Question

Draw the LCAO model of ethylene in the bonding and antibonding orbital. Distinguish the ground state of ethylene from its excited state. Distinguish ${\pi }^2$ from ${\pi }^{\ast }\pi$.

Short answer

The LCAO model of ethylene can be constructed by combining the 2pz orbitals of the carbon atoms. In the bonding orbital ($\pi$), the wavefunctions of 2pz orbitals are added, resulting in constructive interference and electron density above and below the molecular plane. In the antibonding orbital (${\pi }^{\ast }$), the wavefunctions of 2pz orbitals are subtracted, resulting in destructive interference and a nodal plane in the middle. In the ground state, ethylene has a double bond between carbon atoms and its molecular orbital configuration is $({\sigma }_{1s}{)}^2({\sigma }_{1s^{\ast }}{)}^2({\sigma }_{2s}{)}^2({\sigma }_{2s^{\ast }}{)}^2({\sigma }_{2p}{)}^4({\pi }_{2p}{)}^2({\pi }_{2p^{\ast }}{)}^0$. In the excited state, one electron is promoted from the $\pi$ orbital to the ${\pi }^{\ast }$ orbital, weakening the double bond, and the configuration becomes $({\sigma }_{1s}{)}^2({\sigma }_{1s^{\ast }}{)}^2({\sigma }_{2s}{)}^2({\sigma }_{2s^{\ast }}{)}^2({\sigma }_{2p}{)}^4({\pi }_{2p}{)}^1({\pi }_{2p^{\ast }}{)}^1$. The ${\pi }^2$ symbol corresponds to the ground state, while ${\pi }^{\ast }\pi$ corresponds to the excited state of ethylene.

Step-by-step solution

Understand the molecular structure

Ethylene ($C_2{H}_4$) consists of two carbon atoms double bonded to each other and two hydrogen atoms single bonded to each carbon atom. It is a planar molecule with a bond angle of 120° around the carbon atoms.

Identify atomic orbitals to combine

In the LCAO model, we are going to focus on the carbon atoms' atomic orbitals that form the bonding and antibonding pi orbitals. Each carbon atom contributes two types of orbitals: a 2s orbital and a 2pz orbital (both are perpendicular to the plane of the molecule).

Construct the LCAO model for bonding orbital and anti-bonding orbital

To find the bonding combination of atomic orbitals, we add the wavefunctions of the 2pz orbitals on both carbon atoms: ${\varphi }_1={\varphi }_{2pz(C1)}+{\varphi }_{2pz(C2)}$. This creates constructive interference between the orbitals and results in electron density above and below the molecular plane, called a $\pi$ orbital. To find the antibonding combination of atomic orbitals, we subtract the wavefunctions of the 2pz orbitals on both carbon atoms: ${\varphi }_2={\varphi }_{2pz(C1)}-{\varphi }_{2pz(C2)}$. This creates destructive interference between the orbitals and results in a nodal plane in the middle, called a ${\pi }^{\ast }$ orbital.

Draw the orbital energy diagram and distinguish the ground state and excited state

In the ground state of ethylene, each carbon atom has two electrons in the 2s orbital and the remaining two electrons will occupy the $\pi$ orbital (bonding orbital). This results in a double bond between the carbon atoms, and the molecular orbital configuration is written as $({\sigma }_{1s}{)}^2({\sigma }_{1s^{\ast }}{)}^2({\sigma }_{2s}{)}^2({\sigma }_{2s^{\ast }}{)}^2({\sigma }_{2p}{)}^4({\pi }_{2p}{)}^2({\pi }_{2p^{\ast }}{)}^0$. In the excited state of ethylene, one electron is promoted from the $\pi$ orbital (bonding) to the ${\pi }^{\ast }$ orbital (antibonding). This weakens the double bond between the carbon atoms. The molecular orbital configuration is written as $({\sigma }_{1s}{)}^2({\sigma }_{1s^{\ast }}{)}^2({\sigma }_{2s}{)}^2({\sigma }_{2s^{\ast }}{)}^2({\sigma }_{2p}{)}^4({\pi }_{2p}{)}^1({\pi }_{2p^{\ast }}{)}^1$.

Distinguish ${\pi }^2$ from ${\pi }^{\ast }\pi$

The symbols ${\pi }^2$ and ${\pi }^{\ast }\pi$ refer to the electron occupancy of the pi orbitals. The ${\pi }^2$ notation means that both electrons are present in the bonding pi orbital, which corresponds with the ground state of ethylene. On the other hand, ${\pi }^{\ast }\pi$ denotes that one electron is in the bonding pi orbital and the other electron is in the antibonding pi orbital (${\pi }^{\ast }$), which corresponds with the excited state of ethylene.

Key concepts

Bonding and Antibonding Orbitals

In the molecular orbital theory, the concept of bonding and antibonding orbitals is central to understanding how molecules form and interact. When atoms combine to form a molecule, their atomic orbitals mix and overlap to create new orbitals called molecular orbitals.

• Bonding Orbitals: These are formed by constructive interference when atomic orbitals overlap in phase. This results in an increase of electron density between the two nuclei, stabilizing the molecule. In ethylene, the bonding orbital is known as the $\pi$ orbital.
• Antibonding Orbitals: These occur due to destructive interference when atomic orbitals overlap out of phase. This results in a decrease in electron density between nuclei and includes a nodal plane where electron density is zero, destabilizing the molecule. The antibonding orbital in ethylene is the ${\pi }^{\ast }$ orbital.

These orbitals play crucial roles in determining the stability and reactivity of molecules like ethylene.

Ground and Excited States

The ground and excited states of a molecule are differentiated based on the distribution of electrons in molecular orbitals. In the ground state, electrons occupy the lowest energy orbitals available. For ethylene, this means the electrons fill the $\pi$ bonding orbital, resulting in a stable double bond between the carbon atoms.In an excited state, energy is absorbed by the molecule, causing electrons to be promoted to higher energy orbitals. For ethylene, this involves an electron moving from the $\pi$ orbital to the ${\pi }^{\ast }$ antibonding orbital, which weakens the double bond due to less electron density being shared between the carbon atoms. Transitioning between these states influences the molecule's reactivity and properties.

Molecular Orbital Theory

Molecular orbital theory is a cornerstone of quantum chemistry that explains how atomic orbitals combine to form molecular orbitals. This theory provides a comprehensive model for understanding molecular properties and behavior.

• Molecular orbitals are formed by the linear combination of atomic orbitals (LCAO), which mathematically combines the individual atomic orbitals to describe the electron distribution in a molecule.
• They are classified into bonding and antibonding orbitals, influencing molecular stability and shape.
• This model helps in predicting chemical properties such as bond order, magnetism, and electronic configuration.

For ethylene, this means working with the carbon-atom $2s$ and $2p_z$ orbitals to detail how bonding structures and electron distributions yield chemical stability.

Constructive and Destructive Interference

In molecular orbital theory, wave-like properties of electrons result in interference patterns when atomic orbitals overlap. Constructive interference happens when wave functions of overlapping atomic orbitals align in phase, leading to an increased electron density in between nuclei due to the added wave functions. This phenomenon is crucial in forming bonding molecular orbitals since it lowers the overall energy of the molecule. Destructive interference occurs when wave functions are out of phase, thereby canceling each other out. This reduces electron density between atoms, leading to a nodal plane, and results in the formation of antibonding molecular orbitals. Understanding these concepts helps explain the stability differences between bonding and antibonding orbitals in molecules such as ethylene.

Electron Occupancy in Pi Orbitals

Electron occupancy in $\pi$ orbitals greatly affects the chemical characteristics of molecules like ethylene.

• Pi ($\pi$) Orbitals: These bonding orbitals contain electron pairs that stabilize the carbon-carbon double bond by high electron density.
• Pi-star (${\pi }^{\ast }$) Orbitals: Antibonding orbitals don't house electrons in the ground state for ethylene but do in the excited state, resulting in less bond strength.
• Configuration in the ground state is represented as ${\pi }^2$ (two electrons in the bonding orbital).
• In the excited state, one electron shifts to a higher energy ${\pi }^{\ast }$, described as $\pi {\pi }^{\ast }$, decreasing bond density.

Correctly predicting electron occupancy helps clarify molecular reactivity and physical properties.