The Theory of Atomic Spectroscopy: jj Coupling, Intermediate Coupling, and Configuration Interaction C. W. Haigh University College, Swansea, U.K Many introductory textbooks on spectroscopy give satisfactory accounts of Russell-Sannders (LS) coupling, but their treatment o f j coupling is generally very brief. Intermediate coupling will be accorded a t best about a sentence or two, and the topic of configuration interaction is omitted. Many advanced texts discuss these topics in detail, but these treatments are beyond the level of undergraduate chemists. We present a brief account of these ideas that eschews quantum-mechanical technicalities. jj Coupling The LS-coupling scheme is used when spin-orbit interaction i s small compared with the electrostatic repulsion between electrons that results in the separation between terms of a configuration. I assume that the reader is familiar with this scheme and with the concept of the vector model of the atom. The jj-coupling scheme is used when the spin-orbit interaction is relatively large. Then for each electron outside closed subshells i n an atom, its 1 vector i s coupled to its spin-vector s to give its own j vector. Unless 1 = 0 (for a n s electron) there are always two possible values of j. These j values define a term, which for two electrons is denoted, for example, by (id2). We then couple the jlj2 vectors, etc., to obtain a resultant vector J, which specifies a leuel within a term, denoted, for example, by til.j2)&
inequivalent Electrons When the electrons are inequivalent, the normal rules of vector coupling suff~ceto determine all levels without difficulty. Consider the first excited configuration of the neutral group IV atoms, which for lead is 6p7s (omitting closed subshells). For the 6p electron, we have 11 = 1,and hence
In the first, the vector-coupling rules would give us J = 0 or 1.But a J = 1level would imply that both electrons had the same values for all four quantum numbers,
in the MJ = +1state, and this is excluded: Only a J = 0 level is permitted. For the second term, there are no restrictions, and we have
For the third term, a J = 3 level is forbidden a s above, and the vector-coupling rules leave u s the possibility of J values of 0, 1, and 2. The reader would need to apply a slightly deeper level of theory to prove directly that J = 1 was impossible. However, the same result can easily be obtained indirectly as follows. The crucial point is that allowed values of J obtained i n LS and jcoupling must be the same because J i s a truly valid uantum number. Now, in LS coupling (e.g., for carbon 2p41 the levels are 3Po,1,2, ID2, and 'So. Because we already have a J = 1level i n our j coupling scheme, we cannot have a second one. Thus, the remaining levels are
and the two coupling schemes agree in their prediction of J values. Similarly, in a p3 configuration (the ground configuration of neutral bismuth) the LS levels (e.g., for nitrogen) are
For the 7s electron, The hypothetical j-coupling term Thus, there are two terms,
I n the former term, the vector-coupling rules give us J = 1or 2, and in the latter term, J = 0 or 1.Thus, the four levels are denoted
is clearly not allowed because a t least two electrons would have to possess the same set of n, 1,j , and m, values. If one bears in mind the J values allowed for
in the previous paragraph, and also the LS-coupling values above, one easily derives (with the help of the Pauli Principle) the allowed ji-coupling levels a s
Equivalent Electrons When equivalent electrons are present, the Pauli Exclusion Principle will, as in LS coupling, restrict the possibilities. We consider the corresponding ground configuration, for example, 6p2. Because t h e two electrons a r e now equivalent, we just have three terms,
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The Energies But the jj-coupling schemes (and also the LS) are not merely ways of deriving symbolic descriptions of atomic
Table 2. A Highly Excited Configuration of Neutral Ar 3p59s
Table 1. The Ground Configuration of Neutral Pb 6p2 Level
Experimentala
Column
(1)
Calculated (2)
j-couplingb 11
0
(2.210
(0)
Level Experimentala Calculated (3)
lntermediate couplingC
Column
(0)
(1)
3 1
126211.57
(2) jj couplingb (126211.57)
(3)
lntermediate couplingC (126211.57)
(F"2)2
R.M.S. dlffer-
ence
R ' ef
rG,
la.
b ~ e2. f ECai~~lated using ref 46.
Reflb. b~ei2. 5Calculateduslng the method of ref 4c
a
levels: They are mathematical approximations. When one (or the other) is valid, it is possible to write the energies as a simple linear function of quantities representing the spin-rbit coupling and the electrostatic repulsion. Thus, for the np2 case (above) i n j j coupling, we have, omitting a constant term, the following expressions for the energy.
5 is a measure of the spin-rbit coupling, and F 2 is a measure of the relevant electrostatic repulsion. (It would not be in accordance with the aims and level of this article to give their formal definitions.) jj-Coupling Limit
-
I t mieht ~. be thoueht that lead was a suff~cientvheavv ;itom for theji-coupling scheme ro have a good quantiteti!.~ validit\: The data in Table 1 icolumns 1 and 2 clearlv show this noi to be the case. Throughout this article, energies are given in cm? units. (Of course, technically these are wave numbers, El(hc)). Both Herzberg (3)and EdMn (4aa)point out that the first excited configuration of neutral lead, 6p7s, is far closer than this to the ji-coupling limit. However, because the electrostatic splitting ( l a ) of the
C,+J
,1
is 1250.90 and that of the
~ e l s
is 327.34, and because jj coupling predicts the ratio of these splittings to be 2 1 , clearly we are still far from agreement with the jj-coupling formulas. I t is generally true that excited (especially highly excited) configurations are more likely to approach the jj limit than are ground configurations, as is shown by a simple argument. For a series in which the valence configuration is fixed, and the excited electron has fixed 1, but successively higher n, it is easily seen that the electrostatic repulsion between the excited electron and the remaining electrons will rapidly diminish. However, particularly if the excited electron has 1 = 0, the spin-orbit coupling, due to the valence electrons, will be approximately constant. A particularly good example of this is shown in Table 2, columns 1and 2. Since p5 is a full subshell with one hole, the level symbols are the same a s those already obtained for the ps configuration. Although argon (Z = 18) is a very much lighter element than lead (Z = 821, clearly we do have pure jj coupling here. Thus, the terms and levels in jj coupling can be derived a s easily as those in LS coupling, and simple linear expressions exist for the energies. However, pure jj coupling, of which we have exhibited one good example, is very much rarer than pure LS coupling, for reasons subsequently explored. Intermediate Coupling Obviously, if LS coupling is valid for light elements, and ji coupling is approached by heavy atoms and cases such as the last example, there should be a large range of examoles. ~articularlvin the middle of the ~eriodictable. for which neither approximation is valid. Quantum-mechanical methods exist for handling this, and the method of calculation is called intermediate coupling. I t is immaterial whether the symbols of LS or of jj coupling are used because there is a continuous gradation between these limits. To explain the phenomena involved in intermediate coupling, we use the symbols of LS coupling. Only J (and Mjj are now truly valid quantum numbers. For example, in the ground configuration of neutral carbon mentioned above,
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Volume 72 Number 3 March 1995
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Table 3. The Ground Configuration of Neutral Sn 5p2
Level LS
Experimental
Calculated LS couplingb
jj
Table 4. The Ground Configuration of Neutral C 2p2
Level
Ex~erimental~
Calculated
Intermediate jj-couplingb Coupling EdlenC C.&S.*
'Ref i d . b ~ e f ~ . ' Ref 7a.
R.M.S.dif ference
'Ref ic b ~ e f 2 . 'Calculated using ref 46 Calculated using ref 6a.
the two levels with J = 0,labeled 'Po and 'So, are allowed to mix, and neither L nor S are now truly fixed. Thus, the ground level, instead of being a pure triplet, has some singlet character mixed in. Readers familiar with photochemistry, and the theory of dye lasers in particular, know that transitions between singlet and triplet states in molecules are spin-forbidden, but are nevertheless found experimentally and are of great importance. They are weakly allowed because the very small spin-orbit coupling in, for example, the relevant carbon, nitrogen, or oxygen atoms is nevertheless sufficient to mix a little singlet character into the socalled triplets and vice versa. We are here dealing with the same phenomena in atoms. In Table 1, column 3, we see that intermediate coupling gives a better account of the ground configuration of neutral lead than didjj coupling. Table 2, column 3, shows that even when the jj-coupling formulas have high validity, intermediate coupling does give a slight improvement. However, intermediate coupling, simply applied, cannot be regarded as a universal panacea in this field. I n the ground configuration of silicon (3p2), LS coupling with a spin-orbit coupling parameter 5 of 150 and an electrostatic repulsionintegralFz of 1017 ( 5 )gives a r m s difference with experimental line positions of 24 (2). Use of the'standard parameters of ref 4b in intermediate coupling gave a worse rms difference of 61. Of course a least-squares fit in intermediate coupling does slightly improve the LS couplingresnlts. Even in the analogous configuration of germanium (4p2j, LS coupling (with parameters 5 = 969, now quite comparahle with F2= 1026.5) gives a better fit than intermediate coupling with the parameters of ref 4b. The case of the analogous ground configuration of tin (5p2)is particularly instructive and is set out in Table 3. The intermediate coupling calculation with E d l h ' s parameters gives the best fit. But Condon and Shortley's parameters, derived from the same experimental data by a n equally logical method, give a substantially worse fit and a value differing by about 100 cm-'. The jj-coupling calculation is ridiculous and is clearly worse than the LS-coupling calculation. The results of this paragraph show that,
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Journal of Chemical Education
although authors using intermediate coupling may quote parameter values to apparently high precision, their actual reliability may be far lower. The reader may have been surprised t h a t for germanium, where 5 and F2are comparahle, the system is quite close to LS coupling, and for tin, where 5 is over twice Fz, the system is closer to LS than to jj coupling. I have not found the reason explicitly in any of the advanced texts: I t is hidden in the quantum-mechanical formulas. One may legitimately write that the larger the value of the ratio
