3.5. Spectra of Multi-Electron Atoms

While hydrogenic spectra (one-electron system) are simplest, real atoms typically have many electrons. Electron–electron interactions, spin–orbit coupling, and other effects make their spectra more complex. To describe energy levels in multi-electron atoms, we use term symbols and selection rules.

In a hydrogen atom, there is only one electron, resulting in simpler spectral lines. However, in multi-electron atoms, the presence of additional electrons leads to interactions between them, which complicates the spectral patterns observed. Specifically, electrons in different orbitals can influence each other’s energy levels due to repulsion. Terms like 'spin-orbit coupling' refer to the interaction between an electron’s spin and its movement through the electric field created by the nucleus, which further complicates the energy levels. To organize and predict these complex spectra, scientists use 'term symbols'—notations that represent the total spin and orbital angular momentum of all the electrons in an atom—and 'selection rules', which dictate how electrons can transition between energy levels.

Think of a multi-electron atom like a busy office full of employees (electrons). If there's only one employee, it's easy to see what they do and how they interact with the environment (like the simple spectral lines of hydrogen). But as more employees join (more electrons), they have to share resources and space, creating complex interactions that affect everyone's work dynamics. The term symbols and selection rules serve as the office guidelines to manage how these employees (electrons) interact and transition between tasks (energy levels).

A term symbol takes the form ^(2S+1)L_J, where: ● S is the total spin quantum number (sum of the individual electron spins). ● 2S+1 is called the multiplicity. If S = 1, for example, multiplicity = 3, and we write a superscript 3. ● L is the total orbital angular momentum quantum number (sum of the individual ℓ values), designated by letters: • L = 0 → S • L = 1 → P • L = 2 → D • L = 3 → F • L = 4 → G … etc. ● J is the total angular momentum quantum number (vector sum of L and S). Possible values of J go from |L – S| up to L + S in integer steps.

The term symbol gives critical information about the energy levels in complex atoms. Each component indicates specific characteristics of the electrons within the atom. The total spin quantum number (S) sums individual spins, which are crucial for determining how electrons combine. The notation '2S+1' signifies how many different orientations of this spin exist, known as multiplicity. The letter L corresponds to the type of angular momentum generated by the arrangement of all electron orbitals, which can determine the atom's reactivity and magnetic properties. Finally, J includes contributions from both orbitals and spin, dictating how the atom interacts with light and other fields.

Imagine a sports team where each player (electron) has their own position (orbital) and skills (spin). The term symbol is like a team roster that combines all these factors. The total spin (S) shows how players work together (e.g., offense or defense), while L describes their overall strategy (e.g., play style like 'S' for a simple game or 'P' for a more aggressive move). J indicates how flexible the team can be based on players’ positions and skills, helping predict how they’ll react under pressure (like changes when interacting with light).

For electric dipole transitions (the most common type responsible for strong spectral lines), the selection rules are: ● The total spin must not change: ΔS = 0. ● The total orbital angular momentum must change by one unit: ΔL = +1 or –1. ● The total angular momentum J may change by 0 or ±1, except that a transition from J = 0 to J = 0 is forbidden. ● Parity must change: the electron must go from an orbital of one parity to an orbital of the opposite parity (for example from s to p, which is even → odd, or from p to d, which is odd → even).

These selection rules are crucial because they define the allowed transitions that electrons can make when they absorb or emit energy. The first rule, where the total spin must remain unchanged, ensures that energy states remain stable during transitions. The second rule states that changes in orbital momentum must occur, indicating that the electron is moving to a different type of orbital. The conditions on J ensure that interactions stay within specific boundaries, while the parity condition ensures that transitions between energy levels follow physical laws governing symmetry. These rules help explain why certain spectral lines are observed while others are not.

Consider a dance performance. The selection rules are like the choreographer's instructions setting the rules of how dancers (electrons) can move between positions on stage (energy levels). The rules dictate that dancers must keep their formations (spin) constant while changing their dance styles (orbital types) by rehearsing specific moves. Under certain conditions, like the lighting on stage (external fields), some moves (transitions) are allowed while others are not. If a dancer tries to perform a move that stays in the same formation and style, they would be infringing on the choreographer's rules.