Valence Bond Theory

Valence Bond Theory (VBT) was one of the earliest quantum mechanical models applied to coordination compounds, developed primarily by Linus Pauling in the 1930s. It explains the formation of coordinate (dative) bonds between a central metal ion and surrounding ligands by describing how atomic orbitals on the metal hybridize to create a set of equivalent hybrid orbitals that accept lone pairs from the ligands. The geometry of the resulting complex — octahedral, tetrahedral, square planar, and so on — emerges directly from which orbitals participate in hybridization. A critical insight of VBT is the distinction between inner orbital (low-spin) and outer orbital (high-spin) complexes. Strong-field ligands force d-electrons to pair up, freeing inner 3d orbitals for hybridization (d²sp³), while weak-field ligands leave the 3d electrons spread out and hybridization uses outer 4d orbitals (sp³d²). This difference has measurable consequences: inner orbital complexes are generally more stable and less paramagnetic than their outer orbital counterparts. Although VBT has been largely superseded by Crystal Field Theory and Ligand Field Theory for quantitative predictions, it remains a powerful conceptual tool because it connects hybridization — already familiar from organic chemistry — to the three-dimensional structures and magnetic properties of transition metal complexes. Understanding VBT provides a bridge between simple bonding models and more sophisticated approaches.

Key takeaways
  • What Hybridization Means for a Metal Ion
  • Common Hybridization Schemes and Geometries
  • Inner Orbital Complexes (Low-Spin)
  • Outer Orbital Complexes (High-Spin)
  • Magnetic Behavior and Measuring Paramagnetism
  • Limitations of Valence Bond Theory

What Hybridization Means for a Metal Ion

In VBT, the metal ion rearranges (hybridizes) its valence atomic orbitals — combinations of s, p, and d orbitals — to form a new set of equivalent hybrid orbitals, each pointing toward one ligand. The number of hybrid orbitals equals the coordination number. For a six-coordinate complex, six hybrid orbitals are needed; for a four-coordinate complex, four are needed. Each hybrid orbital then overlaps with a filled orbital on a ligand (the lone pair), forming a sigma (σ) coordinate bond.

Common Hybridization Schemes and Geometries

The most important hybridization types are: (1) sp³ — four orbitals, tetrahedral geometry (e.g., [NiCl₄]²⁻); (2) dsp² — four orbitals using one d, one s, and two p orbitals, square planar geometry (e.g., [Ni(CN)₄]²⁻, [PtCl₄]²⁻); (3) d²sp³ — six orbitals using two inner (n−1)d orbitals, giving octahedral geometry with inner d orbitals (e.g., [Co(NH₃)₆]³⁺, [Fe(CN)₆]⁴⁻); (4) sp³d² — six orbitals using two outer nd orbitals, giving octahedral geometry with outer d orbitals (e.g., [CoF₆]³⁻, [FeF₆]³⁻). The key difference between schemes (3) and (4) is which d orbitals are used.

Inner Orbital Complexes (Low-Spin)

When a strong-field ligand (such as CN⁻, NH₃, CO, or NO₂⁻) donates electron pairs to a metal, it exerts enough energy to force the metal's d electrons to pair up into the lower-energy d orbitals. This vacates two inner (n−1)d orbitals, which can then be included in hybridization. The resulting d²sp³ hybridization uses orbitals from the inner shell (n−1), producing a compact, stable complex. Because electrons are maximally paired, inner orbital complexes have fewer unpaired electrons and are weakly paramagnetic or diamagnetic. Example: [Co(NH₃)₆]³⁺ — Co³⁺ has configuration [Ar]3d⁶; under NH₃ all six 3d electrons pair into three 3d orbitals, freeing two 3d orbitals for d²sp³ hybridization → 0 unpaired electrons → diamagnetic.

Outer Orbital Complexes (High-Spin)

When a weak-field ligand (such as F⁻, Cl⁻, Br⁻, I⁻, or H₂O) interacts with the metal, the energy available is insufficient to force d electron pairing. The d electrons remain spread out according to Hund's rule, so the inner d orbitals are occupied and cannot be used for hybridization. Instead, the empty outer nd orbitals are recruited: hybridization is sp³d², using orbitals from the outer shell (n). The resulting complex is less stable than the inner orbital analog. Because more d electrons remain unpaired, outer orbital complexes are strongly paramagnetic. Example: [CoF₆]³⁻ — Co³⁺ with F⁻ ligands; electrons remain in 3d with maximum unpaired electrons → sp³d² hybridization using 4d orbitals → 4 unpaired electrons → strongly paramagnetic.

Magnetic Behavior and Measuring Paramagnetism

Paramagnetism arises from unpaired electrons, each contributing a magnetic moment. The spin-only magnetic moment formula gives an excellent approximation for transition metal complexes: μ = √(n(n+2)) BM, where n is the number of unpaired electrons and BM is the Bohr Magneton (9.274 × 10⁻²⁴ J T⁻¹). A diamagnetic complex (n = 0) is weakly repelled by a magnetic field, whereas a paramagnetic complex (n > 0) is attracted. Experimentally, the magnetic moment is measured using a Gouy balance or modern magnetic susceptibility meters, and comparing the measured value to the spin-only formula allows chemists to determine the number of unpaired electrons and thus infer the hybridization and whether the complex is inner or outer orbital.

Limitations of Valence Bond Theory

VBT cannot explain the color of transition metal complexes (which requires Crystal Field or Ligand Field Theory), nor can it reliably predict which ligands are strong-field or weak-field — it simply assumes this information is already known. It also struggles with cases where the magnetic moment deviates from spin-only values, indicating orbital contributions to magnetism. For these reasons, VBT is best understood as a descriptive, qualitative tool rather than a predictive, quantitative one.

The Key Equation

μ=n(n+2)  BM\mu = \sqrt{n(n+2)} \; \text{BM}
  • \mu — Magnetic moment (BM (Bohr Magneton)): A measure of the strength of the magnetic dipole of the complex; 1 BM = 9.274 × 10⁻²⁴ J T⁻¹
  • n — Number of unpaired electrons (dimensionless): Determined by the electron configuration of the metal ion after accounting for the field strength of the ligands
  • \text{BM} — Bohr Magneton (J T⁻¹): The natural unit of electron magnetic dipole moment, equal to eℏ/2mₑ = 9.274 × 10⁻²⁴ J T⁻¹

Worked Example

Problem. Predict the hybridization, geometry, number of unpaired electrons, and magnetic moment for [Fe(CN)₆]⁴⁻. Fe has atomic number 26. CN⁻ is a strong-field ligand.

μ=0(0+2)=0  BM\mu = \sqrt{0(0+2)} = 0 \; \text{BM}

Answer. d²sp³ hybridization, octahedral geometry, 0 unpaired electrons, μ = 0 BM (diamagnetic).

Figure
Diagram
VBT: Inner vs Outer Orbital Complexes — Fe²⁺ (3d⁶)[Fe(CN)₆]⁴⁻ — Inner Orbital (d²sp³)[FeF₆]⁴⁻ — Outer Orbital (sp³d²)3d orbitals↑↓↑↓↑↓4s orbital4p orbitalsd²sp³ hybrid orbitals (6)Octahedral | n=0 | μ=0 BMDIAMAGNETIC3d orbitals↑↓↑↑↑↑4s orbital4p orbitals4d orbitalssp³d² hybrid orbitals (6)Octahedral | n=4 | μ=4.90 BMPARAMAGNETIC
Origins · 1 minLinus Pauling and how this was discovered
LP
Linus Pauling
1901–1994 · American

Pauling developed Valence Bond Theory and the concept of orbital hybridization in the 1930s, publishing his landmark text 'The Nature of the Chemical Bond' (1939). He applied hybridization to coordination compounds, distinguishing inner and outer orbital complexes and explaining their magnetic properties — work that earned him the 1954 Nobel Prize in Chemistry. Pauling is one of only four people to have won two Nobel Prizes: the 1954 Nobel Prize in Chemistry and the 1962 Nobel Peace Prize for his anti-nuclear activism — making him the only individual to receive two unshared Nobel Prizes.

  • Crystal Field Theory — Crystal Field Theory supersedes VBT for explaining the splitting of d orbitals, color, and stability of coordination compounds, and provides a more quantitative framework for the same phenomena.
  • Coordination Numbers and Geometry — The hybridization scheme in VBT directly determines the geometry of the complex; understanding coordination number is prerequisite to assigning hybridization.
  • Orbital Hybridization (sp, sp², sp³) — The concept of hybridization introduced in organic chemistry (sp, sp², sp³ for carbon) is the same principle extended to d orbitals for transition metals in VBT.
  • Spectrochemical Series — The spectrochemical series ranks ligands by field strength, determining whether a complex is inner (strong-field) or outer (weak-field) orbital — essential knowledge for applying VBT.
  • Electron Configuration of Transition Metals — VBT analysis begins by writing the electron configuration of the metal ion; knowledge of how d electrons fill and pair is fundamental to predicting hybridization and magnetic properties.