Tetrahedral and square-planar crystal fields: Tanabe-Sugano diagrams and d-d transitions

Intuition Beginner

In unit 16.03.02 we saw how six ligands in an octahedron split the five d-orbitals into a higher pair ($e_g$) and a lower trio ($t_{2g}$). But not every complex is octahedral. Two other geometries matter enormously: tetrahedral (four ligands at the corners of a tetrahedron) and square planar (four ligands in one plane).

A tetrahedral complex inverts the octahedral splitting. The doubly degenerate $e$ set ($d_{z^2}$, $d_{x^2-y^2}$) drops to lower energy and the triply degenerate $t_2$ set ($d_{xy}$, $d_{xz}$, $d_{yz}$) rises. The gap ${\Delta }_t$ is smaller — roughly 4/9 of the octahedral splitting ${\Delta }_o$ for the same metal and ligand. Because the gap is so small, tetrahedral complexes are almost always high-spin.

A square-planar complex is what you get by removing the two axial ligands from an octahedron. The $d_{x^2-y^2}$ orbital, which points directly at the four remaining in-plane ligands, shoots up to become the highest-energy orbital. This makes square-planar geometry especially favourable for d${}^8$ metals like Pt${}^{2+}$ and Au${}^{3+}$, where the destabilising $d_{x^2-y^2}$ orbital stays empty.

Tanabe-Sugano diagrams are the tool that ties everything together. Each diagram plots the energies of all electronic states of a d${}^n$ configuration against the crystal field strength. You read off transition energies as vertical lines at the appropriate field strength, directly predicting the positions of d-d absorption bands in the UV-Vis spectrum.

Visual Beginner

Worked example Beginner

Consider the tetrahedral complex $[{\mathrm{CoCl}}_4{]}^{2-}$. Cobalt(II) is d${}^7$. In a tetrahedral field the splitting is inverted ($e$ below $t_2$) and small (${\Delta }_t\approx 3300$ cm${}^{-1}$). Because ${\Delta }_t$ is much smaller than the pairing energy, the complex is high-spin.

Fill the tetrahedral levels: the lower $e$ pair gets two electrons (one each, parallel spins), and the upper $t_2$ trio gets five electrons. By Hund's rule, each of the three $t_2$ orbitals gets one electron first, then the remaining two pair up. The result is three unpaired electrons. The deep blue colour of $[{\mathrm{CoCl}}_4{]}^{2-}$ comes from d-d transitions within the $t_2$ set and from $e\to t_2$ excitations.

Now compare with the square-planar complex $[{\mathrm{PtCl}}_4{]}^{2-}$. Platinum(II) is also d${}^8$. In square-planar geometry the $d_{x^2-y^2}$ orbital sits very high — so high that all eight d-electrons fill the four lower orbitals completely. The complex is diamagnetic (zero unpaired electrons) and pale yellow, absorbing in the near-UV.

Check your understanding Beginner

Formal definition Intermediate+

Tetrahedral splitting. For four identical ligands at the vertices of a regular tetrahedron, the point group is $T_d$ (order 24, no inversion centre). The five d-orbitals decompose under $T_d$ as

$$V_d=e\oplus t_2,$$

where $e=\{d_{z^2},d_{x^2-y^2}\}$ (doubly degenerate, dimension 2) and $t_2=\{d_{xy},d_{xz},d_{yz}\}$ (triply degenerate, dimension 3). Unlike $O_h$, the $g$ subscript is dropped because $T_d$ has no centre of inversion. The splitting is inverted relative to octahedral: $e$ sits lower, $t_2$ sits higher. The tetrahedral splitting parameter is

$${\Delta }_t=E(t_2)-E(e)=\frac{4}{9}{\Delta }_o$$

for the same metal ion and ligand type. The $4/9$ factor arises from two geometric contributions: (a) four ligands instead of six ($2/3$ factor) and (b) no ligand sits directly on a Cartesian axis (additional $2/3$ factor from angular averaging of orbital-ligand overlap), giving $\frac{2}{3}\times \frac{2}{3}=\frac{4}{9}$.

The barycentre rule applies: $2E(e)+3E(t_2)=0$, yielding $E(e)=-\frac{3}{5}{\Delta }_t$ and $E(t_2)=+\frac{2}{5}{\Delta }_t$.

Square-planar splitting. The square-planar geometry ($D_{4h}$ point group, order 16) is derived from octahedral by removing two axial ligands. The five d-orbitals split into four energy levels:

$$d_{x^2-y^2}(b_{1g})\gg d_{xy}(b_{2g})>d_{z^2}(a_{1g})>d_{xz},d_{yz}(e_g).$$

The $d_{x^2-y^2}$ orbital ($b_{1g}$) is far above the others because its lobes point directly at all four equatorial ligands. The energy gap between $b_{1g}$ and $b_{2g}$ is approximately $1.3{\Delta }_o$, exceeding the original octahedral splitting because the in-plane ligands move closer when axial ligands are removed.

Spinels and tetrahedral vs octahedral site preference. In spinel oxides ${\mathrm{AB}}_2{\mathrm{O}}_4$, metal ions occupy both tetrahedral and octahedral sites. The site preference is governed by CFSE: an ion with large octahedral CFSE (d${}^3$: Cr${}^{3+}$, or low-spin d${}^6$: Co${}^{3+}$) prefers octahedral sites; an ion with zero or negative octahedral-site preference (d${}^0$, d${}^5$ HS, d${}^{10}$) accepts tetrahedral sites. A normal spinel has A on tetrahedral sites and B on octahedral sites (e.g., ${\mathrm{MgAl}}_2{\mathrm{O}}_4$). An inverse spinel has half the B ions on tetrahedral sites and A on octahedral sites (e.g., ${\mathrm{Fe}}_3{\mathrm{O}}_4={\mathrm{Fe}}^{3+}[{\mathrm{Fe}}^{2+}{\mathrm{Fe}}^{3+}]{\mathrm{O}}_4$). The inversion is driven by CFSE: Fe${}^{3+}$ (d${}^5$, zero CFSE in either site) has no preference, so Fe${}^{2+}$ (d${}^6$, gains CFSE on the octahedral site) occupies the octahedral positions.

Counterexamples to common slips

• ${\Delta }_t=\frac{4}{9}{\Delta }_o$ is an approximation, not an exact relation. The $4/9$ factor assumes identical metal-ligand distances in both geometries. In practice, tetrahedral complexes often have shorter M-L bonds (four ligands repel each other less than six), which can increase ${\Delta }_t$ somewhat. The qualitative point — tetrahedral splitting is roughly half the octahedral splitting — is robust.

• Square-planar is not simply "flat tetrahedral." The two geometries have different point groups ($D_{4h}$ vs $T_d$), different character tables, and different orbital-ordering patterns. A square-planar complex can be understood as the limit of progressive tetragonal elongation of an octahedron, not as a distorted tetrahedron.

• Tanabe-Sugano diagrams are configuration-specific. There is a different diagram for each d${}^n$ configuration ($d^2$ through $d^8$). You cannot use the $d^3$ diagram to analyse a $d^6$ complex. Each diagram encodes the full set of term energies for that one configuration as a function of ${\Delta }_o/B$.

Key result Intermediate+

Result (Tetrahedral splitting ratio). For a transition-metal ion with a given set of identical ligands, the tetrahedral crystal field splitting is related to the octahedral splitting by ${\Delta }_t=\frac{4}{9}{\Delta }_o$. This relation follows from the geometry of the ligand positions relative to the d-orbital lobes and holds to first order in the point-charge model.

The derivation proceeds by computing the matrix elements of the ligand-field potential in $T_d$ symmetry, analogously to the $O_h$ calculation of unit 16.03.02 but with four ligands at the tetrahedral positions $(a,a,a)$, $(a,-a,-a)$, $(-a,a,-a)$, $(-a,-a,a)$ rather than six on the Cartesian axes. The potential has lower symmetry and fewer ligands; evaluating the crystal field integrals with the $T_d$ character table confirms the $e\oplus t_2$ decomposition and the inverted ordering. The numerical $4/9$ ratio emerges from the angular-overlap integrals.

Corollary (Tetrahedral high-spin rule). Because ${\Delta }_t<P$ for all realistic first-row transition-metal complexes, tetrahedral complexes are always high-spin for d${}^4$ through d${}^7$ configurations.

Worked example: interpreting the spectrum of $[{\mathrm{CoCl}}_4{]}^{2-}$

Cobalt(II) is d${}^7$. In tetrahedral geometry the ground state is ${}^4{A}_2$ (from the ${}^{4}F$ free-ion term). The spin-allowed d-d transitions are:

$${}^4{A}_2\to {}^4{T}_2,{}^4{A}_2\to {}^4{T}_1(F),{}^4{A}_2\to {}^4{T}_1(P).$$

Observed bands for $[{\mathrm{CoCl}}_4{]}^{2-}$: approximately 3500 cm${}^{-1}$ (near-IR, ${}^4{A}_2\to {}^4{T}_2$), 7500 cm${}^{-1}$ (near-IR, ${}^4{A}_2\to {}^4{T}_1(F)$), and 15,000 cm${}^{-1}$ (visible, ${}^4{A}_2\to {}^4{T}_1(P)$, giving the deep blue colour). The lowest transition gives ${\Delta }_t\approx 3500$ cm${}^{-1}$ directly, consistent with $\frac{4}{9}\times {\Delta }_o$ for the corresponding octahedral complex $[\mathrm{Co}({\mathrm{H}}_2\mathrm{O}{)}_6{]}^{2+}$ (${\Delta }_o\approx 9300$ cm${}^{-1}$, giving $\frac{4}{9}\times 9300\approx 4100$ cm${}^{-1}$, within the expected approximation error).

Exercises Intermediate+

Tanabe-Sugano diagrams: construction and interpretation Master

A Tanabe-Sugano diagram for a d${}^n$ configuration plots the energy of every electronic term (in units of the Racah parameter $B$, as $E/B$) against the crystal field strength (${\Delta }_o/B$). The ground state is always plotted as the horizontal axis ($E/B=0$). These diagrams were introduced by Tanabe and Sugano in 1954 ^{[Miessler Ch. 8]} and remain the primary tool for assigning d-d spectra.

Construction. The free-ion terms for each d${}^n$ configuration are derived from Russell-Saunders coupling: the direct products of $n$ spin-orbitals are decomposed into antisymmetrised linear combinations classified by total $L$ and $S$, using the vector coupling scheme and the Pauli principle. Each free-ion term ${}^{2S+1}L$ then splits in $O_h$ (or $T_d$) by subduction of $D^{(L)}$ from SO(3) to the point group, as developed in unit 16.03.02. The resulting sub-terms have point-group symmetry labels ($A_{2g}$, $T_{2g}$, etc.) and multiplicities equal to the dimension of the irrep.

The energies of all sub-terms are then computed as functions of $\Delta /B$ by diagonalising the full d-electron Hamiltonian — including the inter-electron Coulomb repulsion (parameterised by Racah parameters $B$ and $C$) and the crystal field potential (parameterised by $\Delta$) — in the basis of d-orbital Slater determinants. The matrix elements separate into a field-independent part (the free-ion repulsion, set by $B$ and $C$) and a field-dependent part (proportional to $\Delta$). Diagonalising at each value of $\Delta /B$ traces out the curves on the diagram.

Configuration interaction. The key technical subtlety is that sub-terms of the same symmetry and spin multiplicity mix as $\Delta /B$ changes. For example, in d${}^3$ the ${}^4{T}_{1g}$ sub-term arises from both the ${}^{4}F$ and ${}^{4}P$ free-ion terms. These two ${}^4{T}_{1g}$ components are not independent levels — they are coupled by the off-diagonal matrix element of the crystal field, producing two non-crossing curves that repel each other. This configuration interaction is what gives the Tanabe-Sugano diagrams their characteristic curved, non-crossing line patterns. The diagonalisation automatically includes all configuration interaction; the resulting curves cannot intersect where the symmetries and multiplicities match (non-crossing rule).

Reading the diagram. To assign a spectrum, determine $\Delta /B$ for the complex (from the lowest spin-allowed transition, which directly gives $\Delta$). Draw a vertical line at that $\Delta /B$ value. The intersections of this line with the energy-level curves give the transition energies. Compare with observed band positions to extract $B$ (and confirm or determine $C/B$).

The d${}^6$ diagram and spin crossover. The d${}^6$ Tanabe-Sugano diagram is the most instructive for understanding spin crossover. At low ${\Delta }_o/B$, the ground state is the high-spin ${}^5{T}_{2g}$ term ($t_{2g}^4{e}_g^2$, four unpaired electrons). As ${\Delta }_o/B$ increases, the low-spin ${}^1{A}_{1g}$ term ($t_{2g}^6{e}_g^0$, zero unpaired electrons) drops below ${}^5{T}_{2g}$ at a critical value ${\Delta }_o/B\approx 20$ (for $C/B\approx 4.5$, typical of first-row metals). The diagram has a visible discontinuity at this crossover: the energy zero switches from ${}^5{T}_{2g}$ to ${}^1{A}_{1g}$, and all curves change slope. Complexes near this crossover — such as $[\mathrm{Fe}(\mathrm{phen}{)}_2(\mathrm{NCS}{)}_2]$ — can be thermally or optically switched between spin states, a phenomenon central to molecular spintronics.

Hole formalism. The Tanabe-Sugano diagram for d${}^n$ in octahedral geometry is related to d${}^{10-n}$ by the hole formalism: replacing each electron by a "hole" (vacancy) maps d${}^2\leftrightarrow$ d${}^8$, d${}^3\leftrightarrow$ d${}^7$, d${}^4\leftrightarrow$ d${}^6$, with d${}^5$ being self-dual. This is a consequence of electron-hole symmetry in the d-shell: the Coulomb repulsion and exchange terms have the same structure for electrons and holes. The hole formalism also connects octahedral d${}^n$ to tetrahedral d${}^{10-n}$: a d${}^7$ ion in $T_d$ uses the d${}^3$ octahedral Tanabe-Sugano diagram (with ${\Delta }_t$ substituted for ${\Delta }_o$).

Orgel diagrams and the d${}^1$/d${}^9$ case Master

For d${}^1$ and d${}^9$ configurations, only one electron (or one hole) is present, so there is no inter-electron repulsion to complicate the energy levels. The Orgel diagram — a simpler precursor to the Tanabe-Sugano diagram — suffices. It plots the single-electron transition energy as a linear function of $\Delta$:

• d${}^1$ in $O_h$: one transition ${}^2{T}_{2g}\to {}^2{E}_g$, energy $={\Delta }_o$.
• d${}^9$ in $O_h$: one transition ${}^2{E}_g\to {}^2{T}_{2g}$, energy $={\Delta }_o$ (hole picture inverts the levels).
• d${}^1$ in $T_d$: one transition ${}^{2}E\to {}^2{T}_2$, energy $={\Delta }_t$.
• d${}^9$ in $T_d$: one transition ${}^2{T}_2\to {}^{2}E$, energy $={\Delta }_t$.

The Orgel diagram is a straight line — no curvature, no configuration interaction, no term crossings. For d${}^2$ and beyond, inter-electron repulsion introduces the nonlinear behaviour that necessitates the full Tanabe-Sugano treatment.

The d${}^1$ case ($[\mathrm{Ti}({\mathrm{H}}_2\mathrm{O}{)}_6{]}^{3+}$) was discussed in unit 16.03.02 as the simplest example of a d-d spectrum: a single band at ${\Delta }_o$, slightly broadened and asymmetric due to Jahn-Teller distortion of the excited state. The d${}^9$ case ($[\mathrm{Cu}({\mathrm{H}}_2\mathrm{O}{)}_6{]}^{2+}$) is the hole complement: also a single band, but strongly distorted by the permanent Jahn-Teller distortion in the ground state (unit 16.03.04 will develop this).

Selection rules and transition intensities Master

The d-d transitions that produce the colours of transition-metal complexes are subject to two selection rules that determine their intensity.

The Laporte (parity) selection rule. In a centrosymmetric point group ($O_h$, $D_{4h}$), transitions between states of the same parity ($g\to g$ or $u\to u$) are forbidden. All d-d transitions are $g\to g$ and are therefore Laporte-forbidden. They gain intensity through:

1. Vibronic coupling. Asymmetric vibrations temporarily break the inversion centre, allowing the $g\to g$ transition to borrow intensity from an allowed ($g\to u$) charge-transfer transition. This is the dominant mechanism for centrosymmetric complexes.
2. Static distortion. Jahn-Teller-distorted complexes (Cu${}^{2+}$, d${}^9$) lack a strict inversion centre and have intrinsically more intense d-d bands.

Tetrahedral complexes ($T_d$) lack an inversion centre entirely, so the Laporte rule does not strictly apply. The d-d bands of tetrahedral complexes are typically 10–100 times more intense than their octahedral counterparts ($ϵ\approx 100\text{–}1000$ vs $ϵ\approx 10\text{–}100$ M${}^{-1}$cm${}^{-1}$).

The spin selection rule ($\Delta S=0$). Transitions that change the total spin multiplicity are forbidden. Spin-allowed bands ($\Delta S=0$) are orders of magnitude more intense than spin-forbidden ones. For d${}^6$ low-spin (${}^1{A}_{1g}$ ground state), all spin-allowed transitions go to singlet excited states; spin-forbidden transitions to triplet states are detectable but very weak.

The intensity ordering is:

Transition type	$ϵ$ (M${}^{-1}$cm${}^{-1}$)
Spin-allowed, Laporte-allowed (charge transfer)	$10^3$–$10^5$
Spin-allowed, Laporte-forbidden, no inversion centre (tetrahedral d-d)	$10^2$–$10^3$
Spin-allowed, Laporte-forbidden, centrosymmetric (octahedral d-d)	$10$–$10^2$
Spin-forbidden (d-d)	$<1$

The nephelauxetic effect and quantitative fitting Master

The Racah parameter $B$ measured from a complex is always smaller than the free-ion value $B_0$. The ratio $\beta =B/B_0$ quantifies the nephelauxetic effect — the reduction of inter-electron repulsion due to covalent delocalisation of the d-electrons onto the ligand framework. This effect was introduced in unit 16.03.01; here we connect it to the Tanabe-Sugano fitting procedure.

When fitting observed band positions to a Tanabe-Sugano diagram, one extracts both $\Delta$ and $B$ simultaneously. The lowest spin-allowed transition gives $\Delta$ directly. The spacing between the higher transitions depends on $B$: a smaller $B$ compresses the upper levels. For $[\mathrm{Cr}({\mathrm{H}}_2\mathrm{O}{)}_6{]}^{3+}$ (d${}^3$), the three bands are at 17,400, 24,600, and 38,000 cm${}^{-1}$. Fitting to the d${}^3$ Tanabe-Sugano diagram gives ${\Delta }_o=17,400$ cm${}^{-1}$ and $B\approx 918$ cm${}^{-1}$, compared to the free-ion $B_0\approx 1030$ cm${}^{-1}$ for Cr${}^{3+}$. The nephelauxetic ratio $\beta =0.89$ indicates modest covalent delocalisation with water ligands.

The nephelauxetic series for ligands (decreasing $\beta$) is:

$${\mathrm{F}}^{-}>{\mathrm{H}}_2\mathrm{O}>\mathrm{N}{\mathrm{H}}_3>\mathrm{en}>{\mathrm{Cl}}^{-}>{\mathrm{CN}}^{-}>{\mathrm{Br}}^{-}>{\mathrm{I}}^{-}.$$

Ligands at the right produce greater d-electron delocalisation (more covalent bonding) and smaller $B$. The nephelauxetic series does not parallel the spectrochemical series exactly: CN${}^{-}$ is the strongest-field ligand but is not the most nephelauxetic, because $\pi$-back-bonding (which lowers $t_{2g}$ and increases ${\Delta }_o$) is a different interaction from the overall delocalisation that reduces $B$.

The nephelauxetic effect also depends on the metal. Going down a group, $\beta$ decreases (more delocalisation) because 4d and 5d orbitals extend further and overlap more with ligand orbitals. For a given ligand, $\beta (5d)<\beta (4d)<\beta (3d)$.

Quantitative spectrum fitting for tetrahedral complexes follows the same procedure with the appropriate Tanabe-Sugano diagram (or, via the hole formalism, the octahedral d${}^{10-n}$ diagram with ${\Delta }_t$ substituted). The smaller ${\Delta }_t$ means the transitions fall at lower energy (near-IR and visible rather than visible and UV), and the larger $\beta$ typical of the more covalent tetrahedral M-L bond compresses the upper levels further.

Connections Master

• Crystal field splitting in octahedral complexes 16.03.02 supplies the $O_h$ character-table machinery, the $E_g\oplus T_{2g}$ decomposition, the barycentre theorem, and the Tanabe-Sugano diagram framework that this unit extends to $T_d$ and $D_{4h}$ geometries.

• Character of a representation 07.01.03 and character orthogonality 07.01.04 provide the group-theoretic tools used to derive the $e\oplus t_2$ splitting in $T_d$ and the four-level splitting in $D_{4h}$.

• Symmetry and group theory in chemistry 16.02.01 supplies the $T_d$ and $D_{4h}$ character tables needed for the non-octahedral geometry analyses.

• Jahn-Teller distortions 16.03.04 pending (successor unit) connect directly to the Tanabe-Sugano predictions: the diagrams identify which excited states are orbitally degenerate and therefore Jahn-Teller-active, and the distortion modifies the transition energies from their ideal-symmetry values.

• Organometallic 18-electron rule 16.05.01 uses the molecular orbital framework that extends the CFT splitting picture; the square-planar geometry developed here is the coordination geometry of catalytically important d${}^8$ complexes (Vaska's complex, Wilkinson's catalyst).

• Bioinorganic metalloenzymes 16.06.01 (successor unit) encounter both tetrahedral (Zn${}^{2+}$ in zinc fingers) and square-planar (Cu${}^{2+}$ in type I copper centres) geometries whose electronic structure follows from the splitting patterns developed here.

• Stern-Gerlach and spin-1/2 12.01.02 supplies the SU(2) angular-momentum algebra underlying the Russell-Saunders term symbols that label the Tanabe-Sugano diagram energy levels.

Historical notes Master

The Tanabe-Sugano diagrams were published by Yukito Tanabe and Satoru Sugano in two back-to-back papers in the Journal of the Physical Society of Japan in 1954 ("On the absorption spectra of complex ions I and II," JPSJ 9, 753–766 and 766–779) ^{[Miessler Ch. 8]}. Their work systematised the energy-level problem for all d${}^n$ configurations in octahedral symmetry, producing the set of seven diagrams (d${}^2$ through d${}^8$) that remain the standard reference in textbooks. The diagrams were computed by hand diagonalisation of the d-electron Hamiltonian at many values of ${\Delta }_o/B$ — a feat of patient calculation without modern computers.

The Orgel diagrams — simpler energy-level diagrams for d${}^1$ and d${}^9$ — were published by Leslie Orgel in 1955 as part of his broader programme to make ligand field theory accessible to chemists. Orgel's 1960 monograph An Introduction to Transition-Metal Chemistry: Ligand-Field Theory (Methuen) bridged the physics-oriented Bethe-Van Vleck crystal field theory with the chemical intuition of spectrochemical trends.

The nephelauxetic effect was identified and named by Claus Schäffer and Christian Klixbüll Jørgensen in the late 1950s. Jørgensen's systematic measurements of $\beta =B/B_0$ across hundreds of complexes established the nephelauxetic series and connected it to covalent delocalisation — providing one of the earliest quantitative probes of covalent bonding in coordination compounds.

The spin-crossover phenomenon was first reported by Cambi and Szego in 1931 for iron(III) dithiocarbamate complexes, but its full explanation in terms of the Tanabe-Sugano diagram crossover awaited the development of ligand field theory in the 1960s. Modern spin-crossover research, pioneered by Philipp Gütlich and others from the 1980s onward, exploits the bistability of d${}^6$ Fe(II) complexes near the high-spin/low-spin crossover for molecular switches and sensors.

The angular overlap model (AOM) of Schäffer and Jørgensen (1965) provided a parameterisation scheme that transfers seamlessly across geometries — octahedral, tetrahedral, square-planar — using local metal-ligand overlap parameters ($e_{\sigma }$, $e_{\pi }$) rather than global $\Delta$ values. The AOM unified the diverse splitting patterns into a single framework and remains the workhorse for quantitative ligand-field analysis short of full DFT calculations.

Bibliography Master

• Tanabe, Y. & Sugano, S. "On the absorption spectra of complex ions I, II." J. Phys. Soc. Japan 9 (1954), 753–766, 766–779.

• Orgel, L. E. "Spectra of transition-metal complexes." J. Chem. Phys. 23 (1955), 1004–1014.

• Orgel, L. E. An Introduction to Transition-Metal Chemistry: Ligand-Field Theory. Methuen, 1960.

• Ballhausen, C. J. Introduction to Ligand Field Theory. McGraw-Hill, 1962.

• Schäffer, C. E. & Jørgensen, C. K. "The angular overlap model: an attempt to revive the ligand field approaches." Mol. Phys. 9 (1965), 401–412.

• Griffith, J. S. The Theory of Transition-Metal Ions. Cambridge University Press, 1961.

• Jørgensen, C. K. Absorption Spectra and Chemical Bonding in Complexes. Pergamon, 1962.

• Cambi, L. & Szego, L. "Uber die Magnetische Susceptibilitat der komplexen Verbindungen." Ber. Dtsch. Chem. Ges. 64 (1931), 2591–2598.

• Gütlich, P., Hauser, A. & Spiering, H. "Thermal and optical switching of iron(II) complexes." Angew. Chem. Int. Ed. 33 (1994), 2024–2054.

• Miessler, G. L., Fischer, P. J. & Tarr, D. A. Inorganic Chemistry, 5th ed. Pearson, 2014.

• Shriver, D. F. & Atkins, P. W. Inorganic Chemistry, 5th ed. Oxford University Press, 2010.