Obtaining the Electron Angular Momentum Coupling Spectroscopic

Research: Science and Education

Obtaining the Electron Angular Momentum Coupling Spectroscopic Terms, jj Hugo Orofino and Roberto B. Faria* Instituto de Química, Universidade Federal do Rio de Janeiro 21941-972 Rio de Janeiro, RJ, Brazil *[email protected]

Russell-Saunders (LS) coupling is presented in many physical chemistry and inorganic chemistry textbooks to describe the energy levels of an atom. This coupling scheme assumes that total angular momentum, L, couples with total spin angular momentum, S, to produce the total angular momentum for the atom, J. The labels then obtained are used to describe the electronic transitions that produce the line spectra of the chemical elements and to label the low-field electronic states of coordination compounds when using Tanabe-Sugano diagrams. To obtain the LS terms, we have found the procedure proposed by Orchin and Jaffe (1) to be the simplest and most straightforward in the case of electronic configurations containing equivalent electrons. This procedure avoids obtaining terms that violates the Pauli exclusion principle in a very simple way. However, for heavy atoms and many excited states of heavy and light atoms, electron angular momentum coupling, jj, is a better description for the atom electronic states, because the spinorbit interaction becomes greater than the electrostatic interaction. In this case, the orbital angular momentum, l, and the spin angular momentum, s, of each electron couple to give the electron total angular momentum j. Then, these j vectors for each electron couple to give the atom total angular momentum J. As has been shown by Gauerke and Campbell (2), one of the reasons to consider jj coupling is to explain the greater number of lines in the atomic spectra of group 14 heavy elements, compared with the lighter elements (see Table 4 in ref 2). The reason for the additional lines is related to the selection rules. Although electronic transitions between singlet and triplet terms are forbidden for the light group 14 elements, C and Si, these transitions are easily detected for heavier Sn and Pb. This is a good reason to include the jj coupling when presenting the atomic electronic transitions to students. Different ways to obtain the spectroscopic terms considering the jj coupling have been described (2- 6), but these articles are a bit clumsy and do not present a simple template that shows the microstates. When microstates are built, it becomes easier to obtain the spectroscopic terms and also makes this coupling scheme more treatable at an undergraduate level. This approach has been used by Novak (7), who organized his templates by ordering the mj values. However, Novak does not emphasize a systematic filling of the microstates. He shows only the final J values and does not present the symbols for the jj terms. The procedure we present in this article was inspired in the Orchin-Jaffe systematic microstates filling procedure for LS coupling, and we use the Haigh (5) notation for the jj term symbols. Since the LS coupling scheme is presented in many textbooks, there is no need to detail it here and we only present our procedure for obtaining the jj term symbols.

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Obtaining the Terms in the jj Coupling Scheme As indicated earlier, the jj coupling considers that orbital and spin angular momenta for each electron, l and s, couple to give the total momentum j, and these j vectors for each electron couple to give the atom total angular momentum, J. To build the terms for jj coupling, we note that any two electrons in the atom cannot have the same l, s, j, and mj quantum numbers. In the following sections, we present the proposed methodology for the cases of the p2 and p3 electronic configurations. p2 Configuration For this electronic configuration, both electrons have l = 1 and s = 1/2. As the possible j values run from l þ s to |l - s|, we have the possibilities j = 3/2 and j = 1/2 for both electrons. Initially, let us set up a microstate table for the case j1 = 3/2 and j2 = 1/2. As the mj values run from -j to þj, we will have four columns to put the electron 1 and two columns for the electron 2, as shown in Table 1. In this table, each “x” designates an electron with all four quantum numbers, l, s, j, and mj defined by its position in the table. The MJ values that are the sum of the mj values are shown in the last column of Table 1. The number of microstates with the same MJ values is shown in Table 2. Considering that -J e MJ e J and that we have MJ values from 2 to -2, a J = 2 value is found. If we remove one occurrence from each line of Table 2, we will be left with Table 3, which indicates a J = 1 value as MJ values go from 1 to -1. At this point, it is worth pointing out that there is a lack of standard notation in jj coupling. Consequently, it is necessary to make a choice of which notation will be used for the jj terms. This problem is not observed in the LS coupling scheme because there Table 1. Microstates for a p2 Configuration under jj Coupling and Considering j1 = 3/2 and j2 = 1/2 Electron 1, j1 = 3/2

Electron 2, j2 = 1/2

mj values

mj values

-3/2

-1/2

1/2

3/2

x

-1/2

x x x

x

0

x

1

x

x x

r 2010 American Chemical Society and Division of Chemical Education, Inc. pubs.acs.org/jchemeduc Vol. 87 No. 12 December 2010 10.1021/ed1004245 Published on Web 10/01/2010

0

x

x

1 x

_

-1 -1

x

x

MJ = Σmj -2

x

x

_

1/2

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Research: Science and Education Table 5. Microstates for a p2 Configuration under jj Coupling Considering j1 = j2 = 1/2

Table 2. Number of Microstates with the Same MJ Values MJ

Number of Microstates

2

1

1

2

0

2

-1/2

-1

2

-2

x

1

Number of Microstates

2

;

1

1

0

1

-1

1

-2

;


mj values

mj values -1/2

1/2

1/2

MJ = Σmj

x

0

Table 6. Condensed Table of jj Microstates for the Term (3/2, 3/2, 3/2)J of a p3 Electronic Configuration

Table 3. Number of Microstates with the Same MJ Values after Removing the Components of the Projections of J = 2 MJ


Electrons 1, 2 and 3, j1 = j2 = j3 = 3/2 mj values -3/2

-1/2

1/2

x

x

x

x

x

x x

3/2

MJ = Σmj -3/2

x

-1/2

x

x

1/2

x

x

3/2

Table 4. Microstates for a p2 Configuration Using jj Coupling Considering j1 = j2 = 3/2 Electron 1, j1 = 3/2


mj values

mj values


-3/2 -1/2 1/2 3/2 -3/2 -1/2 1/2 3/2 MJ = Σmj x

-1

x

x

x


_

mj values

mj values

0

x

x

x

1

x

x

x

2

x

x

x

x

0

is an agreement between spectroscopists on this matter (8). Unfortunately, there is not a similar agreement when jj coupling terms are considered. Therefore, for the sake of simplicity, we use the terminology defined by Haigh (5), in which, for instance, the term for a two electron configuration is represented by (j1, j2)J, where j1 and j2 are the quantum numbers for the total angular momentum of each electron and J is the quantum number for the total angular momentum of the atom. Then, for this case, we have the terms (3/2, 1/2)2 and (3/2, 1/2)1, which can be indicated together as (3/2, 1/2)2,1. Now, if we consider both electrons with j = 3/2, the microstates are those displayed in Table 4. We notice that in this case both electrons have the same j and this makes any microstate with both electrons with the same mj forbidden with respect to the Pauli principle. In the filling process of Table 4, we have done a left-to-right procedure in such way to avoid repeating some microstates, as the electrons are indistinguishable. Explaining in detail, in the first line, we put electron 1 in the place mj = -3/2. As both electrons have j = 3/2, the second electron cannot have mj = -3/2 because both electrons will have all four quantum numbers l, s, j, and mj equal. The possibilities for the second electron are mj equal -1/2, 1/2, and 3/2, which fills in the first three lines of the table. In the fourth line, as the first electron has 1452

j3 = 1/2

-1/2

x

x

j1 = j2 = 3/2

-3/2

x x

Electron 3

-2

x

x

Electrons 1 and 2

Vol. 87 No. 12 December 2010

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1/2

3/2

-1/2

1/2

-5/2

x x x x

x

x x

x

x

x x

x

x x

x

x

x

-1/2 -1/2

x x

1/2 -1/2

x

x

-3/2 -3/2

x

x

MJ = Σmj

x

1/2

x

3/2

x

5/2

x

1/2

x

3/2

mj = -1/2 the second electron may have mj = -3/2, 1/2, and 3/2 but the microstates with mj values equal to -1/2 and -3/2 for each electron are already in the first line of the table and should not be repeated. The last column in Table 4 shows that MJ values run from 2 to -2, which indicates J = 2, and thus, we have the term symbol (3/2, 3/2)2. As MJ = 0 is the only value that appears twice in Table 4, this means that we have only one additional term with J = 0 and the terms from this table are (3/2, 3/2)2,0. The last possibility for a p2 configuration is the case in which both electrons have mj values equal to 1/2 (Table 5). Again, as both electrons have the same j, only microstates with different mj are allowed, and the unique allowed term is (1/2, 1/2)0.

pubs.acs.org/jchemeduc

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r 2010 American Chemical Society and Division of Chemical Education, Inc.


Collecting all terms for the p2 configuration, we have (3/2, 3/2)2,0; (3/2, 1/2)2,1; and (1/2, 1/2)0. It is worth noting that all J values are the same in both LS and jj coupling schemes, indicating that the number of energy levels is given by the number of J values. p3 Configuration To obtain the jj terms for a p3 configuration, it is a matter of applying the same procedure that we have used for p2, except that the number of microstates and terms is greater. Again, all electrons have l = 1 and s = 1/2 and no two electrons can have the same four quantum numbers l, s, j, and mj. The possibilities for the quantum number j for each electron are again 3/2 and 1/2 Table 8. Number of Microstates with the Same MJ Values for the Term (3/2, 3/2, 1/2)J of a p3 Electronic Configuration

MJ

Number of Number of Total Number of Microstates After Microstates After Microstates (from last Removing J = 5/2 Removing J = 3/2 column of Table 7) Term Term

5/2

1

;

;

3/2

2

1

;

1/2

3

2

1

-1/2

3

2

1

-3/2

2

1

;

-5/2

1

;

;

Conclusion


-3/2

Electron 1

Electrons 2 and 3

j1 = 3/2

j2 = j3 = 1/2

mj values

mj values

-1/2

1/2

3/2

x x x x

s1 s2 p1, p5 p2, p4 p3 d1, d9 d2, d8 d3, d7 d4, d6 d5

-1/2

MJ = Σmj

1/2

as j = l þ s, ..., |l - s|. Thus, we may have the term symbols (3/2, 3/ 2, 3/2)J, (3/2, 3/2, 1/2)J, (3/2, 1/2, 1/2)J, and (1/2, 1/2, 1/2)J, as we are following the Haigh (5) notation (j1, j2, j3)J. We start with the first case (3/2, 3/2, 3/2)J, in which j1 = j2 = j3 = 3/2. It means that all electrons must have different mj values. To make it short, we set up a table similar to Table 4 but in a more compact way. Because the electrons in this case cannot have the same mj value, we do not need to write separate sets of mj values for each electron, and we can set up a table as shown in Table 6. This condensed table could be done in the case of Table 4, too. Table 6 shows that MJ runs from -3/2 through 3/2 without any repetition, which indicates only one allowed J value, which equals 3/2 and that term is (3/2, 3/2, 3/2)3/2. For the (3/2, 3/2, 1/2)J case, we can set up the microstates as shown in Table 7. Because electrons 1 and 2 have the same j and they cannot occupy the same mj “orbitals”, we can put both in the same part of the table. Table 8 is similar to Tables 2 and 3 and shows that we have J values equal to 5/2, 3/2, and 1/2 giving a final term symbol (3/2, 3/2, 1/2)5/2,3/2,1/2. The microstates for the (3/2, 1/2, 1/2)J case are shown in Table 9. As j2 = j3, we can put both in the same part of the table. On the basis of the MJ values, the term symbol is (3/2, 1/2, 1/2)3/2. Because it is not possible to have three electrons using only the mj values -1/2 and 1/2, the term (1/2, 1/2, 1/2)J is forbidden. Thus, all of the jj terms for a p3 configuration are (3/2, 3/2, 3/2)3/2, (3/2, 1/2, 1/2)5/2,3/2,1/2, and (3/2, 1/2, 1/2)3/2.

x

x

-3/2

x

x

-1/2

x

x

1/2

x

x

3/2

Obtaining the jj terms for other electronic configurations using the procedure presented here is as simple as we have shown for p2 and p3. Only the amount of work increases as the number of orbitals and electrons increases, as can be seen in Table 10. As in the case of LS coupling, the jj terms for a more than half-filled subshell containing n empty spin-orbitals (n holes) will be the same as for an electronic configuration with n electrons (p2 and p4, for example), as can be seen in Table 10. Our experience in obtaining the jj terms through this procedure, which explicitly presents the microstates, has shown that students can understand at a deeper level the significance of the spectroscopic terms that allows a discussion in more detail of the similarities and differences between both LS and jj couplings schemes. We hope that the procedure we present here will make the jj coupling scheme more popular in chemistry lectures and chemistry textbooks.

Table 10. Spectroscopic jj Terms for Equivalent Electrons in Orbitals s, p, and d 1 2 1=2 1 1 2, 2 0 3 1 2 3=2 ; 2 1=2 3 3 3 1 1 1 2, 2 2, 0 ; 2, 2 2, 1 ; 2, 2 0 3 3 3 3 3 1 3 1 1 2, 2, 2 3=2 ; 2, 2, 2 5=2, 3=2, 1=2 ; 2, 2, 2 3=2 5 3 2 5=2 ; 2 3=2 5 5 5 3 3 3 2, 2 4, 2, 0 ; 2, 2 4, 3, 2, 1 ; 2, 2 2, 0 5 5 5 5 5 3 5 3 3 3 3 3 2, 2, 2 9=2, 5=2, 3=2 ; 2, 2, 2 11=2, 9=2, 7=2ð2Þ, 5=2ð2Þ, 3=2ð2Þ, 1=2 ; 2, 2, 2 9=2, 7=2, 5=2ð2Þ, 3=2, 1=2 ; 2, 2, 2 3=2 5 5 5 5 5 5 5 3 5 5 3 3 2, 2, 2, 24, 2, 1 ; 2, 2, 2, 2 6, 5, 4ð2Þ, 3ð3Þ, 2ð2Þ, 1ð2Þ, 0 ; 2, 2, 2, 2 6, 5, 4ð3Þ, 3ð2Þ, 2ð4Þ, 1, 0ð2Þ 5 3 3 3 3 3 3 3 , , , ; , , , 2 2 2 2 4, 3, 2, 1 2 2 2 2 0 5 5 5 5 5 5 5 5 5 3 5 5 5 3 3 2, 2, 2, 2, 25=2 ; 2, 2, 2, 2, 2 11=2, 9=2, 7=2ð2Þ, 5=2ð2Þ, 3=2ð2Þ, 1=2 ; 2, 2, 2, 2, 2 13=2, 11=2, 9=2ð3Þ, 7=2ð3Þ, 5=2ð4Þ, 3=2ð3Þ, 1=2ð2Þ 5 5 3 3 3 5 3 3 3 3 2, 2, 2, 2, 2 11=2, 9=2, 7=2ð2Þ, 5=2ð2Þ, 3=2ð2Þ, 1=2 ; 2, 2, 2, 2, 2 5=2


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Acknowledgment The authors thank CNPq, CAPES, and FAPERJ for financial support. Literature Cited 1. Orchin, M.; Jaffe, H. H. Symmetry, Orbitals, and Spectra ( S.O.S.); Wiley-Interscience: New York, 1971; pp 178-184.

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2. 3. 4. 5. 6. 7. 8.

Gauerke, E. S. J.; Campbell, M. L. J. Chem. Educ. 1994, 71, 457–463. Tuttle, E. R. Am. J. Phys. 1980, 48, 539–542. Rubio, J.; Perez, J. J. J. Chem. Educ. 1986, 63, 476–478. Haigh, C. W. J. Chem. Educ. 1995, 72, 206–210. Campbell, M. L. J. Chem. Educ. 1998, 75, 1339–1340. Novak, I. J. Chem. Educ. 1999, 76, 1380–1381. Russell, H. N.; Shenstone, A. G.; Turner, L. A. Phys. Rev. 1929, 33, 900–906.


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Obtaining the Electron Angular Momentum Coupling Spectroscopic

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