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 \(\sigma_{3 d}^{\star}\) 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 \(3d_{z^2}\) orbital can be used to form a σ bond between metal atoms. The σ(3d) bonding MO is formed by constructive overlap, while the σ(3d)* antibonding MO is formed by destructive overlap with a nodal plane between the metal atoms. The bond order in Sc2 is 0, indicating that there might not be a stable bond between Sc atoms.

Step-by-step solution

(a) Identifying the 3d orbital for σ bonding

We know that a σ bond is formed by the direct overlap of orbitals along the internuclear axis, that is, the bond axis. In this case, the bond axis is along the z-axis. There are five 3d orbitals: \(3d_{xy}\), \(3d_{yz}\), \(3d_{xz}\), \(3d_{x^2-y^2}\), and \(3d_{z^2}\). Among these, only the \(3d_{z^2}\) orbital is symmetrical along the z-axis and can be used to form a σ bond between metal atoms.

(b) Sketching σ(3d) and σ(3d)* MOs

To sketch the σ(3d) bonding and σ(3d)* antibonding MOs, we consider the overlap of two \(3d_{z^2}\) orbitals: (1) For the σ(3d) bonding MO: The two \(3d_{z^2}\) orbitals have the same phase and add constructively, leading to a bonding interaction. (2) For the σ(3d)* antibonding MO: The two \(3d_{z^2}\) orbitals have opposite phases and add destructively, leading to an antibonding interaction with a nodal plane between the metal atoms.

(c) Node in σ(3d)* antibonding MO

In the case of the σ(3d)* antibonding MO, the node is generated because the two overlapping \(3d_{z^2}\) orbitals have different phases. The destructive interference causes a region of zero electron density between the two metal atoms, which corresponds to a nodal plane.

(d) Sc2 energy-level diagram

We will now draw the energy-level diagram for the Sc2 molecule, considering only the \(3d_{z^2}\) orbital: (1) Draw separate energy levels for each isolated Sc atom. (2) Add the σ(3d) bonding and σ(3d)* antibonding MOs above the atomic orbitals. (3) Indicate the number of electrons and populate them in the MOs following the Aufbau principle and Hund's rule.

(e) Bond order in Sc2

The bond order represents the strength of a chemical bond. For Sc2, we can calculate the bond order using the formula: Bond order \(= \frac{1}{2} \times (number\ of\ bonding\ electrons - number\ of\ antibonding\ electrons)\) For Sc2, there are 2 electrons in the σ(3d) bonding MO and 2 electrons in the σ(3d)* antibonding MO. Therefore, the bond order can be calculated as: Bond order \(= \frac{1}{2} \times (2 - 2) = 0\) The bond order in Sc2 is 0, which indicates that there might not be a stable bond between the scandium atoms.

Key concepts

σ Bond

In chemistry, a σ bond (sigma bond) is one of the strongest types of covalent chemical bonds and is formed by the head-on or axial overlap of atomic orbitals. When discussing transition metal compounds, the formation of a σ bond between metal atoms involves the orbital that is symmetrical along the bond axis. In our exercise, we look at the overlap of the 3d orbitals on a transition metal, and it's highlighted that the 3d_z^2 orbital is suited for σ bonding as it is symmetric along the z-axis, which we have designated as the bond axis. This directional bonding characteristic is essential for transition metal-metal bonding and greatly influences the properties of the resulting compound.

The σ bonds are usually associated with single bonds between atoms, and in metals, they provide a significant contribution to metals' stability and unique chemistry.

Molecular Orbital Theory

Molecular Orbital Theory (MOT) provides an explanation for the bonding and structure of molecules using quantum mechanics. It extends the concepts of atomic orbitals and electron configurations to molecules by describing the electrons as being delocalized over the entire molecule. In MOT, when two atoms bond, their atomic orbitals combine to form molecular orbitals which are classified as bonding, non-bonding, or antibonding.

For transition metal complexes, MOT helps us understand how d orbitals are involved in bonding. In the context of our exercise, MOT is crucial in explaining the formation of σ(3d) bonding and σ(3d)* antibonding molecular orbitals as the atomic 3d orbitals from two metal atoms overlap. Understanding the electron configuration in these molecular orbitals further helps us predict the bond order, stability, and magnetic properties of the compound.

Bond Order Calculation

Bond order is an index that reflects the overall stability of a bond within a molecule. It is calculated as the difference between the number of electrons in bonding orbitals and antibonding orbitals, divided by two. In the case of the Sc₂ molecule, we have both bonding and antibonding electrons in the same type of orbital, leading to a bond order of zero, as per the calculation provided in the solution.

Typically, a higher bond order corresponds to a stronger and shorter bond. A bond order of zero would imply that there is no net bonding interaction between the two atoms, suggesting that under certain conditions, the dibond might not be stable. Bond order calculations are fundamental in predicting the existence and properties of diatomic species and heavily rely on an accurate depiction of the electron distribution within molecular orbitals.

Transition Metal Chemistry

Transition metal chemistry encompasses the chemical behavior of elements found in the d-block of the periodic table. These metals are characterized by their ability to form a variety of complex compounds, exhibiting diverse oxidation states and coordination geometries. The d orbitals in these elements play a vital role in bonding, particularly in the formation of coordination complexes.

The exercise that we're discussing focuses on a key aspect of transition metal chemistry, which is the metal-metal σ bond. As we've seen, understanding the orbital overlaps and the resulting molecular orbitals is critical in predicting and rationalizing the properties of transition metal compounds. It's essential to appreciate that these elements can exhibit behaviors that are dramatically different from those of main group elements because of the involvement of their d orbitals in bonding.

d Orbital Overlap

The concept of d orbital overlap relates to how the d orbitals of transition metals engage in the formation of chemical bonds. Not all d orbitals can overlap efficiently for σ bonding; it depends on their geometrical orientation relative to the bonding axis. In the context of our exercise, the 3d_z^2 orbital is uniquely suited for σ bonding along the z-axis.

During the formation of a bond, when d orbitals with the same symmetry and proper alignment approach each other, the extent of their overlap significantly impacts the bond strength and the resulting molecular orbital's energy. This kind of directed overlap is essential to understand when considering the complex nature and the vast array of compounds that transition metals can form. The d orbital overlap is a nuanced aspect of transition metal chemistry that explains their rich coordination behavior and the formation of metal-metal bonds in compounds.