How do I tell whether a complex is high spin or low spin?

Compare the crystal field splitting Δo with the electron pairing energy P. If Δo > P the electrons pair in t₂g rather than enter e_g, giving low spin; if Δo < P they occupy all five orbitals singly first, giving high spin. Only octahedral d⁴–d⁷ ions have a choice. Strong-field ligands such as CN⁻, CO, NO₂⁻, bipy and phen usually give low spin, weak-field halides and water usually high spin; Co³⁺ is low spin except in [CoF₆]³⁻; 4d and 5d metals are almost always low spin; tetrahedral complexes, with Δt only about 4/9 of Δo, are almost always high spin.

How is crystal field stabilization energy (CFSE) calculated?

In an octahedral field each t₂g electron contributes −0.4Δo and each e_g electron +0.6Δo; add them up, then add mP for the m extra electron pairs compared with the free ion. High-spin d⁶ (t₂g⁴e_g²) gives 4 × (−0.4) + 2 × 0.6 = −0.4Δo; low-spin d⁶ (t₂g⁶) gives −2.4Δo with two extra pairs, written −2.4Δo + 2P. In a tetrahedral field e lies at −0.6Δt and t₂ at +0.4Δt. Some textbooks count every pair (−2.4Δo + 3P for low-spin d⁶); the difference between the spin states is the same either way.

Which d-electron configurations show a Jahn–Teller distortion?

A distortion occurs when degenerate orbitals are unevenly occupied. It is strong when the e_g set is uneven: high-spin d⁴ (Cr²⁺, Mn³⁺), d⁹ (Cu²⁺) and low-spin d⁷, usually seen as an octahedron elongated along z — in [Cu(H₂O)₆]²⁺ the two axial Cu–O bonds are clearly longer. Uneven t₂g occupation (d¹, d², low-spin d⁴ and d⁵, high-spin d⁶ and d⁷) gives only a weak, usually unobservable effect.

How do I get the colour of a complex from Δo?

A d–d transition absorbs a photon of energy Δ, so λ (nm) = 10⁷ / Δ (cm⁻¹), and the colour you see is the complement of the absorbed colour on the colour wheel. [Ti(H₂O)₆]³⁺ has Δo = 20,300 cm⁻¹, absorbs at about 493 nm (green) and looks purple; [Co(NH₃)₆]³⁺ (22,900 cm⁻¹) absorbs blue light near 437 nm and looks orange-yellow. Only d¹ (and other single-band cases such as d⁹) have a band maximum equal to Δo; other configurations show several bands and need a Tanabe–Sugano diagram.

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🟣Crystal Field Theory Calculator - d-Orbital Splitting Diagram, High/Low Spin, CFSE

Q: How is crystal field stabilization energy (CFSE) calculated?

In an octahedral field each t₂g electron contributes −0.4Δo and each e_g electron +0.6Δo; add them up, then add mP for the m extra electron pairs compared with the free ion. High-spin d⁶ (t₂g⁴e_g²) gives 4 × (−0.4) + 2 × 0.6 = −0.4Δo; low-spin d⁶ (t₂g⁶) gives −2.4Δo with two extra pairs, written −2.4Δo + 2P. In a tetrahedral field e lies at −0.6Δt and t₂ at +0.4Δt. Some textbooks count every pair (−2.4Δo + 3P for low-spin d⁶); the difference between the spin states is the same either way.

Q: Which d-electron configurations show a Jahn–Teller distortion?

A distortion occurs when degenerate orbitals are unevenly occupied. It is strong when the e_g set is uneven: high-spin d⁴ (Cr²⁺, Mn³⁺), d⁹ (Cu²⁺) and low-spin d⁷, usually seen as an octahedron elongated along z — in [Cu(H₂O)₆]²⁺ the two axial Cu–O bonds are clearly longer. Uneven t₂g occupation (d¹, d², low-spin d⁴ and d⁵, high-spin d⁶ and d⁷) gives only a weak, usually unobservable effect.

Q: How do I get the colour of a complex from Δo?

A d–d transition absorbs a photon of energy Δ, so λ (nm) = 10⁷ / Δ (cm⁻¹), and the colour you see is the complement of the absorbed colour on the colour wheel. [Ti(H₂O)₆]³⁺ has Δo = 20,300 cm⁻¹, absorbs at about 493 nm (green) and looks purple; [Co(NH₃)₆]³⁺ (22,900 cm⁻¹) absorbs blue light near 437 nm and looks orange-yellow. Only d¹ (and other single-band cases such as d⁹) have a band maximum equal to Δo; other configurations show several bands and need a Tanabe–Sugano diagram.

Crystal field theory treats ligands as point charges. In an octahedral field the five d orbitals split into a lower t₂g set (−0.4Δo) and an upper e_g set (+0.6Δo); in a tetrahedral field the order is reversed and Δt ≈ 4/9 Δo. If the splitting Δ is larger than the pairing energy P the electrons pair up in the lower level (low spin); if it is smaller they spread out first (high spin). For example Fe²⁺ is d⁶: [Fe(H₂O)₆]²⁺ has Δo ≈ 10,400 cm⁻¹, below P ≈ 17,600 cm⁻¹, so it is high-spin t₂g⁴e_g² with 4 unpaired electrons, a spin-only moment μ = √(4×6) = 4.90 BM and CFSE = −0.4Δo, while [Fe(CN)₆]⁴⁻ (Δo ≈ 32,800 cm⁻¹) is low-spin t₂g⁶, diamagnetic, with CFSE = −2.4Δo + 2P. The tool covers Sc–Zn in their common oxidation states plus Ru, Rh, Pd, Ir, Pt and Au ions (d count = group − oxidation state), 21 ligands in spectrochemical order, and octahedral, tetrahedral, square-planar and trigonal-bipyramidal geometries. Δo can come from measured textbook values, the Jørgensen f·g estimate or your own input in cm⁻¹ or kJ/mol, with free-ion pairing energies as defaults. You get the configuration, unpaired electrons, magnetic moment, CFSE in Δo units and in energy, a Jahn–Teller flag (strong for d⁹ Cu²⁺, high-spin d⁴ and low-spin d⁷) and an energy-level diagram showing both spin states. The absorbed wavelength λ = 10⁷/Δ is placed on a colour wheel to find the complementary colour — [Ti(H₂O)₆]³⁺ with Δo = 20,300 cm⁻¹ absorbs green light at 493 nm and looks purple — and a table lists octahedral high/low-spin CFSE for d¹–d¹⁰.

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How to use Crystal Field Theory Calculator

1Choose the metal ion, ligand and geometry, or click a measured example
2Adjust Δo and the pairing energy P if needed, or force high or low spin
3Read the energy-level diagram, configuration, CFSE, magnetic moment, Jahn–Teller flag, absorbed wavelength and colour, and the worked steps

FAQ

How do I tell whether a complex is high spin or low spin?: Compare the crystal field splitting Δo with the electron pairing energy P. If Δo > P the electrons pair in t₂g rather than enter e_g, giving low spin; if Δo < P they occupy all five orbitals singly first, giving high spin. Only octahedral d⁴–d⁷ ions have a choice. Strong-field ligands such as CN⁻, CO, NO₂⁻, bipy and phen usually give low spin, weak-field halides and water usually high spin; Co³⁺ is low spin except in [CoF₆]³⁻; 4d and 5d metals are almost always low spin; tetrahedral complexes, with Δt only about 4/9 of Δo, are almost always high spin.
How is crystal field stabilization energy (CFSE) calculated?: In an octahedral field each t₂g electron contributes −0.4Δo and each e_g electron +0.6Δo; add them up, then add mP for the m extra electron pairs compared with the free ion. High-spin d⁶ (t₂g⁴e_g²) gives 4 × (−0.4) + 2 × 0.6 = −0.4Δo; low-spin d⁶ (t₂g⁶) gives −2.4Δo with two extra pairs, written −2.4Δo + 2P. In a tetrahedral field e lies at −0.6Δt and t₂ at +0.4Δt. Some textbooks count every pair (−2.4Δo + 3P for low-spin d⁶); the difference between the spin states is the same either way.
Which d-electron configurations show a Jahn–Teller distortion?: A distortion occurs when degenerate orbitals are unevenly occupied. It is strong when the e_g set is uneven: high-spin d⁴ (Cr²⁺, Mn³⁺), d⁹ (Cu²⁺) and low-spin d⁷, usually seen as an octahedron elongated along z — in [Cu(H₂O)₆]²⁺ the two axial Cu–O bonds are clearly longer. Uneven t₂g occupation (d¹, d², low-spin d⁴ and d⁵, high-spin d⁶ and d⁷) gives only a weak, usually unobservable effect.
How do I get the colour of a complex from Δo?: A d–d transition absorbs a photon of energy Δ, so λ (nm) = 10⁷ / Δ (cm⁻¹), and the colour you see is the complement of the absorbed colour on the colour wheel. [Ti(H₂O)₆]³⁺ has Δo = 20,300 cm⁻¹, absorbs at about 493 nm (green) and looks purple; [Co(NH₃)₆]³⁺ (22,900 cm⁻¹) absorbs blue light near 437 nm and looks orange-yellow. Only d¹ (and other single-band cases such as d⁹) have a band maximum equal to Δo; other configurations show several bands and need a Tanabe–Sugano diagram.

Features

d-orbital splitting diagrams for octahedral, tetrahedral, square-planar and trigonal-bipyramidal fields with electrons drawn as arrows
Automatic high/low-spin decision from Δ vs P, with the other spin state drawn alongside
CFSE in Δo units and in cm⁻¹ and kJ/mol, with the pairing-energy convention stated
Unpaired electrons, spin-only magnetic moment μ = √(n(n+2)), para- or diamagnetism
Jahn–Teller distortion flag (uneven e_g or t₂g occupation)
18 measured Δo examples, Jørgensen f·g estimates and free-ion pairing energies
Absorbed wavelength λ = 10⁷/Δ, colour-wheel complement and a simulated band colour
Clickable spectrochemical series and a d¹–d¹⁰ high/low-spin CFSE and magnetic-moment table

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