CHEM 548: Materials Chemistry

Lecture 18: Band Structures of Transition Metal Oxides

Wed 11/04/2026 · Meeting 20

ReO₃ and perovskite band structures, periodic trends, and an introduction to electrical conductivity

Reading: Woodward, Karen, Evans, and Vogt, Solid State Materials Chemistry
Ch. 6; Ch. 10 §10.1

Learning Objectives: TM Oxide Band Structures

Band Structures of Transition Metal Oxides

Describe the crystal structure of ReO₃ and its relationship to perovskite structures
Construct band structure diagrams from molecular orbital diagrams for octahedral transition metal complexes
Identify and interpret oxygen 2p bands and metal d bands (π* and σ*) at special k-points
Extract key energetic parameters from band structures: octahedral ligand field splitting, charge transfer energy, and band gap
Predict periodic trends in band structure when moving across the periodic table (changing electronegativity and covalency)
Analyze the competing effects of spatial overlap versus energetic overlap when moving down the periodic table (3d vs 4d vs 5d)

Electronic Band Structures | Materials Science & Chemistry

Learning Objectives Band Structures of Transition Metal Oxides Describe the crystal structure of ReO₃ and its relationship to perovskite structures Construct band structure diagrams from molecular orbital diagrams for octahedral transition metal complexes Identify and interpret oxygen 2p bands and metal d bands (π* and σ*) at special k-points Extract key energetic parameters from band structures: octahedral ligand field splitting, charge transfer energy, and band gap Predict periodic trends in band structure when moving across the periodic table (changing electronegativity and covalency) Analyze the competing effects of spatial overlap versus energetic overlap when moving down the periodic table (3d vs 4d vs 5d) Electronic Band Structures | Materials Science & Chemistry

Let's start by looking back at where this chapter began.

Electronic Band Structure

The molecular orbital diagram for an octahedron next to the calculated band structure diagram for ReO3.

Rhenium Trioxide

The ReO3 structure as corner-connected octahedra and as a unit cell drawing, the Re=O bonding sketch, and a table comparing the conductivity of ReO3 with Ag, Cu, Al, Ti, and Mn.

Perovskites

The ReO3 unit cell next to the SrTiO3 perovskite unit cell, which adds a large Sr cation at the cell center; treating the large cation as an electron donor makes the two isostructural.

First Brillouin Zone – Primitive Cubic

The first Brillouin zone of a primitive cubic lattice drawn as a cube, with the special points Gamma at the centre, X along kx, M in the kx-ky plane, and R at the cube corner.

label	coordinates
Γ	0a* + 0b* + 0c*
X	(1/2)a* + 0b* + 0c*
M	(1/2)a* + (1/2)b* + 0c*
R	(1/2)a* + (1/2)b* + (1/2)c*

label	wave vector (Cartesian)
Γ	0k_x + 0k_y + 0k_z
X	(π/a)k_x + 0k_y + 0k_z
M	(π/a)k_x + (π/a)k_y + 0k_z
R	(π/a)k_x + (π/a)k_y + (π/a)k_z

The zone boundary lies at half of each reciprocal lattice vector, where $k = \pi/a$. Per Solid State Materials Chemistry, §6.4.

MO Diagram for the ReO₆⁶⁻ Octahedron

The molecular orbital diagram for a ReO6 octahedron with the oxygen 2p SALCs on the left, the rhenium orbitals on the right, and boxes marking the Re 5d bands and the O 2p bands.

Here's the molecular orbital diagram for a rhenium-centered octahedron (Chapter 6).

Neglecting oxygen 2s orbitals (because of energy mismatch): 18 oxygen 2p SALCs (6×3) and five rhenium d-orbitals, plus higher-lying 6s and 6p
Lowest to highest: rhenium-oxygen bonding MOs, then non-bonding oxygen 2p MOs (oxygen SALCs lacking rhenium partners)
Then antibonding rhenium 5d–oxygen 2p states, split by the octahedral crystal field: triply degenerate t2g (π antibonding) and doubly degenerate eg (σ antibonding), and finally antibonding 6s and 6p states
We'll focus near the Fermi level (near HOMO/LUMO): the oxygen 2p and rhenium 5d bands
Note the molecular orbital diagram doesn't reliably predict band count: the Re:O ratio differs between an isolated octahedron and the extended solid, where oxygens are shared between octahedra

Before reading the band structure, let's count how many bands we should expect.

How Many Bands Are There?

Counting bands in ReO3: one rhenium contributes one 6s, three 6p, and five 5d orbitals; three oxygens contribute nine 2p and three 2s orbitals; the 5d and 2p rows are highlighted.

Band Structure of ReO₃

The calculated band structure of ReO3 along X, R, M, Gamma, R with the ReO3 unit cell inset, labeled Re eg sigma-star bands, Re t2g pi-star bands, and O 2p bands, next to the density of states with the Fermi level in the pi-star bands.

The calculated band structure shows nine oxygen 2p bands at the bottom (three oxygens × three p orbitals each).

High-energy regions: oxygen 2p non-bonding states; lower-energy regions reflect bonding with rhenium (bandwidth ~8 eV)
Three bands with rhenium 5d parentage are the t2g bands (from dxy, dxz, dyz) — called the π* bands
Re6+ is d1, so these bands are one-sixth filled, placing the Fermi level within them
Highest energy: more rhenium 5d bands from dx²-y² and dz² — the eg set, σ antibonding
Each oxygen has 6 valence electrons (2s² 2p⁴); three oxygens give 18 electrons, completely filling the nine oxygen 2p-derived bands (each band holds 2 electrons with spin)
The Re6+ ion (d¹) contributes just 1 additional electron; the oxygen 2p bands are already full, so it must occupy the lowest unoccupied states — the t2g (π*) bands

Now let's examine crystal orbitals at various special points.

Orbital Overlap: Re 5d (e_g) σ* Bands

The ReO3 band structure with the sigma-star band circled at Gamma and the weakly antibonding crystal orbital drawn; a click adds the strongly antibonding crystal orbital at M, and a second click labels the dx2-y2 and dz2 bands on the plot.

Starting with the σ* bands (highest energy), at Γ the dx²-y² orbital overlaps with oxygen.

As discussed for the copper-oxygen layer, the two oxygen p orbitals cannot mix with dx²-y² at Γ (net non-bonding)
However, oxygen 2s does contribute (shown here), making this crystal orbital weakly antibonding

[advance: add the M point]

Weak because oxygen 2s lies tens of eV below oxygen 2p and has poor energetic overlap with rhenium 5d
Moving to the M point, energy increases substantially (wide bandwidth); the crystal orbital is now σ antibonding
This large energy increase occurs because at M, oxygen 2p orbitals (which have good energetic overlap with rhenium 5d) can now interact in a σ* fashion, unlike at Γ

[advance: label the two sigma-star bands]

This analysis identifies the discussed band as dx²-y², making the other σ* band dz² by default; they're degenerate at Γ and at R
If these orbital overlap arguments seem fast, review the copper oxide layers lecture — the CuO2 layer overlaps are nearly identical to what occurs here in the xy plane

Now let's do the same analysis for the t two g set.

Orbital Overlap: Re 5d (t_2g) π* Bands

The ReO3 band structure with the t2g pi-star band marked at Gamma and its nonbonding crystal orbital drawn; a click adds the antibonding crystal orbital at the M point.

Orbital Overlap: O 2p Nonbonding Bands

The ReO3 band structure with the highest oxygen 2p bands marked and the nonbonding oxygen crystal orbitals drawn at Gamma and at M.

ReO₃ Band Structure – Key Features

The ReO3 band structure annotated with the octahedral ligand field splitting Delta at R, the charge transfer energy CT at Gamma, and the band gap Eg between the highest O 2p state and the lowest Re 5d state, next to the density of states.

This perovskite band structure reveals key bonding information through specific energy differences:At R, where both t2g and eg orbitals are entirely antibonding with oxygen, their energy splitting equals the octahedral ligand field splitting (Δo) from the molecular orbital diagram.

At Γ, where both oxygen 2p and rhenium 5d states are non-bonding, their energy difference approximates the charge transfer energy from oxygen to rhenium
The σ* bands don't drop to this non-bonding level: antibonding interaction with oxygen 2s raises energy by several eV
The energy difference between the highest oxygen 2p state (at R) and the lowest rhenium 5d state (at Γ) becomes the band gap in d0 compounds like SrTiO3
This band gap is inherently indirect: the valence band maximum and conduction band minimum occur at different k-points

Those three parameters are exactly what we will track as we move around the periodic table.

Periodic Trends: Across the Period

A table comparing ReO3, WO3, KTaO3, and BaHfO3: metal-oxygen distance, electron configuration, sigma-star and pi-star bandwidths, ligand field splitting, charge transfer energy, and band gap; moving from Re to Hf the bandwidths and splitting decrease while charge transfer energy and band gap increase.

Examining four cubic metal-oxygen frameworks moving right to left across the periodic table—Re, W, Ta, Hf with charges 6+, 6+, 5+, 4+ respectively (ReO3, WO3, KTaO3, BaHfO3)—reveals periodic trends.

Band structures are qualitatively similar but differ quantitatively: changing metal electronegativity and bond distances
As effective nuclear charge decreases: bond distances increase, covalency decreases (more ionic character)
Effect (1): antibonding orbitals become less antibonding — narrowing σ* and π* bandwidths; effect (2): longer bonds reduce spatial overlap
The octahedral ligand field splitting (Δo) also decreases
Charge transfer energy increases with decreasing metal electronegativity (Re more electronegative than Hf); Re to Hf, bonds become substantially more ionic (Hf more positively charged)
This increased ionicity enlarges the band gap in d0 systems, since the band gap represents an oxygen-to-metal charge transfer

Let's look at where these four metals sit on the periodic table.

The Periodic Table

Periodic Trends: Down a Group

A table comparing SrTiO3 and BaHfO3: moving down the column from Ti to Hf the metal-oxygen distance, bandwidths, ligand field splitting, charge transfer energy, and band gap all increase, with the bandwidth and splitting increases highlighted in red.

Moving vertically in the periodic table: comparing SrTiO3 with BaHfO3 (Ti and Hf are both group 4).

Charge transfer energy increases from 4.1 eV (SrTiO3) to 5.3 eV (BaHfO3): titanium is more electronegative than hafnium — better energetic overlap between Ti and O
However, BaHfO3 shows larger bandwidths despite poorer energetic overlap
This occurs because 5d orbitals (Hf) extend farther from the nucleus than 3d orbitals (Ti) — greater spatial overlap between metal and oxygen
Competing trends: energetic overlap favors Ti, spatial overlap favors Hf
These competing factors — bandwidth and charge transfer energy — will be crucial for determining whether perovskites exhibit localized or delocalized electrons when discussing electrical properties

Let's check your understanding with a few quick questions.

Check Your Understanding: Counting Bands

Question 3 asking how many bands arise from the oxygen 2p orbitals in the ReO3 unit cell, then its answer: nine bands, three oxygens times three p orbitals.

Check Your Understanding: t_2g Character at Γ

Question 5 asking for the bonding character of the rhenium 5d t2g orbitals at the Gamma point, then its answer: non-bonding, due to symmetry constraints with the oxygen 2p and 2s orbitals.

Check Your Understanding: The σ* Band from Γ to M

Question 6 asking why the sigma-star eg band rises significantly from Gamma to M, then its answer: at M it gains strong sigma antibonding character from interacting with oxygen 2p orbitals.

Question 6. Why does the energy of the sigma star, e g, d x squared minus y squared, band increase significantly when moving from the gamma point to the M point? Option A: at M, the crystal field splitting energy disappears, causing the band to rise. Option B: the orbital becomes non-bonding at M. Option C: at M, it hybridizes with the rhenium 6s orbital, pushing it up in energy. Option D: at M, it gains strong sigma star antibonding character from interacting with oxygen 2p orbitals. Take a moment and think about it.

[advance: reveal the answer]

The answer is D
At gamma it can't mix with oxygen 2p; at M, symmetry allows a strong sigma star interaction with the oxygen 2p orbitals, which have good energy overlap, raising the energy

Question 7 asks what the band structure tells us at the R point.

Check Your Understanding: Reading Δ_o from the Bands

Question 7 asking which chemical parameter equals the t2g to eg splitting at the R point of the perovskite band structure, then its answer: the octahedral ligand field splitting energy.

Check Your Understanding: Periodic Trends

Question 9 asking what happens to the d-band widths and the charge transfer energy as metal-oxygen bonds become more ionic moving left across the periodic table, then its answer: bandwidths decrease and charge transfer energy increases.

Question 9. When moving left across the periodic table, for example from rhenium to hafnium, the metal-oxygen bonds become more ionic. What effect does this have on the d-band widths and the charge transfer energy? Option A: bandwidths decrease and charge transfer energy increases. Option B: bandwidths increase and charge transfer energy decreases. Option C: bandwidths increase and charge transfer energy increases. Option D: bandwidths decrease and charge transfer energy decreases. Take a moment and think about it.

[advance: reveal the answer]

The answer is A
Less covalency narrows the bands; the less electronegative metal, hafnium, raises the d-orbital energy, increasing the gap from oxygen 2p to the metal 5d

With that trend question our band structure story is complete, so now we change topics and start on electrical conductivity.

Learning Objectives: Conductivity and the Drude Model

Electrical Conductivity and the Drude Model

Distinguish between extrinsic (I, V, R) and intrinsic (J, E, σ, ρ) electrical properties
Apply the Drude model to describe electron behavior as an ideal gas in metals
Calculate key parameters: drift velocity, electron mobility, mean free path, and relaxation time
Derive the relationship between conductivity, carrier concentration, and mobility (σ = neμ)
Estimate conductivity parameters for simple metals using the Drude model
Recognize the limitations and failures of the Drude model in predicting temperature dependence and trends across different metals

Electrical Properties of Materials | Materials Science & Chemistry

Ohm's Law

Conductivity

Conductivity of Materials

A logarithmic scale of resistivity and conductivity spanning 28 orders of magnitude, with metals at high conductivity, semiconductors intermediate, and insulators at low conductivity.

Conductivity of Select Materials

A table of conductivities for selected materials, from silver and copper near 6 times 10 to the 7 siemens per meter, through ReO3, SrMoO3, NbN, and doped polyacetylene, down to NiO, Al2O3, SiO2, and Teflon at 10 to the minus 22.

In the upper left, we have elemental metals with the highest conductivity, often used as conductors.

Silver and copper ~6×10^7 S/m; silver is actually the most conductive element at room temperature; aluminum isn't far behind; also rhenium trioxide with quite high conductivity, other oxides, and a nitride like niobium nitride
What makes silver so much more conductive than metals like titanium or lanthanum? It's over an order of magnitude more conducting
What makes some oxides like rhenium trioxide and strontium molybdates so highly conducting, whereas others like SiO2, aluminum oxide, and nickel oxide are highly insulating? You might say for SiO2 and aluminum oxide, all electrons are tied up in covalent bonds with no partially filled bands — and that's true
But nickel oxide: rock salt structure, Ni2+ is d8, partially filled d orbitals; titanium oxide is also rock salt with Ti2+ (d2), yet there's 13 orders of magnitude difference in their conductivity. What explains that?
Semiconductors like silicon and germanium: silicon's intrinsic conductivity isn't very high, yet a hugely important industry is based on it — how do we understand semiconductor conductivity?
Organics: Teflon, unsurprisingly a terrible conductor; but doped polyacetylene has appreciable conductivity, not far behind some metals — how do we get polymers and organic molecules to conduct electricity? These are questions we want to answer by the end of this chapter

That's where we will pick up next time, and it completes today's lecture on the band structures of transition metal oxides.

Practice questions

Work the Lecture 18 practice questions before the next class. They cover the ReO₃ and perovskite structures, the primitive-cubic Brillouin zone, band counting and orbital overlap in the ReO₃ band structure, the ligand field splitting, charge transfer energy and band gap read off it, the periodic trends across and down the table, and the intrinsic description of electrical conductivity.

Open the Lecture 18 practice questions

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