CHEM 548: Materials Chemistry

Lecture 18: Band Structures of Transition Metal Oxides

Wed 11/04/2026 · Meeting 20

ReO₃ and perovskite band structures and periodic trends

Reading: Woodward, Karen, Evans, and Vogt, Solid State Materials Chemistry
Ch. 6

Learning Objectives: TM Oxide Band Structures

Band Structures of Transition Metal Oxides

Describe the crystal structureof ReO₃ and its relationship to perovskite structures

Construct band structure diagramsfrom molecular orbital diagrams for octahedral transition metal complexes

Identify and interpretoxygen 2p bands and metal d bands (π* and σ*) at special k-points

Extract key energetic parametersfrom band structures: octahedral ligand field splitting, charge transfer energy, and band gap

Predict periodic trendsin band structure when moving across the periodic table (changing electronegativity and covalency)

Analyze competing effectsof spatial overlap versus energetic overlap when moving down the periodic table (3d vs 4d vs 5d)

Electronic Band Structure

The calculated band structure of ReO3 plotted as energy against wave vector k through the special points Gamma, X, M, Gamma, R, X, with the Fermi level drawn as a dashed line.

An octahedron's molecular orbital diagram becomes bands running through the special points of the zone. By the end of the lecture we can read this ReO₃ band structure.

Per Solid State Materials Chemistry, §6.6.4, p.231.

Rhenium Trioxide

The ReO3 structure drawn as a block of eight corner-connected ReO6 octahedra with oxygen atoms at every shared corner.

The ReO3 unit cell: rhenium atoms at the corners of a cube and oxygen atoms at the centre of every edge.

The instructor's Re=O bonding sketch: rhenium double-bonded to each of three oxygens.

Substance	Conductivity (S/m)
Ag	62 × 10⁶
Cu	59 × 10⁶
Al	38 × 10⁶
ReO₃	11 × 10⁶
Ti	2.5 × 10⁶
Mn	0.62 × 10⁶

Per Solid State Materials Chemistry, §10.1, p.396.

Perovskites

The ReO3 unit cell: rhenium atoms at the corners of a cube, oxygen atoms at the centre of every edge, and nothing at the cell centre.

ReO₃ — Re at the corners, O at every edge centre, the cage at the cell centre empty.

The SrTiO3 perovskite unit cell: titanium at the corners, oxygen at the edge centres, and a large strontium cation filling the cuboctahedral cage at the cell centre.

SrTiO₃ — the same framework with a large Sr²⁺ filling that cage.

Treat the big cation as an electropositive electron donor — it bonds ionically and, in the limit, simply hands its electrons to the metal–oxide framework — and perovskites and ReO₃ are isostructural, so their band structures are the same problem.

Per Solid State Materials Chemistry, §6.6.5, p.233.

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, §6.4.2, p.219.

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 the instructor's dashed boxes marking the Re 5d bands and the O 2p bands.

O 2p SALCs18 of them — six oxygens × three 2p orbitals — splitting into six σ and twelve π combinations

Re valence orbitalsfive 5d, split by the octahedral field into t_2g and e_g, plus the higher 6s and 6p

Near the Fermi levelonly the nonbonding O 2p and the antibonding Re 5d states matter; the 2s orbitals are too deep to mix

Per Solid State Materials Chemistry, §6.6.4, p.231.

How Many Bands Are There?

orbitals in the unit cell	bands
1 Rhenium × 1 (6s orbitals)	1
1 Rhenium × 3 (6p orbitals)	3
1 Rhenium × 5 (5d orbitals)	5
3 Oxygen × 3 (2p orbitals)	9
3 Oxygen × 1 (2s orbitals)	3

Per Solid State Materials Chemistry, §6.6.4, p.231.

Band Structure of ReO₃

Figure 6.29: the calculated band structure of ReO3 along X, R, M, Gamma, R with the unit cell inset, bracketed as Re dx2-y2 and dz2 sigma-star bands, Re dxy, dxz and dyz pi-star bands, and O 2p bands, beside the density of states with the Fermi level drawn through the pi-star bands.

O 2p bandsnine bands (three oxygens × three 2p), all full: the eighteen oxygen electrons fill every one

Re t_2g (π*) bandsthree bands, and Re⁶⁺ is 5d¹, so they are one-sixth filled — the Fermi level lies inside them

Re e_g (σ*) bandstwo bands, empty, and pushed highest because σ antibonding beats π antibonding

Per Solid State Materials Chemistry, §6.6.4, p.231.

Orbital Overlap: Re 5d $e_g$ $\sigma^*$ Bands

Figure 6.29: the calculated band structure of ReO3 beside its density of states, with the Re sigma-star, Re pi-star and O 2p band families bracketed and the Fermi level marked.

At Γ

SSMC Figure 6.30: the Re 5d(x2-y2)-O sigma-star crystal orbital at Gamma.

no O 2p can mix; only O 2s reaches d_x²-y², so weakly antibonding

At M

SSMC Figure 6.30: the Re 5d(x2-y2)-O sigma-star crystal orbital at M.

O 2p overlaps it σ*, and the band climbs: strongly antibonding

Per Solid State Materials Chemistry, §6.6.4, p.231.

Let's look at the crystal orbitals at the special points, starting with the sigma star bands at the top, at gamma. As we saw for the copper oxygen layer, the oxygen 2p orbitals cannot mix with d x squared minus y squared at gamma. We do get a contribution from the oxygen 2s, so the d x squared minus y squared crystal orbital at gamma is weakly antibonding. The antibonding is weak because the oxygen 2s lies tens of electron volts below the oxygen 2p, so its energetic overlap with rhenium 5d is poor.

[advance: add the M point]

When we move to M the energy rises substantially, so the band is wide. At M the oxygen 2p orbitals, which have good energetic overlap with rhenium 5d, can interact with d x squared minus y squared, so the crystal orbital is sigma antibonding.

Orbital Overlap: Re 5d $t_{2g}$ $\pi^*$ Bands

At Γ

SSMC Figure 6.30: the Re 5d(xy)-O pi-star crystal orbital at Gamma.

d_xy matches no O 2p or 2s by symmetry: pure Re 5d, nonbonding, all three degenerate

At M

SSMC Figure 6.30: the Re 5d(xy)-O pi-star crystal orbital at M.

π overlap with O 2p switches on and the band rises: antibonding

Per Solid State Materials Chemistry, §6.6.4, p.231.

Orbital Overlap: O 2p Nonbonding Bands

The circled band is the top of the O 2p block, the states that stay mostly nonbonding.

At Γ

SSMC Figure 6.30: the O 2p (t1g) nonbonding crystal orbital at Gamma.

neither bonding nor antibonding between oxygens, and no Re s, p or d orbital has the symmetry to overlap

At M

SSMC Figure 6.30: the O 2p (t1g) nonbonding crystal orbital at M.

the phases flip along x and y but nothing changes; a very weak π overlap with Re 6p bends the band slightly

Per Solid State Materials Chemistry, §6.6.4, p.231.

Reading $\Delta$, CT and $E_\text{g}$ off the Bands

The generic d⁰ AMO₃ perovskite, so the brackets read M d; ReO₃'s own values come on the next slide.

Figure 6.31: the band structure of a generic d-zero AMO3 perovskite, annotated with the ligand field splitting Delta as a double arrow at R, the charge transfer energy CT at Gamma, and the band gap Eg with its dashed leader, beside the density of states.

Δ at Rt_2g and e_g are both fully antibonding at R, so their splitting is the octahedral ligand field splitting

CT at ΓO 2p and M d are both nonbonding at Γ, so the gap is the energy to move an electron from O 2p onto the metal: the charge transfer energy

E_g, R to Γhighest O 2p state to lowest M d state: in a d⁰ compound the R-to-Γ energy is the band gap, and it is indirect

Per Solid State Materials Chemistry, §6.6.5, p.233.

At R the t two g and e g orbitals are both fully antibonding, so their splitting is the octahedral ligand field splitting. At gamma the oxygen 2p states and the t two g states are both nonbonding, so the energy between them is almost a perfect charge transfer energy, the energy to move an electron from oxygen 2p onto the metal. Why do the sigma star e g bands not come down to that energy at gamma? They cannot mix with oxygen 2p at gamma, but they mix in an antibonding way with oxygen 2s, which raises them by several electron volts. The highest oxygen 2p state is at R and the lowest rhenium 5d state is at gamma. That energy is not a gap in rhenium trioxide, because the Fermi level cuts through the 5d bands, but in d zero strontium titanate it is the band gap. It is indirect because the two ends sit at different points.

The Periodic Table

A periodic table of the elements with hafnium, tantalum, tungsten and rhenium framed side by side in the 5d row.

The 5d series, in periodic table order:
Hafnium (group 4) · Tantalum (group 5) · Tungsten (group 6) · Rhenium (group 7)

Periodic Trends: Across the Period

	BaHfO₃	KTaO₃	WO₃	ReO₃
M–O distance (Å)*	2.09	1.99	1.95	1.87
e⁻ configuration	5d⁰	5d⁰	5d⁰	5d¹
σ* bandwidth (eV)	5.1	6.2	6.4	7.2
π* bandwidth (eV)	3.5	4.2	4.3	5.0
Δ (eV)	5.5	6.1	6.3	7.2
CT (eV)	5.3	3.8	3.4	2.3
Band gap (eV)*	5.5	3.5	2.4	Metal

*All parameters calculated except M–O distance and band gap, which are experimental values.

Moving left-to-right (Hf → Re) the effective nuclear charge increases

Metal-oxygen bond distancedecreases

Bandwidthincreases (more energetic and spatial overlap)

Δincreases

Charge transfer energydecreases

Band gapdecreases

Per Solid State Materials Chemistry, §6.6.5, p.233.

Here are the four cubic metal oxygen frameworks in periodic table order, barium hafnate, potassium tantalate, tungsten trioxide and rhenium trioxide. The metal is four plus in barium hafnate, five plus in potassium tantalate and six plus in tungsten trioxide and rhenium trioxide. Going from hafnium on the left to rhenium on the right, the effective nuclear charge goes up, and the metal oxygen bond gets shorter and more covalent. The antibonding crystal orbitals are more antibonding and the shorter bonds overlap more, so the sigma star and pi star bands get wider and the ligand field splitting goes up. The charge transfer energy goes down, because the metal is more electronegative. Hafnium oxygen bonds are much more ionic than rhenium oxygen bonds, so the band gap of the d zero compounds is largest at hafnium and smallest at tungsten, because the band gap is roughly an oxygen to metal charge transfer. Next we move down group four.

Periodic Trends: Down a Group

	SrTiO₃	BaHfO₃
M–O distance (Å)*	1.95	2.09
e⁻ configuration	3d⁰	5d⁰
σ* bandwidth (eV)	4.1	5.1
π* bandwidth (eV)	2.4	3.5
Δ (eV)	3.9	5.5
CT (eV)	4.1	5.3
Band gap (eV)*	3.1	5.5

*All parameters calculated except M–O distance and band gap, which are experimental values.

Moving down a column (Ti → Hf)

Metal-oxygen bond distanceincreases

Bandwidthincreases (more spatial overlap)

Δincreases (more spatial overlap)

Charge transferincreases

Band gapincreases

Per Solid State Materials Chemistry, §6.6.5, p.233.

Now we move down group four. We keep barium hafnate and add strontium titanate, since titanium and hafnium are both in group four, with zirconium between them. Down a group it is not obvious how the electronegativity changes, so let's look at the charge transfer energy. It goes from 4.1 electron volts in strontium titanate to 5.3 in barium hafnate, so titanium is more electronegative than hafnium, and its energetic overlap with oxygen is better. Yet the bands of barium hafnate are wider. As we saw in chapter five, 5d orbitals extend much farther from the nucleus than 3d orbitals, so hafnium has more spatial overlap with oxygen, and the extra spatial overlap is enough to widen the bands. Bandwidth and charge transfer energy will decide whether a perovskite has localized or delocalized electrons.

Check Your Understanding: Counting Bands

3. The unit cell of ReO₃ contains one Rhenium and three Oxygen atoms. How many bands are expected to arise from the Oxygen 2p orbitals?

A.6 bands

B.9 bands

C.3 bands

D.5 bands

Answer: B. 9 bands

There are three oxygen atoms, and each has three p-orbitals (p_x, p_y, p_z), leading to 3 × 3 = 9 bands.

Check Your Understanding: $t_{2g}$ Character at Γ

5. At the Gamma (Γ) point (k = 0,0,0) of the Brillouin zone, what is the bonding character of the Rhenium 5d t_2g (d_xy, d_xz, d_yz) orbitals?

A.Strongly sigma anti-bonding (σ*) with Oxygen 2p

B.Weakly anti-bonding with Oxygen 2s

C.Strongly pi anti-bonding (π*) with Oxygen 2p

D.Non-bonding, due to symmetry constraints with Oxygen 2p and 2s orbitals.

Answer: D. Non-bonding, due to symmetry constraints with Oxygen 2p and 2s orbitals.

The lecture explicitly states that at Gamma, the t_2g orbitals ‘cannot overlap at all’ with the oxygen orbitals, making them purely non-bonding.

Check Your Understanding: The $\sigma^*$ Band from Γ to M

6. Why does the energy of the σ* (e_g, d_x²-y²) band increase significantly when moving from the Gamma (Γ) point to the M point?

A.At M, the crystal field splitting energy (Δ_o) disappears, causing the band to rise.

B.The orbital becomes non-bonding at M.

C.At M, it hybridizes with the Rhenium 6s orbital, pushing it up in energy.

D.At M, it gains strong σ* anti-bonding character from interacting with Oxygen 2p orbitals.

Answer: D. At M, it gains strong σ* anti-bonding character from interacting with Oxygen 2p orbitals.

At Gamma, it can't mix with O 2p, but at M, symmetry allows strong σ* interaction with the O 2p orbitals (which have good energy overlap), raising the energy.

Practice questions

Work the Lecture 18 practice questions before the next class.

Open the Lecture 18 practice questions