Bonding in Complexes

Overview: What “bonding in complexes” needs to explain

In the general coordination chemistry chapter, you have already met the basic ideas: central metal ions, ligands, coordination number, and overall structure and formulas of complexes. In this chapter we go a step deeper: what actually holds a metal and its ligands together, and how does this bonding explain colors, magnetism, and reactivity?

We will focus on the main bonding models used for transition‑metal complexes:

• Simple electrostatic ideas
• Crystal field theory (CFT)
• Ligand field theory (LFT, conceptual bridge to MO theory)
• Some basic results from molecular orbital (MO) descriptions

You do not need to know advanced quantum mechanics; we will emphasize pictures and consequences.

Basic bonding picture: coordination as a donor–acceptor interaction

In complexes, ligands bind to a metal centre by providing electron pairs (Lewis bases) to an electron‑poor metal ion (Lewis acid). The resulting interaction is often described as a coordinate (dative) covalent bond:

• The ligand donates an electron pair.
• The metal accepts this pair into an empty orbital.
• After the bond is formed, it is indistinguishable (in many respects) from “normal” covalent bonds.

However, the metal–ligand bond is rarely purely covalent or purely ionic; it usually has mixed character:

• Ionic character: attraction between positively charged metal ion and negatively charged or dipolar ligands.
• Covalent character: overlap of metal and ligand orbitals and electron sharing.

Bonding models differ mainly in how much they emphasize electrostatics vs. covalency.

The electrostatic starting point

As a first approximation, many complexes (especially with highly charged metal ions and hard ligands such as $\ce{F^-}$, $\ce{H2O}$) can be viewed as:

• A point positive charge (metal ion) in an arrangement of point negative charges or dipoles (ligands).

This simple idea:

• Explains that stronger metal–ligand attraction occurs with higher metal charge and more highly charged ligands.
• Underlies crystal field theory, where ligands are treated as point charges or dipoles that interact electrostatically with the metal $d$ orbitals.

However, this approximate picture does not explain:

• Why some ligands give strong color changes while others do not.
• Why some complexes are strongly low‑spin or high‑spin.
• Why bonding strengths do not correlate simply with charge and distance.

For that, we refine the model using $d$‑orbital splitting.

Crystal field theory: splitting of $d$ orbitals

In an isolated metal ion, the five $d$ orbitals ($d_{xy}, d_{yz}, d_{xz}, d_{x^2-y^2}, d_{z^2}$) have the same energy (they are degenerate). When ligands approach, the electrostatic repulsion between the ligand electron density and the electrons in the $d$ orbitals depends on the orientation of these orbitals relative to the ligands. This causes splitting of the $d$‑orbital energies.

The magnitude and pattern of this splitting depends on:

• The geometry of the complex (octahedral, tetrahedral, square‑planar, etc.).
• The nature of the metal ion (charge, identity, oxidation state).
• The nature of the ligands (their crystal field strength).

CFT does not worry about covalent overlap; it only considers electrostatic interactions, but this is already enough to explain many properties of complexes.

Octahedral complexes: $t_{2g}$ and $e_g$

Consider an octahedral complex $\ce{[ML6]^{n+}}$, where six ligands are arranged at the corners of an octahedron along the $x$, $y$, and $z$ axes.

• The $d_{x^2-y^2}$ and $d_{z^2}$ orbitals point directly along the axes towards the ligands.
• The $d_{xy}$, $d_{yz}$, and $d_{xz}$ orbitals lie between the axes, between ligands.

As ligands approach:

• Orbitals pointing directly at ligands experience stronger repulsion and are raised in energy.
• Orbitals pointing between ligands experience less repulsion and are lowered in energy.

Thus the five $d$ orbitals split into two sets:

• Lower‑energy triply degenerate set $t_{2g} = \{d_{xy}, d_{yz}, d_{xz}\}$
• Higher‑energy doubly degenerate set $e_g = \{d_{x^2-y^2}, d_{z^2}\}$

The energy difference between these two sets is the octahedral crystal field splitting $\Delta_\mathrm{o}$ (or $10Dq$).

Key consequences:

• The splitting $\Delta_\mathrm{o}$ is central for understanding:
• Colors (light absorbed corresponds roughly to $\Delta_\mathrm{o}$).
• Magnetic behavior (unpaired vs. paired $d$ electrons).
• Preference for high‑spin or low‑spin configurations.

Details of spin states and specific electron configurations are handled in other chapters; here the main point is: ligand arrangement splits the $d$ levels into energetically distinct sets.

Tetrahedral complexes: smaller splitting, reverse ordering

For a tetrahedral complex $\ce{[ML4]^{n+}}$, four ligands sit between the axes (approximately towards the corners of a tetrahedron).

In this case:

• The $d_{xy}$, $d_{yz}$, and $d_{xz}$ orbitals point more directly towards the ligands.
• The $d_{x^2-y^2}$ and $d_{z^2}$ orbitals point between ligands.

Therefore, the splitting pattern is reversed compared to the octahedral case:

• Higher‑energy set: $t_2 = \{d_{xy}, d_{yz}, d_{xz}\}$
• Lower‑energy set: $e = \{d_{x^2-y^2}, d_{z^2}\}$

The tetrahedral splitting $\Delta_\mathrm{t}$ is:

• Smaller than the octahedral splitting: roughly $\Delta_\mathrm{t} \approx \tfrac{4}{9}\Delta_\mathrm{o}$ for similar metals and ligands.
• Usually too small to cause pairing of electrons, so tetrahedral complexes are typically high‑spin.

Again, the exact spin states and magnetism are treated elsewhere; here we note that geometry controls the pattern and size of $d$‑orbitals splitting.

Square‑planar complexes

Square‑planar complexes are especially common for $d^8$ metal ions (e.g. $\ce{Ni^{2+}}$, $\ce{Pd^{2+}}$, $\ce{Pt^{2+}}$) with strong‑field ligands.

A square‑planar arrangement can be considered as an octahedral geometry from which the two ligands along the $z$ axis are removed.

The main qualitative features:

• $d_{x^2-y^2}$ (pointing directly at the four ligands in the $xy$ plane) becomes very high in energy.
• $d_{z^2}$, which used to interact with axial ligands, drops in energy.
• The $d_{xy}$, $d_{xz}$, and $d_{yz}$ orbitals also shift; their detailed order depends on specifics, but typically:
• $d_{x^2-y^2}$ is highest,
• $d_{z^2}$ next,
• $d_{xy}$, $d_{xz}$, $d_{yz}$ form lower levels in some order.

This large splitting and high energy of $d_{x^2-y^2}$ help explain why square‑planar complexes are often low‑spin and diamagnetic (all electrons paired).

Factors affecting crystal field splitting

The size of $\Delta$ depends on:

1. Oxidation state of the metal
   Higher oxidation state $\Rightarrow$ metal more positively charged $\Rightarrow$ stronger attraction of ligands and closer approach $\Rightarrow$ larger splitting.
2. Position of the metal in the periodic table
   For analogous complexes:
• $\ce{3d}$ metals: smaller $\Delta$
• $\ce{4d}$ and $\ce{5d}$ metals: larger $\Delta$

This is because $4d$ and $5d$ orbitals are more diffuse and interact more strongly with ligands.

3. Nature of the ligands: spectrochemical series
   Experimentally, ligands can be ranked by the splitting they produce. A simplified order (weak‑field to strong‑field):

$$ \ce{I^- < Br^- < Cl^- < F^- < OH^- < H2O < NH3 < en < NO2^- < CN^- \approx CO} $$

• Weak‑field ligands: small $\Delta$, favor high‑spin configurations.
• Strong‑field ligands: large $\Delta$, favor low‑spin configurations (if pairing energy is not too large).

How $\Delta$ influences spin states, magnetism, and color is developed more fully in other sections; here the important notion is the dependence of $d$‑splitting on ligand type and metal.

Beyond pure electrostatics: ligand field and molecular orbital ideas

Crystal field theory treats ligands as point charges or dipoles and neglects covalency in metal–ligand bonds. While this is often useful, it has limitations:

• It cannot fully explain trends in bonding strength.
• It does not give detailed information on the shapes and symmetries of bonding vs. antibonding orbitals.
• It treats $\pi$ interactions only in an ad hoc way.

To address this, ligand field theory (LFT) and molecular orbital (MO) descriptions treat the metal–ligand bond as genuinely covalent, using orbital overlap and symmetry.

Basic MO picture for octahedral complexes

For an octahedral $\ce{[ML6]^{n+}}$ complex:

1. Metal orbitals
• Valence orbitals: $4s$, $4p$, $3d$ (for a $\ce{3d}$ metal).
• These have certain symmetries in the octahedral field ($a_{1g}$, $t_{1u}$, $e_g$, $t_{2g}$).
2. Ligand donor orbitals
• Often lone pairs on the ligands, directed towards the metal.
• These combine to form group orbitals that match the symmetries of metal orbitals.
3. Bonding and antibonding combinations
• Matching symmetries can mix to form bonding and antibonding molecular orbitals.
• For $\sigma$ bonding in an octahedral field, the main metal orbitals involved are:
• $4s$ ($a_{1g}$)
• $4p$ ($t_{1u}$)
• $d_{x^2-y^2}$ and $d_{z^2}$ (together $e_g$)

The result:

• Strong metal–ligand $\sigma$ bonds involve combinations of ligand lone pairs with the metal $s$, $p$, and $e_g$ $d$ orbitals.
• The resulting antibonding combination at high energy corresponds roughly to the $e_g$ set in CFT.

4. Nonbonding and $t_{2g}$ orbitals
• The $t_{2g}$ ($d_{xy}$, $d_{yz}$, $d_{xz}$) orbitals do not point directly at ligands in a purely $\sigma$ picture. They interact weakly with $\sigma$ ligand orbitals and can be considered approximately nonbonding in a purely $\sigma$‑bonding complex.
• When $\pi$ interactions are included, $t_{2g}$ can become bonding or antibonding (see below).

In this way, MO theory naturally reproduces:

• A lower set of mostly metal $t_{2g}$‑like orbitals.
• A higher set of metal–ligand antibonding $e_g^*$ orbitals.
• A splitting related to what CFT calls $\Delta_\mathrm{o}$, but now with explicit covalent character.

$\sigma$‑ vs. $\pi$‑bonding ligands

In MO‑based ligand field descriptions, ligands can interact with the metal not only through $\sigma$ donation (lone pairs along the metal–ligand axis) but also through $\pi$ systems:

• $\sigma$ donors: donate electron density along the bond axis. Most common ligands (e.g. $\ce{H2O}$, $\ce{NH3}$, $\ce{Cl^-}$) are at least $\sigma$ donors.
• $\pi$ donors: possess filled $p$ or $\pi$ orbitals that can overlap with metal $d$ orbitals laterally (e.g. $\ce{Cl^-}$, $\ce{Br^-}$, $\ce{I^-}$, $\ce{OH^-}$, $\ce{O^{2-}}$). They can donate electron density into metal $d$ orbitals of appropriate symmetry, especially $t_{2g}$ in octahedral complexes.
• $\pi$ acceptors: possess empty $\pi^*$ orbitals that can accept electron density from filled metal $d$ orbitals (notably $\ce{CO}$, $\ce{CN^-}$, $\ce{NO^+}$). This is often called backbonding or $\pi$ back‑donation.

The effect on the $d$‑orbital energies:

• $\pi$ donor ligands:
• Push up the energy of metal $t_{2g}$ orbitals (because of additional antibonding character), reducing the $t_{2g}$–$e_g$ gap.
• Typically produce smaller overall splitting (weak‑field ligands).
• $\pi$ acceptor ligands:
• Lower the energy of $t_{2g}$ orbitals by forming bonding combinations with them, thereby increasing the gap to the $e_g^*$ orbitals.
• Typically give larger splittings (strong‑field ligands).

This $t_{2g}$ involvement in $\pi$ bonding is a key refinement beyond simple CFT and helps rationalize the spectrochemical series qualitatively.

Types of ligands and their bonding roles

Different ligands interact with metal centres in characteristic bonding ways. Some important classes:

Purely $\sigma$‑donor ligands

Examples: $\ce{NH3}$, $\ce{H2O}$, $\ce{R3P}$ (phosphines, often approximate $\sigma$ donors with some $\pi$‑acceptor character depending on substituents).

Features:

• Main interaction: donation of a lone pair along the M–L axis.
• Metal–ligand bond is primarily a single, directed $\sigma$ bond.
• They are often “middle‑field” ligands in the spectrochemical series.

These ligands mainly influence the $e_g$ set and the overall strength of the metal–ligand bond, but introduce minimal $\pi$ effects in the $t_{2g}$ set.

$\pi$‑donor ligands

Examples: $\ce{Cl^-}$, $\ce{Br^-}$, $\ce{I^-}$, $\ce{OH^-}$, $\ce{O^{2-}}$, $\ce{NR2^-}$ (amido ligands).

Features:

• Have filled $p$ or $\pi$ orbitals that can overlap sideways with metal $d$ orbitals.
• Form metal–ligand bonds with both $\sigma$‑ and $\pi$‑donor character.
• Often lead to:
• Relatively small $d$‑orbital splitting (weak‑field).
• High‑spin configurations when possible.

Chemically, they are usually “hard” ligands; bonding retains higher ionic character, but $\pi$ donation adds covalency.

$\pi$‑acceptor (backbonding) ligands

Examples: $\ce{CO}$, $\ce{CN^-}$, $\ce{NO^+}$, phosphines with strong $\pi$‑acceptor substituents.

Features:

• Have low‑lying empty $\pi^*$ orbitals that can accept electron density from filled metal $d$ orbitals.
• The bonding picture:
• Ligand donates $\sigma$ electron density to the metal.
• Metal donates $\pi$ electron density back to the ligand.

This synergic bonding:

• Strengthens the metal–ligand bond (both partners stabilize each other).
• Stabilizes lower metal oxidation states (because metal $d$ electrons are partially delocalized onto the ligand).
• Increases the $d$‑orbital splitting and usually yields low‑spin configurations.
• Shortens M–L bond lengths and often weakens internal ligand bonds (e.g. the C–O bond in $\ce{CO}$ ligands), affecting vibrational frequencies (spectroscopic consequence).

Backbonding is central to understanding organometallic complexes (covered in other chapters), but its bonding principle belongs here.

Bonding differences across geometries

Because the symmetry and directions of metal–ligand bonds differ in octahedral, tetrahedral, and square‑planar complexes, the detailed bonding patterns differ as well:

• Octahedral: six equivalent $\sigma$ donors; $e_g$ strongly $\sigma$ antibonding, $t_{2g}$ can be nonbonding or participate in $\pi$ bonding.
• Tetrahedral: four ligands between axes; commonly pure $\sigma$ donors; weaker and reversed $d$‑splitting compared to octahedral.
• Square‑planar: strong interaction in the $xy$ plane, leading to $d_{x^2-y^2}$ as the strongly antibonding orbital; other $d$ orbitals interact less strongly or primarily via $\pi$ effects.

The bonding patterns, in turn, influence:

• Which orbitals are occupied by electrons.
• Metal–ligand bond strengths.
• Reactivity pathways (for example, which orbitals are available for bond formation or cleavage).

Detailed property discussions are handled elsewhere; for this chapter, the essential idea is that geometry directly controls which metal orbitals engage most strongly in bonding.

Relationship between bonding models and observable properties

Although explanations of colors, magnetism, and reactivity belong in other sections, it is useful to summarize how bonding models connect to these phenomena:

• Color:
  Transitions of electrons between split $d$ levels (for example, from $t_{2g}$ to $e_g^*$) often absorb visible light. The energy of light absorbed corresponds roughly to the splitting $\Delta$ determined by bonding.
• Magnetism:
  The number of unpaired $d$ electrons (high‑spin vs. low‑spin) depends on the relative sizes of $\Delta$ and the pairing energy. Both are influenced by the bonding environment (metal, ligands, geometry).
• Bond length and strength:
  Stronger $\sigma$ and $\pi$ bonding, especially with $\pi$‑acceptor ligands, produces shorter and stronger metal–ligand bonds. The presence of strong antibonding interactions leads to longer, weaker bonds.
• Oxidation state stabilization:
  $\pi$‑acceptor ligands stabilize low oxidation states through backbonding; strong $\pi$ donors and hard $\sigma$ donors stabilize higher oxidation states.

Summary of key bonding concepts in complexes

• Metal–ligand bonds combine ionic and covalent character and are best viewed as coordinate covalent bonds.
• Crystal field theory uses a purely electrostatic model to explain $d$‑orbital splitting in different geometries (octahedral, tetrahedral, square‑planar).
• Splitting patterns ($t_{2g}$/$e_g$ ordering and spacing) depend on geometry, metal oxidation state, metal identity, and ligand type.
• Ligand field theory and MO theory refine this picture by treating bonding as covalent orbital interactions, including $\sigma$ and $\pi$ bonding.
• Ligands can be:
• Purely $\sigma$ donors,
• $\pi$ donors,
• $\pi$ acceptors (backbonding partners),
  each affecting $d$‑orbital energies and splitting differently.
• These bonding features underpin the characteristic colors, magnetism, bond strengths, and redox behavior of coordination complexes.