NIST: Atomic Spectros. - Different Coupling Schemes

9. Notations for Different Coupling Schemes

The following examples make clear the meaning of the different coupling-scheme notations. Not all the configurations in the examples have been identified experimentally, and some of the examples of a particular coupling scheme given for heuristic purposes may be physically inappropriate. Cowan [3] describes the physical conditions for the different coupling schemes and gives experimental examples.

LS Coupling (Russell-Saunders Coupling)
Some of the examples given below indicate notations bearing on the order of coupling of the electrons.
1. 3d ⁷ ⁴F_7/2
2. 3d ⁷(⁴F)4s4p(³P°) ⁶F°_9/2
3. 4f ⁷(⁸S°)6s6p²(⁴P) ¹¹P°₅
4. 3p ⁵(²P°)3d ²(¹G) ²F°_7/2
5. 4f ¹⁰(³K2)6s6p(¹P°) ³L°₆
6. 4f ⁷(⁸S°)5d (⁷D°)6p ⁸F°_13/2
7. 4f ⁷(⁸S°)5d (⁹D°)6s (⁸D°)7s ⁹D°₅
8. 4f ⁷(⁸S°)5d (⁹D°)6s6p(³P°) ¹¹F₈
9. 4f ⁷(⁸S°)5d ²(¹G) (⁸G°)6p ⁷F₀
10. 4f(²F°) 5d ²(¹G)6s (²G) ¹P°₁
In the second example, the seven 3d electrons couple to give a ⁴F term, and the 4s and 4p electrons couple to form the ³P° term; the final ⁶F° term is one of nine possible terms obtained by coupling the ⁴F grandparent and ³P° parent terms. The next three examples are similar to the second. The meaning of the index number 2 following the ³K symbol in the fifth example is explained in the section LS Coupling.
The coupling in example 6 is appropriate if the interaction of the 5d and 4f electrons is sufficiently stronger than the 5d-6p interaction. The ⁷D° parent term results from coupling the 5d electron to the ⁸S° grandparent, and the 6p electron is then coupled to the ⁷D° parent to form the final ⁸F term. A space is inserted between the 5d electron and the ⁷D° parent to emphasize that the latter is formed by coupling a term (⁸S°) listed to the left of the space. Example 7 illustrates a similar coupling order carried to a further stage; the ⁸D° parent term results from the coupling of the 6s electron to the ⁹D° grandparent.
Example 8 is similar to examples 2 through 5, but in 8 the first of the two terms that couple to form the final ¹¹F term, i.e., the ⁹D° term, is itself formed by the coupling of the 5d electron to the ⁸S° core term. Example 9 shows an ⁸G° parent term formed by coupling the ⁸S° and ¹G grandparent terms. A space is again used to emphasize that the following (⁸G°) term is formed by the coupling of terms listed before the space.
A different order of coupling is indicated in the final example, the 5d ² ¹G term being coupled first to the external 6s electron instead of directly to the 4f core electron. The 4f (²F°) core term is isolated by a space to denote that it is coupled (to the 5d ²(¹G)6s ²G term) only after the other electrons have been coupled. The notation in this particular case (with a single 4f electron) could be simplified by writing the 4f electron after the ²G term to which it is coupled. It appears more important, however, to retain the convention of giving the core portion of the configuration first.
The notations in examples 1 through 5 are in the form recommended by Russell, Shenstone, and Turner [10], and used in both the Atomic Energy States [11] and Atomic Energy Levels [8,] [12] compilations. The spacings used in the remaining examples allow different orders of coupling of the electrons to be indicated without the use of additional parentheses, brackets, etc.
Some authors assign a short name to each (final) term, so that the configuration can be omitted in tables of classified lines, etc. The most common scheme distinguishes the low terms of a particular SL type by the prefixes a, b, c, ..., and the high terms by z, y, x, ... [12].
jj Coupling of Equivalent Electrons
This scheme is used, for example, in relativistic calculations. The lower-case j indicates the angular momentum of one electron (j = l ± ¹/₂) or of each electron in an l_j^N group. Various ways of indicating which of the two possible j values applies to such a group without writing the j-value subscript have been used by different authors; we give the j values explicitly in the examples below. We use the symbols J_i and j to represent total angular momenta.
1. (6p ²_1/2)₀
2. (6p ²_1/2 6p_3/2)°_3/2
3. (6p ²_1/2 6p ²_3/2)₂
4. 4d ³_5/2 4d ²_3/2 (⁹/₂, 2)_11/2
The relatively large spin-orbit interaction of the 6p electrons produces jj-coupling structures for the 6p ², 6p ³, and 6p ⁴ ground configurations of neutral Pb, Bi, and Po, respectively; the notations for the ground levels of these atoms are given as the first three examples above. The configuration in the first example shows the notation for equivalent electrons having the same j value l_j^N, in this case two 6p electrons each having j = ¹/₂. A convenient notation for a particular level (J = 0) of such a group is also indicated. The second example extends this notation to the case of a 6p ³ configuration divided into two groups according to the two possible j values. A similar notation is shown for the 6p ⁴ level in the third example; this level might also be designated (6p ^-2_3/2)₂, the negative superscript indicating the two 6p holes. The (J₁, J₂)_J term and level notation shown on the right in the fourth example is convenient because each of the two electron groups 4d ³_5/2 and 4d ²_3/2 has more than one allowed total J_i value. The assumed convention is that J₁ applies to the group on the left (J₁ = ⁹/₂ for the 4d ³_5/2 group) and J₂ to that on the right.
J₁ j or J₁ J₂ Coupling
1. 3d ⁹(²D_5/2)4p_3/2 (⁵/₂, ³/₂)°₃
2. 4f ¹¹(²H°_9/2 2)6s6p(³P°₁) (⁹/₂, 1)_7/2
3. 4f ⁹(⁶H°)5d (⁷H°₈)6s6p(³P°₀) (8,0)₈
4. 4f ¹²(³H₆) 5d(²D)6s6p(³P°) (⁴F°_3/2) (6, ³/₂)°_13/2
5. 5f ⁴(⁵I₄)6d_3/2 (4,³/₂)_11/27s7p(¹P°₁) (¹¹/₂, 1)°_9/2
6. 5f ⁴_7/25f ⁵_5/2 (8,⁵/₂)°_21/27p_3/2 (²¹/₂, ³/₂)₁₀
7. 5f ³_7/2 5f ³_5/2 (⁹/₂, ⁹/₂)₉7s7p(³P°₂) (9,2)°₇
The first five examples all have core electrons in LS coupling, whereas jj coupling is indicated for the 5f core electrons in the last two examples. Since the J₁ and J₂ values in the final (J₁, J₂) term have already been given as subscripts in the configuration, the (J₁, J₂) term notations are redundant in all these examples. Unless separation of the configuration and final term designations is desired, as in some data tables, one may obtain a more concise notation by simply enclosing the entire configuration in brackets and adding the final J value as a subscript. Thus, the level in the first example can be designated as [3d ⁹(²D_5/2) 4p_3/2]°₃. If the configuration and coupling order are assumed to be known, still shorter designations may be used; for example, the fourth level above might then be given as [(³H₆) (³P°) (⁴F°_3/2)]_13/2 or (³H₆, ³P°, ⁴F°_3/2)_13/2. Similar economies of notation are of course possible, and often useful, in all coupling schemes.
J₁l or J₁L₂ Coupling (J₁K Coupling)
1. 3p ⁵(²P°_1/2)5g ²[⁹/₂]°₅
2. 4f ²(³H₄)5g ²[3]_5/2
3. 4f ¹³(²F°_7/2)5d ²(¹D) ¹[⁷/₂]°_7/2
4. 4f ¹³(²F°_5/2)5d6s(³D) ³[⁹/₂]°_11/2
The final terms in the first two examples result from coupling a parent-level J₁ to the orbital angular momentum of a 5g electron to obtain a resultant K, the K value being enclosed in brackets. The spin of the external electron is then coupled with the K angular momentum to obtain a pair of J values, J = K ± ¹/₂ (for K ≠ 0). The multiplicity (2) of such pair terms is usually omitted from the term symbol, but other multiplicities occur in the more general J₁L₂ coupling (examples 3 and 4). The last two examples are straightforward extensions of J₁l coupling, with the L₂ and S₂ momenta of the "external"' term (¹D and ³D in examples 3 and 4, respectively) replacing the l and s momenta of a single external electron.
LS₁ Coupling (LK Coupling)
1. 3s ²3p(²P°)4f G ²[⁷/₂]₃
2. 3d ⁷(⁴P)4s4p(³P°) D° ³[⁵/₂]°_7/2
The orbital angular momentum of the core is coupled with the orbital angular momentum of the external electron(s) to give the total orbital angular momentum L. The letter symbol for the final L value is listed with the configuration because this angular momentum is then coupled with the spin of the core (S₁) to obtain the resultant K angular momentum of the final term (in brackets). The multiplicity of the [K] term arises from the spin of the external electron(s).
Coupling Schemes and Term Symbols
The coupling schemes outlined above include those now most frequently used in calculations of atomic structure [3]. Any term symbol gives the values of two angular momenta that may be coupled to give the total electronic angular momentum of a level (indicated by the J value). For configurations of more than one unfilled subshell, the angular momenta involved in the final coupling derive from two groups of electrons (either group may consist of only one electron). These are often an inner group of coupled electrons and an outer group of coupled electrons, respectively. In any case the quantum numbers for the two groups can be distinguished by subscripts 1 and 2, so that quantum numbers represented by capital letters without subscripts are total quantum numbers for both groups. Thus, the quantum numbers for the two vectors that couple to give the final J are related to the term symbol as follows:

Coupling
Scheme Quantum numbers for vectors
that couple to give J Term
Symbol

LS L,S ^2S+1L

J₁ J₂ J₁, J₂ (J₁, J₂)

J₁ L₂(→ K) K, S₂ ^2S₂+1[K]

LS₁(→ K) K, S₂ ^2S₂+1[K]

The parity is indicated by appended degree symbols on odd parity terms.

Coupling Scheme	Quantum numbers for vectors that couple to give J	Term Symbol


LS	L,S	^2S+1L
J₁ J₂	J₁, J₂	(J₁, J₂)
J₁ L₂(→ K)	K, S₂	^2S₂+1[K]
LS₁(→ K)	K, S₂	^2S₂+1[K]

9. Notations for Different Coupling Schemes

LS Coupling (Russell-Saunders Coupling)

jj Coupling of Equivalent Electrons

J1 j or J1 J2 Coupling

J1l or J1L2 Coupling (J1K Coupling)

LS1 Coupling (LK Coupling)

Coupling Schemes and Term Symbols

J₁ j or J₁ J₂ Coupling

J₁l or J₁L₂ Coupling (J₁K Coupling)

LS₁ Coupling (LK Coupling)