Question

What type of molecular orbital would result from the in-phase combination of two \(d_{x z}\) atomic orbitals shown below? Assume the \(x\) -axis is the internuclear axis.

Short answer

The resulting molecular orbital from the in-phase combination of two \(d_{x z}\) atomic orbitals would be a bonding molecular orbital, represented as \(\sigma_{d_{xz}}\).

Step-by-step solution

Understand the shape of the d_{x z} atomic orbital

A \(d_{x z}\) atomic orbital has a shape where four lobes are arranged in the xz plane, with two lobes on the positive half of the x-axis and the other two lobes on the negative half of the x-axis. The lobes are also symmetric about the z-axis. Keep in mind that the atomic orbital has a nodal plane, which is a plane in which the probability of finding electrons is zero, between the positive and negative lobes, that's the yz plane.

Visualize the in-phase combination of two d_{x z} atomic orbitals

In an in-phase combination, the lobes with the same phase (sign of the wave function) are overlapped, which results in a constructive interference, and thus, an increase in the electron probability density. In this case, we have two \(d_{x z}\) atomic orbitals. Since the x-axis is the internuclear axis, the positive lobes will be pointing toward each other, and the negative lobes will be pointing away from each other.

Predict the resulting molecular orbital

In the in-phase combination of two \(d_{x z}\) atomic orbitals, the positive lobes merge constructively to form an enlarged lobe along the internuclear axis (x-axis). Similarly, the negative lobes also combine constructively to form enlarged lobes along the x-axis. The resulting molecular orbital is a bonding molecular orbital, as the increased electron density between the two nuclei helps to hold them together. This bonding orbital is often represented as \(\sigma_{d_{xz}}\), where the sigma indicates a symmetric molecular orbital along the internuclear axis.

Key concepts

d orbitals

The concept of "d orbitals" is essential for understanding the molecular orbital theory, especially in transition metals. The "d" in "d orbitals" stands for diffuse, referring to the shape and size of the orbitals in the electron cloud. These orbitals come into play after the s and p orbitals in an electron configuration.

There are five different types of d orbitals labeled as:

• \(d_{xy}\)
• \(d_{yz}\)
• \(d_{zx}\)
• \(d_{x^2-y^2}\)
• \(d_{z^2}\)

Each of these orbitals has a distinct shape, which can be important when predicting how they overlap with orbitals from other atoms to form molecular orbitals. In general, the d orbitals have more complex geometric shapes compared to the s and p orbitals. For example, the \(d_{xz}\) orbital, which is crucial in this problem, has four lobes, two pointing toward each x and z-axis, and it is perpendicular to the yz plane. There’s a node in the plane where no electrons can be found because the probability of electron presence is zero. These special characteristics of d orbitals allow for unique and various bonding scenarios and are essential in understanding the structure of more complex molecules.

in-phase combination

An in-phase combination in the context of molecular orbitals occurs when two or more atomic orbitals combine constructively, meaning their wave functions align such that the same phases (positive or negative) overlap.

Constructive overlap leads to an increase in electron density between the atomic nuclei, which strengthens the connection between the atoms, fostering a stable molecular structure. This superposition of wave functions is similar to the phenomenon you might have seen with waves, where constructive interference results in a larger wave.

For \(d_{xz}\) orbitals, when two of these orbitals come together in-phase — that is, with positive lobes overlapping with positive lobes and negative with negative — the result is a higher electron density along the internuclear axis (i.e., along the line connecting the nuclei). This constructive overlap eventually forms a bonding molecular orbital characterized by this increased density. The in-phase combination fundamentally changes the energy landscape of the atoms involved by creating a zone of expanded electron density that can stabilize the resulting molecular structure.

bonding molecular orbital

The formation of a bonding molecular orbital is perhaps the most exciting result of the in-phase combination of atomic orbitals, such as two \(d_{xz}\) orbitals.

A bonding molecular orbital (BMO) is a type of molecular orbital that ensures increased stability and lower energy for the molecule compared to its constituent atoms. This is due to the greater electron density found between the two nuclei, which effectively "pulls" the atoms together and creates a bond. The bonding molecular orbital exhibits a sigma (\(\sigma\)) designation when the electron density is symmetrically distributed along the internuclear (x) axis.

When a \(d_{xz}\) orbital combines in-phase as described, this forms a \(\sigma_{d_{xz}}\) bonding orbital. Notably, it's this increased electron density in the newly formed molecular orbital that binds the atoms together more robustly than if they were separate. So, BMOs are not only a hallmark of molecular orbital theory but also a vital component explaining molecular stability and the overall geometry of molecules.