Repeat Exercise 12.51 for a linear chain of eight lithium atoms. (f) How does the HOMO–LUMO energy gap for this case compare to that of the four-atom case?

Verified step by step guidance

1. insert step 1: Understand the concept of HOMO and LUMO. The Highest Occupied Molecular Orbital (HOMO) and the Lowest Unoccupied Molecular Orbital (LUMO) are important in determining the electronic properties of molecules.
2. insert step 2: Recognize that the problem involves a linear chain of lithium atoms. In a linear chain, the atomic orbitals combine to form molecular orbitals.
3. insert step 3: Recall that as the number of atoms in a chain increases, the number of molecular orbitals also increases, leading to a smaller energy gap between HOMO and LUMO.
4. insert step 4: Compare the eight-atom chain to the four-atom chain. With more atoms, the molecular orbitals are more closely spaced, which typically results in a smaller HOMO-LUMO gap.
5. insert step 5: Conclude that the HOMO-LUMO energy gap for the eight-atom lithium chain is expected to be smaller than that of the four-atom chain due to the increased number of molecular orbitals and closer energy levels.

Key Concepts

HOMO-LUMO Energy Gap

The HOMO-LUMO energy gap refers to the energy difference between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO). This gap is crucial in determining the electronic properties of a molecule, including its stability and reactivity. A smaller gap often indicates a greater likelihood of electronic transitions, which can affect the material's conductivity and optical properties.

Molecular Orbital Theory

Molecular Orbital Theory describes how atomic orbitals combine to form molecular orbitals that can be occupied by electrons. In a linear chain of atoms, the overlap of atomic orbitals leads to the formation of delocalized molecular orbitals, which can significantly influence the HOMO-LUMO gap. Understanding this theory helps in predicting how changes in the number of atoms in a chain affect electronic properties.

Size Dependence in Nanostructures

In nanostructures, the size and arrangement of atoms can greatly influence their electronic properties. As the number of atoms in a linear chain increases, the HOMO-LUMO energy gap typically decreases due to increased orbital overlap and delocalization of electrons. This size dependence is a key factor in materials science and nanotechnology, affecting how materials behave at the nanoscale.