Question

How is the number of molecular orbitals approximated by a linear combination of atomic orbitals related to the number of atomic orbitals used in the approximation?

Short answer

The number of molecular orbitals formed is equal to the number of atomic orbitals used in the linear combination during the approximation process.

Step-by-step solution

Understanding the Linear Combination of Atomic Orbitals (LCAO)

The Linear Combination of Atomic Orbitals (LCAO) is a method used in quantum chemistry to approximate molecular orbitals by combining atomic orbitals. Each molecular orbital formed is approximated as a linear combination of atomic orbitals from the constituent atoms.

Relating the Number of Molecular Orbitals to Atomic Orbitals

According to the LCAO approach, when atomic orbitals combine, they form an equal number of molecular orbitals. Therefore, if 'n' atomic orbitals are combined, 'n' molecular orbitals are formed. This includes bonding, antibonding, and possibly nonbonding molecular orbitals.

Conclusion

The number of molecular orbitals formed when applying LCAO is equal to the number of atomic orbitals used in the approximation. This is consistent with the quantum mechanical principle of conservation of orbital.

Key concepts

Molecular Orbitals

When delving into the intricate world of quantum chemistry, one key concept is that of molecular orbitals (MOs). These are not merely theoretical constructs; they represent the probable locations where electrons reside in a molecule.

Molecular orbitals come into existence when atoms come together to form a molecule. This combination is not random but abides by specific principles, leading to the formation of orbitals that can span the entire molecule. In essence, a molecular orbital is the region in a molecule where there is a high probability of finding an electron. Understanding this concept is vital as it explains many properties of molecules, such as their reactivity and bond strength.

In our textbook exercise, we learn that molecular orbitals are a result of the Linear Combination of Atomic Orbitals (LCAO). By superimposing the atomic orbitals (AOs) from individual atoms, new hybridized orbitals emerge. These can be bonding, where the probability of electron density between the atomic nuclei increases, or antibonding, where this probability is decreased, which can lead to destabilization of the molecule. Sometimes, non-bonding orbitals are created, resembling the atomic orbitals of the isolated atoms but within the molecule.

Quantum Chemistry

At the core of quantum chemistry lies the behavior of subatomic particles, which includes electrons and their interactions with each other and atomic nuclei. Quantum chemistry combines principles from chemistry and physics to explain phenomena at the molecular scale.

This science is not intuitive for it goes beyond the realm of what we can see or feel, dipping into the nature of matter on the smallest scales. Quantum chemistry postulates that particles like electrons exhibit both particle-like and wave-like characteristics. This wave-particle duality is critical for understanding molecular behavior, especially when it comes to orbital formation and electron distribution.

Using mathematical models and computer simulations, quantum chemists can predict how molecules will act and how they will interact with one another. The LCAO method, discussed in our exercise, is one of these models and is particularly useful for approximating the shape of molecular orbitals and the energy levels of electrons within those orbitals.

Conservation of Orbitals

The principle of conservation of orbitals offers a deeper insight into molecular formation. Stemming from the foundational rules of quantum mechanics, it states that during the formation of molecular orbitals via the LCAO method, the number of molecular orbitals created is always equal to the number of atomic orbitals combined.

Why is this important? It maintains the fundamental principle of conservation in physics, which in this context, dictates that no atomic orbital is lost or gained in the process of molecule formation; they are merely transformed. The exercise highlights this beautifully: if you combine 'n' atomic orbitals, you end up with 'n' molecular orbitals. This one-to-one correspondence ensures that all electrons in an atom remain accounted for when they transition from atomic to molecular states.

The conservation of orbitals principle has implications for predicting the shape, stability, and reactivity of molecules. When students grasp the LCAO approach and how it adheres to conservation laws, they unlock the ability to anticipate the behaviors of molecules in more complex reactions.