Question

Show how a \(d_{x z}\) atomic orbital and a \(p_{z}\) atomic orbital combine to form a bonding molecular orbital. Assume the \(x\) -axis is the internuclear axis. Is a \(\sigma\) or a \(\pi\) molecular orbital formed? Explain.

Short answer

The \(d_{x z}\) and \(p_{z}\) atomic orbitals, both with nodal planes in the xy-plane, combine in a head-on overlap along the x-axis (internuclear axis), resulting in a σ (sigma) molecular orbital. The σ molecular orbital is formed due to the head-on overlap, creating a bonding molecular orbital with cylindrical symmetry around the internuclear axis. A π molecular orbital would require a sideways overlap, which does not occur between these specific atomic orbitals.

Step-by-step solution

Shapes and orientations of atomic orbitals

The \(d_{x z}\) atomic orbital is shaped like a four-leaf clover with two lobes along the x-axis and two lobes along the z-axis. The \(p_{z}\) atomic orbital has a simple dumbbell shape, with one lobe above the xy-plane (positive z) and one lobe below the xy-plane (negative z). Both orbitals have a nodal plane in the xy-plane. Step 2: Analyze the overlap between these atomic orbitals to form a molecular orbital

Overlap of atomic orbitals

Now that we know the shapes and orientations of the orbitals, we must determine how they overlap. Since the x-axis is the internuclear axis, the orbitals approach each other head-on, with the \(d_{x z}\) lobes along the x-axis overlapping with the \(p_{z}\) lobes. Step 3: Identify the type of molecular orbital formed, whether it is a σ or π orbital

Type of molecular orbital formed

Due to the head-on overlap between the \(d_{x z}\) and \(p_{z}\) orbitals along the x-axis (the internuclear axis), the resulting molecular orbital is a σ (sigma) orbital. Step 4: Provide an explanation for the result

Explanation for the σ molecular orbital formation

The σ molecular orbital is formed as a result of the head-on overlap between the \(d_{x z}\) and \(p_{z}\) orbitals. This type of overlap creates a bonding molecular orbital with cylindrical symmetry around the internuclear axis (x-axis in this case). In contrast, a π molecular orbital would require a sideways overlap of the orbitals, which does not occur between these specific atomic orbitals.

Key concepts

Atomic Orbitals Overlap

To understand molecular bonding, one of the fundamental concepts is atomic orbitals overlap. Atomic orbitals are regions in an atom where electrons reside and move. When atoms come close together to form a molecule, their atomic orbitals can overlap, providing the potential for bond formation.

The extent and manner of this overlap directly influence the strength and type of the chemical bond. For example, if the orbitals overlap directly, as seen with an s orbital from one atom and a p orbital from another, or a d and p orbital along the same axis, this head-on overlap leads to the formation of a strong bond known as a sigma bond.

In the exercise provided, we consider the overlap between a d_xz orbital and a p_z orbital. They have specific geometric orientations, such that when they approach each other along the internuclear x-axis, they overlap in a direct end-to-end fashion. This kind of interaction lays the foundation for strong sigma molecular orbital formation.

Sigma Molecular Orbital

A sigma molecular orbital (σ MO) is a result of end-to-end overlap of atomic orbitals, which typically produces a bonding molecular orbital. This bonding orbital allows for the sharing of electrons between the two atoms, which can significantly lower the energy of the system, creating a more stable molecule.

The identifier 'sigma' comes from the Greek letter σ, the first letter of the word 'sphaira,' meaning sphere. Indeed, the sigma bond has cylindrical symmetry about the internuclear axis. In the context of the exercise's result, the combination of d_xz and p_z atomic orbitals produces a σ MO because of the head-on overlap along the internuclear axis, the x-axis.

This direct overlap means that the electron density is highest in the region between the two nuclei, characteristic of sigma bonds, as opposed to being above and below the bond axis, which would be indicative of a pi (π) bond.

Chemical Bonding

The concept of chemical bonding is central to understanding how molecules are formed and why certain configurations are more stable than others. Chemical bonds arise from the attraction between atoms that results in the sharing or transfer of electrons. The three primary types of chemical bonds are ionic, covalent, and metallic bonds, each with distinct electron sharing or transferring characteristics.

In our exercise, the focus is on covalent bonding, where atoms share electrons in molecular orbitals. The formation of the sigma molecular orbital is a classic case of covalent bonding. The overlapping atomic orbitals allow atoms to share electrons and achieve a lower, more stable energy state.

Covalent bonds can vary in strength and length, and the overlap of orbitals dictates this variation. By understanding how atomic orbitals interact to form molecular orbitals, we gain insight into the structure and properties of molecules, enabling us to predict how they might react during chemical interactions.