Question

Many compounds of the transition-metal elements contain direct bonds between metal atoms. We will assume that the \(z\)-axis is defined as the metal-metal bond axis. (a) Which of the \(3 d\) orbitals (Figure \(6.23\) ) can be used to make a \(\sigma\) bond between metal atoms? (b) Sketch the \(\sigma_{3 d}\) bonding and \(\sigma_{3}^{*} d\) antibonding MOs. (c) With reference to the "Closer Look" box on the phases of orbitals, explain why a node is generated in the \(\sigma_{3 d}^{*} \mathrm{MO}\). (d) Sketch the energy-level diagram for the \(\mathrm{Sc}_{2}\) molecule, assuming that only the \(3 d\) orbital from part (a) is important. (e) What is the bond order in \(\mathrm{Sc}_{2}\) ?

Short answer

The 3dz² orbital is most likely to form a sigma bond between metal atoms. The σ3d bonding MO is formed by the constructive overlap of in-phase 3dz² orbitals, while the σ3d* antibonding MO is formed by the destructive overlap of out-of-phase 3dz² orbitals, resulting in a node along the bond axis. The energy-level diagram for Sc2 shows that two electrons occupy the σ3d bonding MO and two electrons occupy the σ3d* antibonding MO. The bond order in Sc2 is 0, indicating no stable bond between Sc atoms in this simple model.

Step-by-step solution

Identify the orbital for sigma bond formation

Among the five 3d orbitals (3dxy, 3dxz, 3dyz, 3dx²-y², and 3dz²), the 3dz² orbital is most likely to make a sigma bond between metal atoms. This choice is based on the fact that it has a large lobe along the z-axis and can thus form strong head-to-head overlap along the metal-metal bond axis (assumed to be the z-axis).

Sketch the sigma bonding and antibonding MOs

To sketch the σ3d bonding and the σ3d* antibonding molecular orbitals, we visualize the constructive and destructive overlap of the 3dz² orbitals on the metal atoms. For the σ3d bonding MO, we have constructive overlap, which means the phases of the 3dz² orbitals are in-phase. The large lobes on the z-axis overlap, resulting in increased electron density along the bond axis. For the σ3d* antibonding MO, we have destructive overlap, which means the phases of the 3dz² orbitals are out-of-phase. The large lobes on the z-axis overlap, resulting in a node where the electron density is reduced to zero.

Explain the presence of a node in the antibonding MO

The node arises in the σ3d* antibonding MO due to the destructive interference between the orbitals with opposite phases. When the phases are out of sync, the electron density cancels out between the orbitals, generating a nodal plane with no electron density along the bond axis. The presence of a node along the bond axis weakens the bond, typical for antibonding orbitals.

Sketch the energy-level diagram for Sc2

For Sc2, we assume that only the 3dz² orbital is important for the bond formation. Elemental scandium ([Ar] 3d¹ 4s²) has one electron in the 3d orbital and two in the 4s orbital. To sketch the energy-level diagram, place the σ3d bonding molecular orbital at a lower energy level than the 3dz² atomic orbitals and the σ3d* antibonding MO at a higher energy level. Fill in the electrons, with two electrons from each Sc atom for a total of four. The first two electrons will occupy the σ3d bonding MO, and the other two will occupy the σ3d* antibonding MO.

Calculate the bond order in Sc2

Bond order is calculated as the difference between the number of electrons in bonding molecular orbitals and antibonding molecular orbitals, divided by two: Bond Order = (No. of electrons in bonding MOs - No. of electrons in antibonding MOs) / 2 Sc2 has two electrons in the σ3d bonding MO and two electrons in the σ3d* antibonding MO. Thus, the bond order in Sc2 is: Bond Order = (2 - 2) / 2 = 0. Therefore, the bond order in Sc2 is 0, indicating no stable bond between the Sc atoms in this simple model. However, other orbitals might be important in reality, and the simplification made in this exercise might not accurately represent the actual bonding situation in Sc2.

Key concepts

Molecular Orbitals

Molecular orbitals (MOs) are formed when atomic orbitals from different atoms combine during the formation of a molecule. In the case of transition metals, these MOs play a crucial role in understanding the types of bonds formed between metal atoms.

The formation of molecular orbitals depends on the linear combination of atomic orbitals (LCAO). When atomic orbitals combine, they can overlap constructively or destructively, leading to bonding or antibonding molecular orbitals, respectively. This is true for all types of M-O systems, and especially significant in understanding the bonding in transition metals.

Molecular orbitals provide a global perspective on how electrons are distributed within a molecule, and this concept is essential in explaining the chemical bonding and properties of transition metal compounds.

Sigma Bond

A sigma bond ( σ bond) is the strongest type of covalent bond and is formed by the head-to-head overlap of atomic orbitals. For transition metals, one of the common orbitals that form a sigma bond is the 3dz² orbital due to its orientation. It lies along the z-axis, which is often considered the bond axis in metal complexes.

In the context of metal-metal bonding, the lobes of the 3dz² orbitals from two different metal atoms overlap linearly along the bond axis. This overlap increases electron density along the axis, contributing to a strong bond stabilization.

It's important to note that the strength of the sigma bond is highly dependent on the amount of orbital overlap. The more in-phase the orbitals are during overlap, the stronger the σ bond formed.

Antibonding Orbital

Antibonding orbitals ( σ* MOs) are formed due to destructive interference between the phases of atomic orbitals. In transition metal complexes, such as the Sc_2 molecule, the antibonding molecular orbital plays a critical role in determining the overall stability of the molecule.

When two atomic orbitals combine out of phase, an antibonding molecular orbital results. This out-of-phase combination creates a nodal plane where the electron density cancels out, ominously reducing the bonding interaction along the metal-metal axis. This nodal plane effectively weakens the stability of the bond.

The antibonding orbital is higher in energy than both the parent atomic orbitals and the bonding molecular orbital. It is generally unfavorable to have electrons in this orbital. For Sc_2, placing electrons in the antibonding σ3d* orbital results in no net gain for bonding, highlighting the critical nature of electron distribution in molecular orbitals.

Energy-Level Diagram

An energy-level diagram is a visual representation of the energy states of the electrons in atomic or molecular orbitals. For the Sc_2 molecule, the energy-level diagram showcases how the 3d and other pertinent orbitals mix and split into bonding and antibonding states.

The σ3d bonding molecular orbital is lower in energy compared to the 3dz² atomic orbitals, while the σ3d* antibonding MO is higher in energy. Electrons are filled into these orbitals starting from the lowest energy level, following principles like the Aufbau principle, Hund’s rule, and the Pauli exclusion principle.

In Sc_2, with four electrons, the energy-level diagram reveals that two electrons fill the lower-energy σ3d bonding orbital and another two fill the higher-energy σ3d* antibonding orbital. Consequently, as the bond order—a calculation defined as the number of bonding electrons minus the number of antibonding electrons divided by two—shows zero, indicating a lack of stable bonding due to equal occupation of bonding and antibonding orbitals.