In the Classroom
When Do Arrows Not Have Tips? Igor Novak Department of Chemistry, Faculty of Science, National University of Singapore, Singapore 119260, Singapore;
[email protected] Background Spin–orbit coupling (SOC) is important for understanding electronic structure and electronic states and is thus an integral part of many quantum chemistry or advanced inorganic chemistry courses. The graphical method of boxes-andarrows is usually used to describe electronic states and SOC in atoms. This method, as it stands, can only be applied to light atoms (LS-coupling). We present an analogous version of the method that can be applied to heavy atoms (the case of jjcoupling). The use of the same graphical method emphasizes the identical physical nature of the energy terms in both light and heavy atoms: Coulombic energy of charge distributions and spin–orbit coupling (magnetic energy of the interaction between angular momenta). Students are usually introduced to only one of the two SOC schemes: Russell-Saunders (or LS) coupling. The other ( jj-coupling) is usually not discussed, except for a vague remark that it is more difficult to use and applies to heavy atoms. However, since the boundary between “light” and “heavy” atoms is somewhat arbitrary, students are left wondering which scheme to apply in specific cases. Several nongraphical methods have been described for SOC (1, 2). These methods are a rehash of Condon and Shortley’s classical work (3). Still, undergraduates find the graphical method of “boxes-andarrows” most accessible and we present here a graphical method that is based on well-known tables in Gauerke and Campbell (2) and Condon and Shortley (3). Method We shall illustrate the application of the “boxes-andarrows” method to LS and jj schemes for the np2 electron configuration. In this configuration, the orbital angular momentum quantum numbers l1, l 2 both have value 1 and ml 1, ml 2 each take values 1,0, {1. L and S designate the total orbital and spin angular momentum quantum numbers.
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We assume that electron interaction energy is larger than SOC interactions. Arrow-up signifies ms = 1/2; arrow-down ms = {1/2. The Pauli principle allows at most two arrows with opposite orientation to be placed in any one box. The order of orientations is immaterial. Each box corresponds to a microstate with projections mL and mS. Conventionally, a Pauli-consistent pair of L,S values is identified by first selecting the microstate with the highest value of mL and then seeking the one with the highest mS; the associated (2L + 1)(2S + 1) degenerate microstates are then eliminated from the tabulation. The
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Scheme I
jj-Coupling (Schemes IIa–IIc) 1.
We assume that SOC energy is larger than electron interaction energy.
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Each electron is represented by a tipless arrow. The reason for arrows being tipless is that in the jj scheme spin angular momentum is added to orbital angular momentum for each electron
LS-Coupling (Scheme I) 1.
whole process is repeated until all permitted L,S terms have been realized. States with different L and S have different energies because of different electron interactions. The combination of 15 possible microstates (Scheme I) gives 1S , 1D, and 3P states. SOC induces additional small changes in energy for all three states (1S , 1D , and 3P ). States with different J values have different energies. Finally, the permitted J values associated with each L,S term are found from the formula J = L + S, L + S – 1, …, |L – S|, giving 1S0, 1D2, 3 P2,1,0 states.
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j=l+s which leaves only j and → mj as the good quantum numbers, as the z-component of j is mj h⁄ and its magnitude is given → by the magnitude of j = h⁄ [ j( j + 1)]1/2. 3.
The Pauli principle applies and requires that not all quantum numbers can be equal.
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The addition of spin and orbital angular momentum for an individual electron yields resultant j states according to the rule j = l + s, …, |l – s|. Here the rule gives j = 3/2 or 1/2 for each of the two electrons and allows the enumeration of microstates.
Journal of Chemical Education • Vol. 76 No. 10 October 1999 • JChemEd.chem.wisc.edu
In the Classroom
a. Case { j1 = 3/2, j2 = 3/2} puts two tipless arrows (electrons) into 4 boxes, resulting in 6 microstates. The number of boxes is 4 because there are 4 possible projections mj of j = 3/2, as shown in the diagram. We find the corresponding J = Σi ji from the projections mJ, and these give J = 2,0. Because j1 = j 2, the Pauli principle requires that mj1 ≠ mj2 and hence only a single tipless arrow is allowed per box (Scheme IIa). [mJ ] mj
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Scheme IIa 6 microstates for j1 = 3/2 j2 = 3/2
b. Case { j1 = 3/2, j2 = 1/2} comprises two nonequivalent electrons j1 ≠ j2 and hence the Pauli principle allows mj1 = mj2; i.e., up to two tipless arrows may fill a box. NOTE: mj1 = 3/2,1/2, {1/2, {3/2; mj2 = 1/2, {1/2. The two electrons are not equivalent and hence there two different microstates with mJ = 0 (Scheme IIb), corresponding to J = 2,1 here. [mJ ] mj
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Scheme IIb 8 microstates for j1 = 3/2 and j 2 = 1/2
c. Case { j1 = 1/2, j2 = 1/2} has two arrows in two boxes, giving only a single possible microstate and thus J = 0. Only two boxes are available because there are only two possible projections for j = 1/2. Because j1 = j2 the Pauli
principle requires mj1 ≠ mj2 and hence there can be only one tipless arrow per box. [mJ ] mj
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Scheme IIc 1 microstate for j1 = 1/2, j2 = 1/2
The combination of IIa–IIc gives the final result, which is J = 2, 2, 1, 0, 0. Note that LS coupling (Scheme I) produces the same values of J, but here they are associated with specific LS states. Conclusion Our unified approach to spin-orbit coupling has several pedagogical advantages. First, it clarifies the meanings of boxes and arrows. Students tend to associate boxes with orbitals and think, for example, that electrons in p orbitals should always be represented by three boxes because there are three orbitals of the p type (p1, p0, p{1). Second, the use of the same graphical method emphasizes that spin–orbit coupling has identical physical meanings in the LS and jj schemes and there is no conceptual difference between light and heavy atoms in this respect. Third, the method can be readily extended to analysis of moments of any kind (rotational, magnetic, etc.), because it is not tied to the orbital concept. Fourth, an alternative single tipless scheme could be used for both LS and jj cases. Each microstate is described by six boxes, one for each spin–orbital. The spin–orbitals (boxes) are labeled by mJ = mL + mS (i.e., 1 + 1/2, 1 – 1/2, 0 + 1/2, 0 – 1/2, {1 + 1/2, {1 – 1/2). Each of these boxes contains at most a single tipless arrow and a single scheme would describe both LS and jj coupling. This radical enumeration scheme circumvents the potentially confusing problem of how antiparallel spin arrangements (for two electrons) are components of both a triplet state and a singlet state. However, it leads to the abolition of arrows as well as the notions of LS and jj coupling. In view of the fact that LS and jj coupling cases are commonly distinguished in textbooks (although LS and jj are of course extreme cases) we choose not to pursue this suggestion further. The scheme can be easily derived from our preceding discussion. Literature Cited 1. Arias, F.; Sagues, F. Educ. Chem. 1990, 27, 83. 2. Gauerke, S. J.; Campbell, M. L. J. Chem. Educ. 1994, 71, 457. 3. Condon, E. U.; Shortley, G. H. The Theory of Atomic Spectra; Cambridge University Press: Cambridge, 1959.
JChemEd.chem.wisc.edu • Vol. 76 No. 10 October 1999 • Journal of Chemical Education
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