In this problem we'll construct the molecular orbitals of...

In this problem we'll construct the molecular orbitals of the dioxygen molecule $\left(\mathrm{O}_{2}\right) .$ Imagine bringing two oxygen atoms to within a bonding distance along the $x$ axis (the horizontal axis in the plane of the page), and imagine the molecular orbitals that would form from overlap of the valence atomic orbitals. (a) Sketch the bonding and antibonding combinations $(2 s \sigma$ atomic orbitals interact. Show the nodes. (Hint: See Problem 1.44.) (b) Sketch the bonding and antibonding combinations $(2 p \sigma$ and $2 p \sigma^{*}$ ) that result when the $2 p_{x}$ atomic orbitals overlap. (Note again that the $x$ axis is the horizontal axis in the plane of the page.) Show the nodes. (Hint: See Problem 1.45.) (c) Sketch the bonding and antibonding combinations $\left(2 p \pi_{y}\right.$ and $\left.2 p \pi_{y}^{*}\right)$ that result when the $2 p_{y}$ atomic orbitals overlap. (Assume that the $y$ axis is the vertical axis in the plane of the page.) Show the nodes. (Hint: See Problem 1.46.) (d) Show that the bonding and antibonding combinations $\left(2 p \pi_{z}\right.$ and $\left.2 p \pi_{z}^{*}\right)$ that result when the $2 p_{z}$ atomic orbitals overlap are identical to the $2 p \pi_{y}$ and $2 p \pi_{y}$ MOs in (c), but oriented at $90^{\circ}$. (Assume that the $z$ axis is the vertical axis perpendicular to the plane of the page.) (e) The energy order of these MOs and the component atomic orbitals is $2 s \sigma<2 s$ (atomic orbitals) $<2 s \sigma^{\text {? }}<2 p \sigma<2 p \pi_{y}=$ $2 p \pi_{z}<2 p$ (atomic orbitals) $<2 p \pi_{y}^{*}=2 p \pi_{z}^{*}<2 p \sigma^{*}$ Construct an orbital interaction energy diagram showing the energy levels of the atomic orbitals along with the energies of these MOs. Add the available valence electrons from two oxygen atoms.

In this problem we'll construct the molecular orbitals of the dioxygen molecule $\left(\mathrm{O}_{2}\right) .$ Imagine bringing two oxygen atoms to within a bonding distance along the $x$ axis (the horizontal axis in the plane of the page), and imagine the molecular orbitals that would form from overlap of the valence atomic orbitals.
(a) Sketch the bonding and antibonding combinations $(2 s \sigma$
atomic orbitals interact. Show the nodes. (Hint: See Problem 1.44.)
(b) Sketch the bonding and antibonding combinations $(2 p \sigma$ and $2 p \sigma^{*}$ ) that result when the $2 p_{x}$ atomic orbitals overlap. (Note again that the $x$ axis is the horizontal axis in the plane of the page.) Show the nodes. (Hint: See Problem 1.45.)
(c) Sketch the bonding and antibonding combinations $\left(2 p \pi_{y}\right.$ and $\left.2 p \pi_{y}^{*}\right)$ that result when the $2 p_{y}$ atomic orbitals overlap. (Assume that the $y$ axis is the vertical axis in the plane of the page.) Show the nodes. (Hint: See Problem 1.46.)
(d) Show that the bonding and antibonding combinations $\left(2 p \pi_{z}\right.$ and $\left.2 p \pi_{z}^{*}\right)$ that result when the $2 p_{z}$ atomic orbitals overlap are identical to the $2 p \pi_{y}$ and $2 p \pi_{y}$ MOs in
(c), but oriented at $90^{\circ}$. (Assume that the $z$ axis is the vertical axis perpendicular to the plane of the page.)
(e) The energy order of these MOs and the component atomic orbitals is
$2 s \sigma<2 s$ (atomic orbitals) $<2 s \sigma^{\text {? }}<2 p \sigma<2 p \pi_{y}=$
$2 p \pi_{z}<2 p$ (atomic orbitals) $<2 p \pi_{y}^{*}=2 p \pi_{z}^{*}<2 p \sigma^{*}$
Construct an orbital interaction energy diagram showing the energy levels of the atomic orbitals along with the energies of these MOs. Add the available valence electrons from two oxygen atoms.

Key Concepts

Molecular Orbital Theory

Molecular Orbital (MO) Theory describes how atomic orbitals combine to form molecular orbitals that are spread over the entire molecule. This theory is used to explain the bonding, electronic structure, and properties of molecules by considering the constructive and destructive interference of atomic wave functions.

Orbital Overlap and Symmetry

The formation of molecular orbitals relies on the overlap between atomic orbitals that have compatible symmetry and orientation. In this context, the proper alignment of 2s and 2p orbitals along specific axes allows them to combine either constructively or destructively, leading to bonding or antibonding interactions, respectively.

Bonding and Antibonding Interactions

When atomic orbitals overlap, they can form bonding molecular orbitals that have lower energy due to constructive interference, and antibonding molecular orbitals with higher energy resulting from destructive interference. The presence of nodes in the resulting orbitals is indicative of a change in the phase of the wave function and is a hallmark of antibonding interactions.

Nodal Planes in Molecular Orbitals

Nodes are regions in a molecular orbital where the probability density of finding an electron is zero. The number and position of nodes are critical in distinguishing between bonding and antibonding orbitals, with antibonding orbitals generally having one or more nodal planes, which affects the stability and energy of the MO.

Energy Ordering of Molecular Orbitals

The relative energies of atomic orbitals and the molecular orbitals formed from them determine the stability and electronic configuration of a molecule. An orbital energy diagram ranks the orbitals based on their energy levels, showing how electrons fill the bonding and antibonding orbitals according to the Aufbau principle and Hund’s rule.

Orbital Interaction Diagrams

Orbital interaction diagrams graphically represent the interactions between atomic orbitals and the formation of molecular orbitals. These diagrams not only show energy levels but also the contributions of various overlapping orbitals, making them a useful tool for visualizing the distribution of electrons in molecules and understanding chemical bonding.

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