4.3 LS and jj Coupling Schemes

Angular momentum coupling in multi-electron atoms can be tricky. LS and jj coupling schemes help us understand how electrons interact. LS coupling works better for lighter atoms, while jj coupling suits heavier ones.

The choice between LS and jj depends on spin-orbit interaction strength. As atomic number increases, we gradually shift from LS to jj coupling. For some atoms, we might need to use both schemes to get the full picture.

LS vs jj Coupling Schemes

Angular Momentum Coupling Schemes

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• LS coupling (Russell-Saunders coupling) and jj coupling describe angular momentum coupling in multi-electron atoms
• The choice between LS and jj coupling depends on the relative strength of spin-orbit interaction compared to electrostatic interactions between electrons
• The transition from LS to jj coupling as atomic number increases is gradual
• For atoms with intermediate atomic numbers, a combination of both coupling schemes may be necessary to accurately describe electronic states and their relative energies

Differences in Coupling Mechanisms

• In LS coupling:
  • Orbital angular momenta (l) of individual electrons couple to form total orbital angular momentum (L)
  • Spin angular momenta (s) of individual electrons couple to form total spin angular momentum (S)
  • Total angular momentum (J) is obtained by coupling L and S
• In jj coupling:
  • Orbital angular momentum (l) and spin angular momentum (s) of each electron couple to form total angular momentum (j) for each electron
  • Total angular momentum (J) of the atom is obtained by coupling individual j values of electrons

LS Coupling for Lighter Atoms

Applicability and Term Symbols

• LS coupling is more appropriate for describing electronic states of lighter atoms (typically up to the first two rows of the periodic table)
• In these atoms, spin-orbit interaction is relatively weak compared to electrostatic interactions between electrons
• Term symbol for an electronic state in LS coupling is written as $^{2S+1}L_J$
  • $S$ is total spin angular momentum quantum number
  • $L$ is total orbital angular momentum quantum number (S, P, D, F, ... for L = 0, 1, 2, 3, ...)
  • $J$ is total angular momentum quantum number

Determining Term Symbols in LS Coupling

1. Determine electron configuration of the atom and number of unpaired electrons
2. Calculate total spin angular momentum quantum number (S) by adding individual spin quantum numbers (ms) of unpaired electrons
3. Calculate total orbital angular momentum quantum number (L) by adding individual orbital angular momentum quantum numbers (ml) of unpaired electrons
4. Determine possible values of total angular momentum quantum number (J) using triangular inequality: $|L - S| \leq J \leq L + S$
5. Write term symbol as $^{2S+1}L_J$, with appropriate letter for L and possible values of J as subscripts

Ground State Term Symbol and Hund's Rules

• Ground state term symbol has the highest spin multiplicity (2S+1) and highest L value within that multiplicity, according to Hund's rules
• Hund's rules for determining relative energies of electronic states in LS coupling:
  1. State with highest spin multiplicity (2S+1) has lowest energy
  2. Among states with same spin multiplicity, one with highest total orbital angular momentum (L) has lowest energy
  3. For states with same S and L values:
     • One with lowest J has lowest energy if shell is less than half-filled
     • One with highest J has lowest energy if shell is more than half-filled

jj Coupling for Heavier Atoms

Applicability and Term Symbols

• jj coupling is more appropriate for describing electronic states of heavier atoms (typically from the third row onwards)
• In these atoms, spin-orbit interaction becomes stronger and comparable to electrostatic interactions between electrons
• Term symbol for an electronic state in jj coupling is written as $(j_1, j_2, ..., j_N)_J$
  • $j_1, j_2, ..., j_N$ are total angular momentum quantum numbers of individual electrons
  • $J$ is total angular momentum quantum number of the atom

Determining Term Symbols in jj Coupling

1. Determine electron configuration of the atom and quantum numbers (n, l, j) for each electron
2. Couple j values of individual electrons to obtain possible values of total angular momentum quantum number (J) using triangular inequality: $|j_1 - j_2| \leq J \leq j_1 + j_2$
3. Write term symbol as $(j_1, j_2, ..., j_N)_J$, with possible values of J as subscripts

Ground State Term Symbol and Landé Interval Rule

• In jj coupling, ground state term symbol is the one with lowest value of J, according to Landé interval rule
• Landé interval rule: State with lowest value of total angular momentum quantum number (J) has the lowest energy

Electronic State Energies: LS vs jj

Energy Differences in LS Coupling

• In LS coupling, energy differences between electronic states are primarily determined by electrostatic interactions between electrons
• Relative energies of electronic states are determined by Hund's rules (as mentioned earlier)
  • Examples of Hund's rules:
    • Carbon atom (1s²2s²2p²) ground state term symbol is $^3P_0$ (highest spin multiplicity and highest L within that multiplicity)
    • Nitrogen atom (1s²2s²2p³) ground state term symbol is $^4S_{3/2}$ (highest spin multiplicity)

Energy Differences in jj Coupling

• In jj coupling, energy differences between electronic states are primarily determined by spin-orbit interaction
• Relative energies of electronic states are determined by Landé interval rule (as mentioned earlier)
  • Examples of Landé interval rule:
    • Lead atom (1s²2s²2p⁶3s²3p⁶3d¹⁰4s²4p⁶4d¹⁰4f¹⁴5s²5p⁶5d¹⁰6s²6p²) ground state term symbol is $(1/2, 1/2)_0$ (lowest J value)
    • Bismuth atom (1s²2s²2p⁶3s²3p⁶3d¹⁰4s²4p⁶4d¹⁰4f¹⁴5s²5p⁶5d¹⁰6s²6p³) ground state term symbol is $(3/2, 3/2)_0$ (lowest J value)