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The lithium ion \((\text{Li}^{2+})\) is a fascinating subject to explore, especially when it involves understanding its energy level diagram. When we refer to the lithium ion \((\text{Li}^{2+})\), we're talking about a lithium atom that has lost two of its three electrons. This occurs because the lithium atom strives to achieve a stable electronic configuration by removing its outer electrons.
In a completely neutral lithium atom, denoted simply as lithium or Li, the atomic number \(Z\) is 3. This tells us that in its neutral state, lithium has three protons and three electrons. However, for a \((\text{Li}^{2+})\) ion, two electrons are missing, leaving it with only one electron.
Understanding this is crucial because the behavior and properties of ions like \((\text{Li}^{2+})\) dramatically change due to their electron loss, affecting how they interact with electromagnetic radiation and consequently how they are depicted in energy level diagrams.

Quantum numbers play a key role in the world of quantum mechanics. They are the 'address' or the parameters we use to describe the specific energy states of an electron within an atom. Think of them as the numerical tags helping scientists pinpoint where an electron might be in its cloud-travel around the nucleus.
One of the most important quantum numbers is the principal quantum number, represented by \(n\). This number is crucial because it determines the electron's energy level, essentially giving us a "ladder-rung" number of sorts.
\(n\) can be any positive integer (1, 2, 3,...), and as \(n\) increases, the energy level gets higher and the electron is found further from the nucleus. In our exercise of dealing with a \((\text{Li}^{2+})\) ion, each different \(n\) value represents a different potential energy state for the single remaining electron. This is one of the reasons understanding quantum numbers is so integral to building and interpreting energy level diagrams.

Calculating energy levels is a fascinating journey that bridges physics and chemistry, allowing us to predict an electron's possible places around an atom's nucleus. For hydrogen-like ions such as \((\text{Li}^{2+})\), the energy level can be calculated using the formula: \[E = -\frac{{13.6 \cdot Z^{2}}}{n^{2}} \, \text{eV}\] This formula shows just how \(E\), the energy, is dependent on the principal quantum number \(n\) and the atomic number \(Z\).
The negative sign signifies that we are considering the bound states of electrons, meaning electrons are trapped by the nucleus's electrostatic potential. In practical terms, lower energy values (more negative) signify more stability and closer proximity to the nucleus.
Using this formula, for the \((\text{Li}^{2+})\) ion with \(Z=3\), you can calculate and plot energy levels for different \(n\) values. This calculation will allow you to construct an energy level diagram, which visually represents where the electron can "exist" and "transition" between different energy states while emitting or absorbing photons in the process.