Question

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

Short answer

Bonding orbitals have lower energy than antibonding orbitals.

Step-by-step solution

Understand the Molecular Orbitals

Molecular orbitals are formed by the combination of atomic orbitals. In our case: \( \sigma_{1s} \) and \( \sigma_{1s}^{*} \) orbitals are formed by the overlap of \(1s\) atomic orbitals, while \(\pi_{2p}\) and \(\pi_{2p}^{*}\) are formed by the overlap of \(2p\) atomic orbitals.

Visualize \(\sigma_{1s}\) and \(\sigma_{1s}^{*}\) Orbitals

The \(\sigma_{1s}\) orbital is a bonding molecular orbital, formed by the constructive interference of two \(1s\) orbitals from different atoms. It is cylindrically symmetrical around the bond axis. The \(\sigma_{1s}^{*}\) orbital is an antibonding orbital, formed by the destructive interference of the same \(1s\) orbitals, having a nodal plane perpendicular to the bond axis.

Visualize \(\pi_{2p}\) and \(\pi_{2p}^{*}\) Orbitals

For \(\pi_{2p}\) orbitals, the lobes of \(2p\) orbitals overlap sideways, creating a bonding interaction above and below the plane of the atoms. For \(\pi_{2p}^{*}\) orbitals, the overlap results in an antibonding interaction with nodal planes between the nuclei.

Compare Energies of Orbitals

Bonding molecular orbitals (\(\sigma_{1s}\) and \(\pi_{2p}\)) are lower in energy than their corresponding antibonding orbitals (\(\sigma_{1s}^{*}\) and \(\pi_{2p}^{*}\)) because antibonding orbitals introduce extra energy through destructive overlap.

Key concepts

Bonding Molecular Orbitals

Bonding molecular orbitals are crucial in understanding how molecules form. These orbitals arise when atomic orbitals from different atoms combine constructively. When this happens, the electron density between the nuclei increases, which leads to a stabilizing force that holds the atoms together. The increased electron density in the bond area means that the atoms are lower in potential energy compared to when they are isolated.

Some key characteristics of bonding molecular orbitals include:

• Lower energy compared to the original atomic orbitals
• Increased electron density between the atomic nuclei
• Aiding in the formation of stable molecules

The energy of a bonding molecular orbital, such as \( \sigma_{1s} \), is lower than its antibonding counterpart. This energy difference is one of the main reasons why molecules are stable at specific configurations.

Antibonding Orbitals

In contrast to bonding orbitals, antibonding orbitals arise from destructive interference when atomic orbitals overlap. An antibonding orbital is characterized by having a region of zero electron density, known as a nodal plane, which occurs between the atomic nuclei. Because of this destructive interference, the electron density between the nuclei is reduced, which destabilizes the molecule.

Key features of antibonding molecular orbitals include:

• Higher energy than either of the contributing atomic orbitals
• Presence of a nodal plane that decreases electron density between nuclei
• Contributing to the potential destabilization of a molecule when populated by electrons

An example is the \( \sigma_{1s}^{*} \) orbital, which has a higher energy due to the presence of a nodal plane that minimizes electron density at the bond axis. Typically, molecules are more stable when electrons occupy bonding rather than antibonding orbitals.

Sigma Orbitals

Sigma (\(\sigma\)) orbitals are a type of molecular orbital that form when atomic orbitals overlap end-to-end. This head-on overlap leads to a cylindrical shape of electron cloud distribution around the axis connecting the two bonded nuclei. This cylindrical symmetry is an easily recognizable characteristic of \(\sigma\) orbitals.

Some examples and features of sigma orbitals include:

• Formation from orbitals like \(s\) or \(p\) orbitals overlapping along the bond axis
• Characterized by having no nodal plane except for the one at infinity
• Include bonding (\(\sigma\)) and antibonding (\(\sigma^{*}\)) types

\(\sigma_{1s}\) is a classic example of a bonding \(\sigma\) orbital, whereas \(\sigma_{1s}^{*}\) represents its antibonding form. These orbitals play a significant role in determining the nature and strength of a chemical bond, particularly in simple diatomic molecules.

Pi Orbitals

Pi (\(\pi\)) orbitals are another type of molecular orbital formed by the lateral, or side-by-side, overlap of atomic orbitals, typically the \(p\) orbitals. Unlike sigma orbitals, \(\pi\) orbitals do not have cylindrical symmetry around the bond axis. Instead, they are located above and below the bond axis, creating regions where electron density is concentrated.

Characteristics of \(\pi\) orbitals include:

• Side-to-side overlap of \(p\) or \(d\) orbitals
• Presence of a nodal plane on the bond axis
• Can be bonding (\(\pi\)) or antibonding (\(\pi^{*}\))

The \(\pi_{2p}\) orbital demonstrates a bonding interaction, whereas \(\pi_{2p}^{*}\) indicates an antibonding scenario with nodal planes that disrupt direct bonding interactions between nuclei. Pi bonding is typically crucial in compounds with double or triple bonds, enhancing the molecule's stability and influencing chemical reactivity.