapted linear combinations (SALCs) to polyatomic molecules (e.g., BeH₂, CH₄).

HOMO and LUMODefine HOMO and LUMO and explain their importance in determining chemical reactivity.

Conjugated π systemsIdentify how conjugated π systems (e.g., benzene) generate delocalized bonding and characteristic energy levels.

Molecular Orbitals: LCAO

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

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

LCAOWe 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

antibonding MO

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

Now the other combination, the antibonding orbital.

[advance: circle the antibonding orbital]

We also get an antibonding molecular orbital, formed by destructive interference between the two hydrogen 1s wave functions.

Shading = sign of the wave function, dark blue positive, pale negative
Node between the atoms: zero density, electrons pushed out of the internuclear region → energy above the separate atoms

Now let me draw the same thing the way we will draw it from here on.

[advance: redraw the diagram in symbolic style]

We can redraw this using the symbolic style we’ll use going forward.

Same sign → constructive interference and a bonding MO; 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

If we plot the molecular orbital wavefunction along the internuclear axis, we see clear interference patterns.

$\psi_+ = 0.55(\psi_1 + \psi_2)$ $\psi_- = 1.17(\psi_1 - \psi_2)$

Bonding MOConstructive interference · Increases e⁻ between nuclei · Lower energy

Antibonding MODestructive interference · Nodal plane between nuclei · Higher energy

MO Diagram: HHe

stabilization only ~1.8 eV

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

There is one more thing to notice about how far the bonding orbital actually drops.

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

Another key point is that the stabilization of the bonding molecular orbital decreases when the atomic orbitals differ in energy.

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

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

Bottom of the diagram first: the 2s block
The two 2s overlap → σ(2s) bonding and σ*(2s) antibonding, like H₂
Two 2s electrons per O → both full → no net bonding, as in H₂

Now the 2p orbitals, which sit well above the 2s.

[advance: bring up the O 2p levels]

The 2p orbitals are considerably higher in energy than the 2s orbitals, as we saw when we ordered the orbital energies.

z = internuclear axis: p_z points at the other O; p_x, p_y perpendicular to the bond
Four 2p electrons per O: a pair in p_y, one each in p_z and p_x

First the pair that overlaps head-on, the two p_z orbitals.

[advance: bring up the head-on sigma pair]

Along the internuclear (z) axis, the p_z orbitals overlap head-on, forming σ (sigma) bonding and antibonding MOs.

In phase: inner lobes reinforce between the nuclei → σ(2p)
Out of phase: inner lobes cancel between the nuclei → σ*(2p)
Head-on overlap is large → the two levels split far apart

The other four 2p orbitals meet side-on.

[advance: bring up the side-on pi pairs]

The p_x and p_y orbitals overlap side-on, forming π (pi) bonding and antibonding MOs with less overlap.

p_x and p_y give identical pairs → π(2p) and π*(2p) each two levels at one energy
Six 2p MOs split 1–2–2–1: σ(2p), two π, two π*, σ*(2p)
Greater spatial overlap → larger bonding–antibonding gap → σ stronger than π
Fill 12 valence electrons from the bottom → last two one per π*, same spin → two unpaired electrons
The true diagram also needs σ–σ mixing of 2s and 2p, which slightly alters energies and ordering

Let's take that mixing on its own first.

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

When we introduce s–p mixing, the 2s and 2p σ bonding orbitals interact.

Bonding orbitals first: the bottom row
Adding the wave functions constructively → more density between the nuclei → stabilizes
Mostly 2s → σ(2s) more bonding, E down

Now the same two orbitals combined with the opposite sign.

[advance: subtract the sigma pair]

Subtracting them (changing the sign of one orbital) reduces density between the nuclei and increases it on the molecular periphery, raising the energy.

Subtracting = changing the shading on one orbital
Mostly 2p → σ(2p) less bonding, E up
Mixing stabilizes the orbital with more 2s character, destabilizes the one with more 2p

The same pattern occurs for the corresponding σ antibonding orbitals.

[advance: add the sigma-star pair]

For the antibonding pair, adding σ*(2s) to σ*(2p) shrinks the two inner lobes on either side of the node.

Less density against the node → less antibonding
Mostly 2s → σ*(2s) stabilized, E down

Subtracting does the opposite.

[advance: subtract the sigma-star pair]

Subtracting grows the two inner lobes, so the combination is more antibonding.

Mostly 2p → σ*(2p) destabilized, E up

With that in hand, we can put the mixing back into the O₂ diagram.

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

Returning to the MO diagram, the dashed lines mark where each σ level sat without mixing, and the arrows show how far it moves.

1σ gains p character → stabilized, E down; 2σ gains s character → destabilized, E up
The antibonding pair shifts the same way: 1σ* drops slightly, 2σ* rises
π and π* stay put: the s orbitals have the wrong symmetry to mix with them
Net effect: smaller 2σ–π gap; in lighter diatomics like N₂ or C₂, mixing lifts 2σ above the π

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

Beyond Diatomics: SALCs

s p_x p_y p_z

Central Atom

Linear (BeH₂)

Tetrahedral (CH₄)

Octahedral (SH₆)

0 nodal planes 1 nodal plane 2 nodal planes

Treat the ligand orbitals as symmetry adapted linear combinations (SALCs)

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

Start with the simplest case, linear BeH₂.

[advance: add the linear BeH₂ SALCs]

The symmetric SALC can overlap with the Be 2s orbital, while the antisymmetric SALC aligns with the Be 2pₓ orbital.

The remaining 2p_y, 2p_z: no net overlap, no interaction
Formally a group-theory result, here evident by visual inspection of orbital symmetry

Next is the tetrahedral case, methane.

[advance: add the tetrahedral CH₄ SALCs]

In a tetrahedral molecule like methane (CH₄), the four hydrogen atoms form four ligand SALCs.

Each combination matches the symmetry of one of carbon’s valence orbitals
Totally symmetric SALC → carbon 2s; the other three → the three 2p → four equivalent σ bonds, tetrahedral

And one more, an octahedral molecule with six ligands around the central atom.

[advance: add the octahedral SH₆ SALCs]

In an octahedral molecule like hypothetical S–H₆, six ligands each contribute one orbital, forming six ligand SALCs.

Their shapes require group theory, but once identified: the first matches the sulfur s, the next three the sulfur p
The remaining two have no matching symmetry with sulfur’s valence orbitals → nonbonding

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

BeH₂: MO Diagram and Frontier Orbitals

HOMO

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

Two of these orbitals matter more than the rest, so let's name them.

[advance: label the HOMO and LUMO]

In an MO diagram, certain orbitals dominate a molecule’s behavior—these are the frontier orbitals.

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

Delocalized π Bonding in Benzene

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 MOsarise from constructive interference (increased electron density between nuclei → stabilization).

Antibonding MOsarise from destructive interference (nodal plane between nuclei → destabilization).

Energy splittingbetween bonding and antibonding orbitals increases with orbital overlap and decreases with energy mismatch.

Polarityarises when orbitals of unequal energy mix, giving partial ionic character.

σ and π bondsdiffer by overlap geometry: σ bonds form along the internuclear axis, π bonds form side-on.

s–p mixingalters MO energy ordering, notably in lighter diatomics (C₂, N₂).

SALCsFor polyatomic molecules, ligand orbitals combine into SALCs that mix with central-atom orbitals of matching symmetry.

HOMO and LUMOdefine the frontier orbitals controlling reactivity and optical transitions.

Conjugated π systemsIn 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.

Open the Lecture 12 practice questions