Some of the examples given below indicate notations bearing on the order of coupling of the electrons.*LS*Coupling (Russell-Saunders Coupling)

In the second example, the seven 3- 3
*d*^{7}^{4}F_{7/2} - 3
*d*^{7}(^{4}F)4*s*4*p*(^{3}P°)^{6}F°_{9/2} - 4
*f*^{7}(^{8}S°)6*s*6*p*^{2}(^{4}P)^{11}P°_{5} - 3
*p*^{5}(^{2}P°)3*d*^{2}(^{1}G)^{2}F°_{7/2} - 4
*f*^{10}(^{3}K2)6*s*6*p*(^{1}P°)^{3}L°_{6} - 4
*f*^{7}(^{8}S°)5*d*(^{7}D°)6*p*^{8}F°_{13/2} - 4
*f*^{7}(^{8}S°)5*d*(^{9}D°)6*s*(^{8}D°)7*s*^{9}D°_{5} - 4
*f*^{7}(^{8}S°)5*d*(^{9}D°)6*s*6*p*(^{3}P°)^{11}F_{8} - 4
*f*^{7}(^{8}S°)5*d*^{2}(^{1}G) (^{8}G°)6*p*^{7}F_{0} - 4
*f*(^{2}F°) 5*d*^{2}(^{1}G)6s (^{2}G)^{1}P°_{1}

*d*electrons couple to give a^{4}F term, and the 4*s*and 4*p*electrons couple to form the^{3}P° term; the final^{6}F° term is one of nine possible terms obtained by coupling the^{4}F grandparent and^{3}P° parent terms. The next three examples are similar to the second. The meaning of the index number 2 following the^{3}K symbol in the fifth example is explained in the section*LS*Coupling.The coupling in example 6 is appropriate if the interaction of the 5

*d*and 4*f*electrons is sufficiently stronger than the 5*d*-6*p*interaction. The^{7}D° parent term results from coupling the 5*d*electron to the^{8}S° grandparent, and the 6*p*electron is then coupled to the^{7}D° parent to form the final^{8}F term. A space is inserted between the 5*d*electron and the^{7}D° parent to emphasize that the latter is formed by coupling a term (^{8}S°) listed to the left of the space. Example 7 illustrates a similar coupling order carried to a further stage; the^{8}D° parent term results from the coupling of the 6*s*electron to the^{9}D° 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

^{11}F term, i.e., the^{9}D° term, is itself formed by the coupling of the 5*d*electron to the^{8}S° core term. Example 9 shows an^{8}G° parent term formed by coupling the^{8}S° and^{1}G grandparent terms. A space is again used to emphasize that the following (^{8}G°) term is formed by the coupling of terms listed before the space.A different order of coupling is indicated in the final example, the 5

*d*^{2}^{1}G term being coupled first to the external 6*s*electron instead of directly to the 4*f*core electron. The 4*f*(^{2}F°) core term is isolated by a space to denote that it is coupled (to the 5*d*^{2}(^{1}G)6*s*^{2}G term) only after the other electrons have been coupled. The notation in this particular case (with a single 4*f*electron) could be simplified by writing the 4*f*electron after the^{2}G 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].- 3
*jj*Coupling of Equivalent Electrons*j*indicates the angular momentum of one electron (*j = l*±^{1}/_{2}) or of each electron in an*l*group. Various ways of indicating which of the two possible_{j}^{N}*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*and_{i}*j*to represent total angular momenta.- (6
*p*^{2}_{1/2})_{0}

- (6
*p*^{2}_{1/2}6*p*_{3/2})°_{3/2}

- (6
*p*^{2}_{1/2}6*p*^{2}_{3/2})_{2}

- 4
*d*^{3}_{5/2}4*d*^{2}_{3/2}(^{9}/_{2}, 2)_{11/2}

The relatively large spin-orbit interaction of the 6

*p*electrons produces*jj*-coupling structures for the 6*p*^{2}, 6*p*^{3}, and 6*p*^{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 6*p*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 6*p*^{3}configuration divided into two groups according to the two possible*j*values. A similar notation is shown for the 6*p*^{4}level in the third example; this level might also be designated (6*p*^{-2}_{3/2})_{2}, the negative superscript indicating the two 6*p*holes. The (*J*_{1},*J*_{2})_{J}term and level notation shown on the right in the fourth example is convenient because each of the two electron groups 4*d*^{3}_{5/2}and 4*d*^{2}_{3/2}has more than one allowed total*J*_{i}value. The assumed convention is that*J*_{1}applies to the group on the left (*J*_{1}=^{9}/_{2}for the 4*d*^{3}_{5/2}group) and*J*_{2}to that on the right.- (6
*J*_{1}*j*or*J*_{1}*J*_{2}Coupling- 3
*d*^{9}(^{2}D_{5/2})4*p*_{3/2}(^{5}/_{2},^{3}/_{2})°_{3} - 4
*f*^{11}(^{2}H°_{9/2}2)6*s*6*p*(^{3}P°_{1}) (^{9}/_{2}, 1)_{7/2} - 4
*f*^{9}(^{6}H°)5*d*(^{7}H°_{8})6*s*6*p*(^{3}P°_{0}) (8,0)_{8} - 4
*f*^{12}(^{3}H_{6}) 5*d*(^{2}D)6*s*6*p*(^{3}P°) (^{4}F°_{3/2}) (6,^{3}/_{2})°_{13/2} - 5
*f*^{4}(^{5}I_{4})6*d*_{3/2}(4,^{3}/_{2})_{11/2}7*s*7*p*(^{1}P°_{1}) (^{11}/_{2}, 1)°_{9/2} - 5
*f*^{4}_{7/2}5*f*^{5}_{5/2}(8,^{5}/_{2})°_{21/2}7*p*_{3/2}(^{21}/_{2},^{3}/_{2})_{10} - 5
*f*^{3}_{7/2}5*f*^{3}_{5/2}(^{9}/_{2},^{9}/_{2})_{9}7*s*7*p*(^{3}P°_{2}) (9,2)°_{7}

The first five examples all have core electrons in

*LS*coupling, whereas*jj*coupling is indicated for the 5*f*core electrons in the last two examples. Since the*J*_{1}and*J*_{2}values in the final (*J*_{1},*J*_{2}) term have already been given as subscripts in the configuration, the (*J*_{1},*J*_{2}) 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 [3*d*^{9}(^{2}D_{5/2}) 4*p*_{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 [(^{3}H_{6}) (^{3}P°) (^{4}F°_{3/2})]_{13/2}or (^{3}H_{6},^{3}P°,^{4}F°_{3/2})_{13/2}. Similar economies of notation are of course possible, and often useful, in all coupling schemes.- 3
or*J*_{1}*l*Coupling (*J*_{1}*L*_{2}Coupling)*J*_{1}*K*- 3
*p*^{5}(^{2}P°_{1/2})5*g*^{2}[^{9}/_{2}]°_{5} - 4
*f*^{2}(^{3}H_{4})5*g*^{2}[3]_{5/2} - 4
*f*^{13}(^{2}F°_{7/2})5*d*^{2}(^{1}D)^{1}[^{7}/_{2}]°_{7/2} - 4
*f*^{13}(^{2}F°_{5/2})5*d*6*s*(^{3}D)^{3}[^{9}/_{2}]°_{11/2}

The final terms in the first two examples result from coupling a parent-level

**J**_{1}to the orbital angular momentum of a 5*g*electron to obtain a resultant, the**K***K*value being enclosed in brackets. The spin of the external electron is then coupled with theangular momentum to obtain a pair of**K***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*J*_{1}*L*_{2}coupling (examples 3 and 4). The last two examples are straightforward extensions of*J*_{1}*l*coupling, with the**L**_{2}and**S**_{2}momenta of the "external"' term (^{1}D and^{3}D in examples 3 and 4, respectively) replacing theand**l**momenta of a single external electron.**s**- 3
*LS*_{1}Coupling (*LK*Coupling)- 3
*s*^{2}3*p*(^{2}P°)4*f*G^{2}[^{7}/_{2}]_{3} - 3
*d*^{7}(^{4}P)4*s*4*p*(^{3}P°) 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

. The letter symbol for the final**L***L*value is listed with the configuration because this angular momentum is then coupled with the spin of the core (**S**_{1}) to obtain the resultantangular momentum of the final term (in brackets). The multiplicity of the [**K***K*] term arises from the spin of the external electron(s).- 3
### 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

SchemeQuantum numbers for vectors

that couple to give*J*Term

Symbol*LS**L,S*^{2S+1}*L**J*_{1}*J*_{2}*J*_{1},*J*_{2}( *J*_{1},*J*_{2})*J*_{1}*L*_{2}(→*K*)*K, S*_{2}^{2S2+1}[*K*]*LS*_{1}(→*K*)*K, S*_{2}^{2S2+1}[*K*]The parity is indicated by appended degree symbols on odd parity terms.