Question

Sketch the shapes of these molecular orbitals: \(\sigma_{1 s}\), \(\sigma_{1 s}^{\star}, \pi_{2 p},\) and \(\pi_{2 p}^{\star}\) How do their energies compare?

Short answer

The \(\sigma_{1s}\) and \(\sigma_{1s}^{\star}\) molecular orbitals consist of symmetrical lobes along the internuclear axis, with the antibonding orbital featuring a node in the middle. The \(\pi_{2p}\) and \(\pi_{2p}^{\star}\) molecular orbitals consist of lobes above and below the plane of the atoms, with additional nodes for the antibonding orbital. In terms of energy, the order from lowest to highest is \(\sigma_{1s}\), \(\sigma_{1s}^{\star}\), \(\pi_{2p}\), \(\pi_{2p}^{\star}\).

Step-by-step solution

Sketch the \(\sigma_{1s}\) molecular orbital

The \(\sigma_{1s}\) orbital results from the overlap of two 1s atomic orbitals along the internuclear axis. The sketch should represent two lobes of equal size symmetrically positioned along a connecting axis, looking almost like a dumbbell or a figure eight shape.

Sketch the \(\sigma_{1s}^{\star}\) molecular orbital

The \(\sigma_{1s}^{\star}\) is the antibonding counterpart of the \(\sigma_{1s}\). This orbital also consists of two lobes, but there is a node, or a planar region with no electron density, at the nucleus. This gives it a donut conformation around the nuclear axis, with positive and negative phases (usually shown with different colors or shading) on either side of the node.

Sketch the \(\pi_{2p}\) molecular orbital

The \(\pi_{2p}\) orbital is formed by the side-on overlap of two 2p atomic orbitals. This should be sketched as two vertically aligned lobes on each atom, indicating the location of electron density above and below the plane of the atoms.

Sketch the \(\pi_{2p}^{\star}\) molecular orbital

The \(\pi_{2p}^{\star}\) is the antibonding version of the \(\pi_{2p}\). There are two nodes, creating three lobes for this orbital. The central lobe lies between the two atomic nuclei, and the other two lobes project above and below this central lobe.

Compare the energies of the molecular orbits

In terms of energy, antibonding orbitals (\(\sigma_{1s}^{\star}\) and \(\pi_{2p}^{\star}\)) are higher in energy than their bonding counterparts (\(\sigma_{1s}\) and \(\pi_{2p}\)). In addition, 2p orbitals (both bonding and antibonding) are higher than the 1s orbitals due to the increased energy level and distance from the nucleus. Thus, the order of increasing energy is \(\sigma_{1s}\), \(\sigma_{1s}^{\star}\), \(\pi_{2p}\), and \(\pi_{2p}^{\star}\).

Key concepts

Orbital Shapes

The shape of molecular orbitals is crucial for understanding how atoms bond and the electronic structure of molecules. The molecular orbital known as \( \sigma_{1s} \) arises from the head-on overlap of two \( 1s \) atomic orbitals along the line that connects the nuclei of the bonded atoms. This orbital typically resembles a figure-eight or dumbbell shape, oriented along the internuclear axis. It consists of two lobes, where electron density is concentrated.
In contrast, the \( \sigma_{1s}^{\star} \) orbital is an antibonding orbital and has a significantly different shape. This molecular orbital also features two lobes, but with a nodal plane at the nucleus where there is no electron density. The nodal plane creates a symmetrical division across the axis, often represented with differentiated colors to show positive and negative phases of wave function, forming a sort of lobe pattern with a central void or node.
The \( \pi_{2p} \) molecular orbital represents side-to-side overlap of \( 2p \) atomic orbitals. Each lobe is positioned above and below the plane formed by the atomic nuclei, resulting in a dumbbell-like structure vertically aligned to the internuclear axis.
Meanwhile, the \( \pi_{2p}^{\star} \) indicates an antibonding orbital of \( \pi_{2p} \), combining two \( 2p \) orbitals with two nodes, dividing the orbital into three separate lobes. The configuration exhibits a central lobe flanked by lobes both above and below, which causes a distinct arrangement when compared to its bonding counterpart.

Energy Comparison

When discussing molecular orbitals, understanding their energy levels is key to grasping chemical bonding. Bonding orbitals, such as \( \sigma_{1 s} \) and \( \pi_{2 p} \), tend to be lower in energy because electrons are situated between the nuclei where they can help hold the atoms together.
Antibonding orbitals, like \( \sigma_{1s}^{\star} \) and \( \pi_{2p}^{\star} \), have electrons located in regions which do not contribute to bonding. They possess areas without electron density that interfere with the stabilizing interactions between nuclei, thereby making them higher in energy.
Additionally, within the same type (either bonding or antibonding), orbitals deriving from higher energy atomic orbitals—such as the \( 2p \) orbitals—are higher in energy than those from lower energy atomic orbitals, like \( 1s \). For instance, a \( \pi_{2 p} \) orbital will inherently have a greater energy than a \( \sigma_{1s} \), reflected in the increased distance of their respective electrons from the nucleus. Hence, the order from lowest to highest energy is \( \sigma_{1s} \), \( \sigma_{1s}^{\star} \), \( \pi_{2p} \), and \( \pi_{2p}^{\star} \).

Antibonding and Bonding Orbitals

In molecular orbital theory, bonding and antibonding orbitals play essential roles in forming the bonds between atoms. Bonding orbitals, such as \( \sigma_{1s} \) and \( \pi_{2 p} \), facilitate bonding by allowing electron density to be concentrated between atomic nuclei. This electron density helps to pull the nuclei toward each other, forming a stable bond.
Antibonding orbitals, like \( \sigma_{1s}^{\star} \) and \( \pi_{2 p}^{\star} \), offer a stark contrast. They possess regions called nodes, where electron density is absent. This lack of overlapping electron density between nuclei results in a higher energy that counteracts bonding effects and can reduce overall molecular stability.
When comparing different molecular orbitals, keep in mind that the presence of electrons in antibonding orbitals typically weakens bonds. What's interesting is that these orbitals can sometimes be occupied in molecular excitation processes, impacting the chemical characteristics and reactivity of molecules. Both bonding and antibonding orbitals contribute to the complete set of possible states electrons can occupy, influencing the nature of chemical bonds and the framework of molecular structures.