LS and jj Coupling Schemes

Hello! Welcome to the final lesson in our module on Advanced Angular Momentum and Spin.

In the previous lesson, we analyzed the fine structure of the hydrogen atom, which involved coupling the single electron's orbital angular momentum $\vec{L}$ and its spin $\vec{S}$ to form a total angular momentum $\vec{J}$. This was our first look at how internal interactions (specifically, spin-orbit coupling) determine the "correct" basis for describing energy states.

Today, we generalize this idea to multi-electron atoms. When multiple electrons are present, we have several angular momenta to consider (${\vec{l}}_1,{\vec{s}}_1,{\vec{l}}_2,{\vec{s}}_2,\dots$). The crucial question becomes: in what order do we combine them? The answer depends on the relative strengths of two key interactions: the electrostatic repulsion between electrons and the spin-orbit interaction for each electron. This leads to two distinct pictures, or coupling schemes: LS-coupling for lighter atoms and jj-coupling for heavier atoms.

By the end of this lesson, you will be able to apply both LS-coupling and jj-coupling schemes to determine the term symbols that label the quantum states of multi-electron atoms.

1. Two Competing Interactions, Two Coupling Schemes

For a multi-electron atom, the Hamiltonian contains the simple central field term $H_0$, plus two significant perturbations:

1. Residual Coulomb Interaction ($H_{ee}$): This is the portion of the electron-electron electrostatic repulsion that is not captured by the averaged central potential. It's a purely orbital interaction and tends to couple the electrons' orbital motions.
2. Spin-Orbit Interaction ($H_{SO}$): As we saw last time, this term is of the form ${\sum }_i\xi (r_i){\vec{l}}_i\cdot {\vec{s}}_i$. It's a relativistic magnetic interaction that couples each electron's own spin to its own orbit.

The competition between these two effects determines the coupling scheme. A clear explanation of this is provided in the following text.

In summary:

• LS-coupling (Russell-Saunders coupling): For light atoms (low Z), $H_{ee}\gg H_{SO}$. The strong electrostatic repulsion forces the orbital momenta ${\vec{l}}_i$ to couple together and the spin momenta ${\vec{s}}_i$ to couple together first.
• jj-coupling: For heavy atoms (high Z), relativistic effects become more pronounced, making the spin-orbit interaction for each electron very strong. Here, $H_{SO}\gg H_{ee}$. Each electron's ${\vec{l}}_i$ and ${\vec{s}}_i$ couple together first.

Let's now explore each scheme in detail.

2. LS-Coupling: The Light Atom Regime

In this scheme, we follow a three-step process to find the total angular momentum $\vec{J}$:

1. Combine all individual orbital angular momenta to get the total orbital angular momentum $\vec{L}$:
   $\vec{L}={\sum }_i{\vec{l}}_i$
   The possible values for the quantum number $L$ range from $|l_1-l_2|$ to $l_1+l_2$ for two electrons.

2. Combine all individual spin angular momenta to get the total spin angular momentum $\vec{S}$:
   $\vec{S}={\sum }_i{\vec{s}}_i$
   For two electrons, the total spin quantum number $S$ can be 0 (singlet) or 1 (triplet).

3. Finally, combine the total orbital and total spin angular momenta to get the grand total angular momentum $\vec{J}$:
   $\vec{J}=\vec{L}+\vec{S}$
   The quantum number $J$ can take values from $|L-S|$ to $L+S$ in integer steps.

The resulting energy levels are classified by term symbols of the form ${}^{2S+1}{L}_J$, where:

• $2S+1$ is the spin multiplicity (1=singlet, 2=doublet, 3=triplet, etc.).
• $L$ is the total orbital angular momentum, represented by a capital letter (S for L=0, P for L=1, D for L=2, F for L=3, ...).
• $J$ is the total angular momentum quantum number.

Case 1: Non-Equivalent Electrons

When electrons are in different subshells (e.g., a 2p electron and a 3p electron), they are "non-equivalent." In this case, finding the term symbols is a straightforward application of adding angular momenta, as the Pauli exclusion principle does not impose extra restrictions.

The following video demonstrates this for an sp configuration.

Case 2: Equivalent Electrons and Hund's Rules

The more interesting and common case involves "equivalent" electrons, which are in the same subshell (e.g., the two 2p electrons in the ground state of Carbon, 1s²2s²2p²). Here, the Pauli exclusion principle is critical: no two electrons can have the same set of four quantum numbers ($n,l,m_l,m_s$). This rule forbids some of the term symbols that would otherwise be allowed.

To find the allowed terms, we must construct a list of all possible unique microstates consistent with the Pauli principle and then group them into $LS$ multiplets. This is a fundamental technique in atomic physics. The ground state of carbon (2p²) is the classic example.

The next video provides an exceptionally clear, step-by-step guide through this process and then introduces Hund's rules to determine the ordering of the energy levels. Given your background, the detailed derivation will be very instructive.

Hund's Rules are a set of empirical rules that determine the term symbol of the ground state:

1. Maximize Spin Multiplicity $S$: The term with the highest value of $S$ has the lowest energy. This is due to the exchange interaction, which keeps electrons with parallel spins further apart, reducing their Coulomb repulsion.
2. Maximize Orbital Angular Momentum $L$: For terms with the same maximum $S$, the one with the highest value of $L$ is lowest in energy. Heuristically, a higher $L$ corresponds to electrons orbiting in the same direction, allowing them to stay further apart.
3. Determine Total Angular Momentum $J$: For the ground state term (with maximum $S$ and $L$), the value of $J$ is:
   • $J=|L-S|$ if the subshell is less than half-full.
   • $J=L+S$ if the subshell is more than half-full.

This third rule is a direct consequence of the sign of the spin-orbit coupling energy.

3. jj-Coupling: The Heavy Atom Regime

For heavy atoms, the strong nuclear charge makes electrons move at relativistic speeds, dramatically increasing the spin-orbit interaction. This interaction now dominates over the residual Coulomb repulsion. The coupling order is inverted:

1. For each electron, couple its orbital and spin angular momenta to get its individual total angular momentum ${\vec{j}}_i$:
   ${\vec{j}}_i={\vec{l}}_i+{\vec{s}}_i$
   For a p-electron ($l=1,s=1/2$), the possible $j$ values are 1/2 and 3/2.

2. Combine all the individual total angular momenta ${\vec{j}}_i$ to get the grand total angular momentum $\vec{J}$:
   $\vec{J}={\sum }_i{\vec{j}}_i$
   The quantum number $J$ can take values from $|j_1-j_2|$ to $j_1+j_2$ for two electrons.

Notice that total $L$ and total $S$ are not good quantum numbers in this scheme; the energy levels are first split according to the $\{j_i\}$ configuration and then further split by the weaker electron-electron interaction into states of definite $J$.

The following resource provides a concise worked example for the 6p² configuration of Lead (Pb, Z=82), a classic case of jj-coupling.

4. Correlation Between Coupling Schemes

LS and jj coupling are idealized limits. Real atoms often lie somewhere in between. A correlation diagram is a powerful tool for visualizing the transition from one scheme to the other as the atomic number $Z$ increases.

The key principle is that the total angular momentum $J$ and the parity are always good quantum numbers, regardless of the coupling scheme. The number of states with a given value of $J$ for a specific configuration must be the same in both limits.

This diagram beautifully illustrates that the two coupling schemes are just different basis sets for describing the same underlying physical states. Which basis is more "natural" (i.e., closer to being diagonal) depends on the atom's properties.

Conclusion

Today, we've explored the methods used to classify the complex energy levels of multi-electron atoms. We saw that the competition between electron-electron repulsion and spin-orbit interaction gives rise to two distinct but related pictures.

Key Takeaways:

• LS-coupling is used for light atoms (low Z) where electrostatic forces dominate. We first find total $L$ and $S$, then combine them to get $J$. The states are labeled by ${}^{2S+1}{L}_J$.
• jj-coupling is used for heavy atoms (high Z) where relativistic spin-orbit effects dominate. We first find individual $j_i=l_i+s_i$, then combine them to get the total $J$. States are labeled by their $(j_1,j_2,\dots {)}_J$ configuration.
• The Pauli exclusion principle is crucial for determining the allowed terms for equivalent electrons. The tabular method of analyzing microstates is the standard procedure.
• Hund's rules provide a recipe for identifying the ground state term symbol within the LS-coupling scheme.
• LS and jj-coupling are two limits of a continuous transition. Total angular momentum J is always a good quantum number.

Preview of the next lesson:
This lesson concludes our deep dive into angular momentum and its applications in atomic structure. We now shift our focus from static structure to dynamics. In the next module, we will begin our study of Approximation Methods for Quantum Dynamics, starting with one of the most important tools in a physicist's arsenal: time-dependent perturbation theory. We will learn how to calculate transition rates between quantum states when a time-varying potential, such as an electromagnetic wave, is applied.