Question

Which of the following combinations is not allowed in the LC \(\Lambda O\) method for the formation of a MO (consider the \(z\) -axis as the molecular axis?? (1) \(\mathrm{s}+\mathrm{p}_{2}\) (2) \(\mathrm{s}-\mathrm{p}_{x}\) (3) \(\mathrm{p}_{x}-\mathrm{p}_{x}\) (4) \(\mathrm{p}_{2}+\mathrm{p}_{z}\)

Short answer

The combination \(s-p_x\) is not allowed.

Step-by-step solution

Understand the LC \(\/Lambda O\) method

In the Linear Combination of Atomic Orbitals (LCAO) method for molecular orbital (MO) formation, atomic orbitals combine based on their symmetry with respect to the molecular axis. Orbitals must have the same symmetry properties to overlap effectively.

Analyze the combinations

Each given combination involves atomic orbitals. Evaluate if their symmetries are compatible for combination.

Check each option

1. Combination \(s+p_2\): s-orbitals are spherically symmetric, while \(p_2\) orbitals (rotated in the yz-plane) maintain cylindrical symmetry about the z-axis. This combination seems plausible. 2. Combination \(s-p_x\): s-orbitals have spherical symmetry, while \(p_x\) orbitals have cylindrical symmetry about the x-axis, incompatible for overlap. 3. Combination \(p_x-p_x\): Both \(p_x\) orbitals share the same symmetry properties about the x-axis, making this combination feasible. 4. Combination \(p_2+p_z\): \(p_2\) orbitals (in yz-plane) revolve around the z-axis and \(p_z\) orbitals also have rotational symmetry around the z-axis, making this combination plausible.

Identify the incorrect combination

From the analysis, the only combination which does not match symmetry considerations for overlap into molecular orbitals is the second option: \(s-p_x\).

Key concepts

Molecular Orbitals

Molecular orbitals (MOs) are regions in a molecule where electrons are likely to be found. They are formed by the combination of atomic orbitals from individual atoms. The critical idea is that when atomic orbitals combine, they create new orbitals that belong to the entire molecule, rather than to a single atom. These molecular orbitals can be bonding or anti-bonding, influencing the molecule's stability. For instance, bonding MOs lower the energy of the system, leading to a more stable molecule, while anti-bonding MOs can increase the energy. The combination of atomic orbitals to form MOs follows specific rules to ensure the proper overlap and bonding possibilities.

Symmetry of Atomic Orbitals

Symmetry is a crucial concept when determining if atomic orbitals can combine to form molecular orbitals. Each atomic orbital has a specific symmetry, described by how it behaves under various molecular axis transformations. For example:

• s-orbitals have spherical symmetry, meaning they look the same in all directions.
• p_x, p_y, and p_z orbitals each have cylindrical symmetry about the x, y, and z axes respectively.

For atomic orbitals to combine effectively, their symmetries with respect to the molecular axis must match. When evaluating combinations like in the exercise, the symmetries play a key role. If the symmetries are incompatible, the orbitals will not overlap correctly, and no stable molecular orbital will form.

Linear Combination of Atomic Orbitals

The Linear Combination of Atomic Orbitals (LCAO) method is a mathematical approach used to describe the formation of molecular orbitals. In this method, atomic orbitals are added together to construct new molecular orbitals, which can be bonding or anti-bonding. The LCAO method helps in predicting the shapes and energies of molecular orbitals. Key points include:

• Atomic orbitals must combine with the same symmetry to contribute to molecular orbitals.
• The combination can be constructive (leading to bonding MOs) or destructive (leading to anti-bonding MOs).
• Molecular orbitals formed will have different energy levels, with bonding MOs having lower energy than the corresponding atomic orbitals, and anti-bonding MOs having higher energy.

This approach is fundamental in understanding the electronic structure of molecules and predicting their chemical behavior.

Incompatible Orbital Combinations

Not all atomic orbitals can combine to form molecular orbitals. Incompatibilities usually arise due to symmetry mismatches. For instance, consider the example from the exercise:

• The combination of an s-orbital with a p_x orbital is incompatible because an s-orbital has spherical symmetry, while a p_x orbital has cylindrical symmetry about the x-axis.

This mismatch in symmetry means that these orbitals cannot overlap in a way that forms a stable molecular orbital. In contrast:

• Two p_x orbitals can combine because they share the same symmetry along the x-axis, allowing effective overlap.
• Similarly, p_2 and p_z orbitals can combine as their symmetries can align with respect to the molecular axis, especially when considering the yz-plane and the z-axis.

Understanding these incompatibilities is essential when predicting which atomic orbitals can interact to form molecular orbitals, thus providing insights into the electronic structure and reactivity of molecules.