Myoglobin O$_2$ Binding

Myoglobin is a protein found in muscle tissues that stores oxygen. It contains a heme group, which is a porphyrin ring with an iron (Fe) atom at its center. The oxygen molecule (O$_2$) binds directly to this central Fe atom. The question specifies that the heme group is in the xy-plane. This means the Fe atom lies in this plane, and the O$_2$ molecule typically approaches and binds along an axis perpendicular to this plane, which we consider the z-axis.

Formation of the $\sigma$-Bond

A chemical bond between two atoms or molecules involves the overlap of their atomic or molecular orbitals. A $\sigma$-bond is a type of covalent bond where the atomic or molecular orbitals overlap head-on along the internuclear axis, resulting in electron density concentrated along this axis.

In the context of O$_2$ binding to the Fe atom in myoglobin, a $\sigma$-bond is formed by the direct overlap of a suitable orbital from the Fe atom and a suitable molecular orbital from the O$_2$ molecule along the Fe-O$_2$ axis (the z-axis).

Iron Atomic Orbitals

The Fe atom in the heme group has several atomic orbitals available for bonding, including d-orbitals. Since the O$_2$ binds along the z-axis (perpendicular to the xy-plane of the heme), the Fe d-orbital that is primarily oriented along this axis is the $\text{d}_{z^2}$ orbital. The $\text{d}_{z^2}$ orbital has lobes pointing directly along the ±z axes, making it ideally positioned for head-on ($\sigma$) overlap with an orbital from the approaching O$_2$ molecule along the z-axis.

Other d-orbitals, such as $\text{d}_{xz}$, $\text{d}_{yz}$, $\text{d}_{xy}$, and $\text{d}_{x^2-y^2}$, have different orientations relative to the z-axis and are typically involved in $\pi$ bonding or other types of interactions depending on their symmetry and overlap with the O$_2$ orbitals.

O$_2$ Molecular Orbitals

The O$_2$ molecule has a set of molecular orbitals formed from the combination of atomic orbitals of the two oxygen atoms. The options provided refer to the $\pi$ (bonding) and $\pi^*$ (antibonding) molecular orbitals of O$_2$. These $\pi$ and $\pi^*$ orbitals are degenerate pairs with electron density located away from the internuclear O-O axis, either above/below or in front/behind the axis. They have $\pi$ symmetry with respect to the O-O axis.

Orbitals Involved in the Fe-O$_2$ $\sigma$-Bond

For a $\sigma$-bond to form along the Fe-O$_2$ axis (the z-axis), there must be overlap between Fe and O$_2$ orbitals that have $\sigma$ symmetry relative to this axis.

• The Fe $\text{d}_{z^2}$ orbital is clearly suitable for $\sigma$ overlap along the z-axis due to its shape and orientation.
• The O$_2$ $\pi^*$ molecular orbitals, in a strictly linear Fe-O-O arrangement, would have $\pi$ symmetry with respect to the Fe-O bond axis (which aligns with the O-O axis). However, O$_2$ binding to Fe in myoglobin typically occurs in an end-on fashion with a bent Fe-O-O angle. This bending changes the symmetry of the O$_2$ molecular orbitals relative to the Fe-O bond axis. In the bent configuration, one of the O$_2$ $\pi^*$ orbitals can have electron density oriented in a way that allows for significant overlap with the Fe $\text{d}_{z^2}$ orbital along the Fe-O bond axis, contributing to the $\sigma$-bond.

Therefore, the primary σ interaction involves the Fe $\text{d}_{z^2}$ atomic orbital. Based on theoretical models and the options provided, the interacting O$_2$ molecular orbital is the $\pi^*$ orbital, which contributes to the $\sigma$ bond via overlap with Fe $\text{d}_{z^2}$ in the bent configuration.

Contrast with $\pi$ Bonding

In addition to the $\sigma$-bond, there is also $\pi$ bonding between Fe and O$_2$. This involves the overlap of Fe $\text{d}_{xz}$ or $\text{d}_{yz}$ orbitals (which lie in the xz or yz planes and have $\pi$ symmetry with respect to the z-axis) with the O$_2$ $\pi^*$ molecular orbitals (which also have appropriate $\pi$ symmetry relative to the z-axis in the bent configuration). This $\pi$ interaction is often described as back-donation of electron density from filled Fe d-orbitals to the empty or partially filled O$_2$ $\pi^*$ orbitals.

However, the question specifically asks about the orbitals involved in the $\sigma$-bond formation.

Conclusion

Based on the analysis of orbital symmetries and the common description of the Fe-O$_2$ interaction in myoglobin (particularly considering the end-on bent geometry), the atomic orbital of Fe and the molecular orbital of O$_2$ involved in the formation of the $\sigma$-bond are the $\text{d}_{z^2}$ and $\pi^*$, respectively.

Thus, the orbitals involved are $\pi^*$ (O$_2$) and $\text{d}_{z^2}$ (Fe).