# 5.1.3: Molecular orbitals from d orbitals

In transition metals and other heavier elements, the $$d$$ orbitals may combine with other orbitals of compatible symmetry (and energy) to form molecular orbitals. generally, there are three types of bonding and antibonding interactions that may occur with $$d$$ orbitals; $$\sigma$$, $$\pi$$, and $$\delta$$ bonds. Figure $$\PageIndex{1}$$: The five $$3d$$ orbitals are shown. The orientation of the axes is consistent and $$z$$ axis is horizontal for convinience in drawing bonding along the $$z$$ axis (see examples below). (CC-BY-NC-SA; Kathryn Haas)

## $$\sigma$$ bonding with $$d$$ orbitals

$$sigma$$ bonds are symmetric with respect to the inter-nuclear axis (in a diatomic molecule, this is the $$z$$ axis). An example of a $$\sigma$$ bond formed by $$\d$$ orbitals is that of two $$d_{z^2}$$ orbitals (see Figure $$\PageIndex{2}$$). If a bonded atom is in a position other than on the $$z$$ axis (in an octahedral geometry, for example), sigma bonds can also form. For example, two $$d_{x^2-y^2}$$ orbitals on atoms bonded along the $$x$$ or $$y$$ axes could also form a $$\sigma$$ bond.

$$d$$ orbitals can also form $$\sigma$$ bonds with other types of orbitals with the appropriate symmetry. Examples of orbitals with appropriate symmetry are the $$s$$ orbital and certain $$p$$ orbitals on another atom, as shown below in Figure $$\PageIndex{2}$$. Figure $$\PageIndex{2}$$: Selected examples of $$\sigma$$ bonds involving $$d$$ orbitals along the $$z$$ internuclear axis (shown in bold horizontal line) between two atoms. (CC-BY-NC-SA; Kathryn Haas)

## $$\pi$$ bonding with $$d$$ orbitals

$$\pi$$ bonds are those with one node that is in-plane with the internuclear axis. $$\pi$$ bonds can for between two $$d$$ orbitals or between $$d$$ orbitals and other types of orbitals with comparable symmetry. An example of a $$\pi$$ bond between two $$d$$ orbitals is that formed by two $$d_{xz}$$ orbitals along the $$z$$ axis (shown in Figure $$\PageIndex{3}$$). $$d$$ orbitals can also form $$\pi$$ bonds using $$p$$ orbitals with compatible symmetry, as shown in Figure $$\PageIndex{3}$$. Figure $$\PageIndex{3}$$: Selected examples of $$\pi$$ bonds involving $$d$$ orbitals along the $$z$$ internuclear axis (shown in bold horizontal line) between two atoms. (CC-BY-NC-SA; Kathryn Haas)

## $$\delta$$ bonding with $$d$$ orbitals

$$\delta$$ bonds are those with two nodes that are in-plane with the internuclear axis. $$\delta$$ bonds can form between two $$d$$ orbitals with appropriate symmetry. For example, when two atoms bond along the $$z$$ axis, the $$d_{xy}$$ orbitals and the two $$d_{x^2-y^2}$$ orbitals can form $$\delta$$ bonds Figure $$\PageIndex{4}$$ Figure $$\PageIndex{4}$$: Selected examples of $$\delta$$ bonds involving $$d$$ orbitals along the $$z$$ internuclear axis (shown in bold horizontal line) between two atoms. (CC-BY-NC-SA; Kathryn Haas)

## Incompatible orbitals

In the descriptions above, we focused on how bonds (and antibonds) can be formed with $$d$$ orbitals. All bonding and non-bonding interactions require that orbitals have compatible symmetry to form productive interactions. It is worth mentioning that orbitals with symmetry that is incompatible with the $$d$$ orbitals will not form bonding and antibonding interactions. The figure below shows several sets of orbitals that are incompatible for bonding. Figure $$\PageIndex{5}$$: Several incompatible pairs of orbitals. (CC-BY-NC-SA; Kathryn Haas)

Modified or created by Kathryn Haas (khaaslab.com)