A Simple, Systematic Method for Determining J Levels for jj Coupling Ensign Steven J. Gauerke, USN, and Mark L. Campbell United States Naval Academy, Annapolis, MD 21402-5026 The accurate description of atomic structure relies heavily on the coupling of angular momenta in the valence electrons of the atom. Such coupling is normally described in two limiting representations: LS or jj coupling. At the undergraduate level the student is usually taught only RussellSaunders, or LS, coupling. Indeed, current textbooks in physical (1)and inorganic (2)chemistry discuss the Russell-Saunders case almost exclusively. The book by Atkins (la)does attempt to handle the jj-coupled case for the p2configuration but does so incorrectly. A predominant reason for the neglect textbook authors have shown for jj coupling is the perceived difficulty in finding the terms and levels associated with this scheme. In this Journal, Ruhio and Perez (3) describe a method that they proclaim as simple. Unfortunately, their method is difficult to follow, and its proper use is not obvious except with the simplest electron configurations. The only other method we have found in the literature was described by Tuttle (4).Again, this treatment is more complicated than is desirable for instruction at the undergraduate level. In this paper, we describe a process that is based on the same concepts as Thttle's but is simpler and more systematic, allowing for application a t the undergraduate level. Angular Momentum Coupling In LS coupling, the valence electrons'individual orhital angular momenta e's couple to yield the total orhital angular momentum L, which is a constant of the motion. In terms of the classical model of the precession of vectors, this means that the Ys' precess more rapidly around their mutual field than around any other field. Similarly, the individual spins s's couple to yield the total spin angular momentum S, which is also a constant of the motion. States represented by a particular combination of L and S are called a term. The determination of LS term symbols for different electron configurations has been covered extensively in this Journal (5-7).In general, several LS terms will result from any one electron configuration. For each term the spin-orbit interaction is treated as a small perturbation yielding a representation in which the total electronic angular momentum J of the atom is the vector sum of the orbital and spin angular momenta. Its quantum numbers cover a range. L+S,L+S-1, ... IL-SI The result is that each term is split into levels that consist of states with the same value of J that are (21+ 1)-fold degenerate corresponding to the possible values of MJ. Limitations of L S Coupling LS wupling is an appropriate description when the noncentral electrostatic energy t e r n for the valence electrons are much larger than the spin-orbit terms. The electrostatic interaction normally predominates in ground-state electronic configurations for light to moderately heavy atoms. However, in ground configurations of heavy atoms and in many excited configurations of light and heavy atoms, the spin-orbit energy of the valence electrons contrib-
utes to a larger fraction of the energy perturbation than the residual electrostatic energy. LS wupling inadequately describes the observed states in these cases. The levels given by ji coupling may then be used to explain the absences of multiplets predicted by LS coupling, as well as the presence of LS-forbidden transitions that are allowed by jj-coupling selection rules. J Coupling a s the Preferred Scheme
In cases where jj wupling is the more appropriate coupling scheme, the electrons appear to move independently of one another and in these circumstances the individual values j, t , and s are the good quantum numbers. Each electron's t and s couple to give j, the electron's total angular momentum. As with LS coupling, several terms result from the different ways in which each electron can couple its angular momenta. For each term the electrostatic interaction is treated as a small perturbation yielding a representation in which the total electronic angular momentum J of the atom is the vector sum of each electron's angular momentum.
The result is that each term is split into levels, which again consist of states with the same value of J that are (21+ 1)-fold degenerate. Due to a correlation of states, for a given electron configuration one finds the same J levels in jj coupling as in LS coupling although there are generally many more terms with the LS coupling case (except in the case of p" and s" configurations). Preferred Notation Although the notation used for LS coupling is universally standardized, the notation for jj coupling is not. For jj-coupled terms, we prefer the notation
where 1 and 1' are the letter designations for the type of orbitals (i.e., s, p, d, etc.), and e, e' are the numerical values of the angular momentum quantum numbers for the electrons. In the other common notation system (3,4,8) the terms are designated solely by thej's, which obscures important information needed to determine allowed spectral transitions between jj-coupled states. As an illustration of the ambiguity that may result consider the forbidden transition
which is incorrectly characterized in ref 8 as an allowed transition. Using the notation
Volume 71 Number6 June 1994
457
one might erroneously conclude that this is an allowed transition based on j-coupling selection rules (9). However, with the preferred notation, it is apparent that the transition is forbidden because this transition requires that two electrons change their quantum numbers.
iidoupled States for an In Configuration . . The method for determiningj-coupled states presented in this section is based on the application of the Pauli exclusion principle in governing-the possible microstates that compose an electron configuration. A microstate is a articular auantum state in which each electron in the atom is associated with a set of unique quantum numbers. The Pauli exclusion ~ r i n c i ~is l emost often intemreted in terms of the strong held case in which the e's i d s's for each electron are uncoupled from each other, so they are all space-quantized with good quantum numbers me and m,. Use of the mj Representation However, the weak field case, in which an individual electron's t and s couple to givej and its space quantization m, is the more appropriate representation when discussing jj coupling. The Pauli exclusion principle still applies in this case: No two electrons within the atom can have the same complete set (n, e, s, j, m,) of quantum numbers. Because the spin quantum number for an electron has the value V2. the ~ossiblevalues of i for anv eiven electron can for an"2ectron configuraonly be ln'and t - 112. tion of the t m 1" there will be a t most two tvoes of electrons. ~ a c of L these forms a subset of equrGalent electrons--equivalent electrons being defined a s having the same value of n, e, s, and j. According to the exclusion principle, for each subset of equivalent electrons, only those microstates in which the mj values are different will be allowed. The following method allows the determination ofjj-coupled states by determining all possible combinations of m:s for each subset ofequivaient electrons and then using cokelation of states to detfrmme the allowed J states for a given term. Determining Energy Levels and Possible n r m s Energy levels forj-coupled terms can be determined by canying out the following procedure.
Step 2.
Determine the possible %-coupledterms for the given electron configuration. For each term found above. set uo a table of all possible combinations of mjva~udsfor the term consistent with the Pauli exclusion principle and the indistinguishabilityof electrons.
Step 3.
Determine the value of MJ(= Em,) for each microstate (row)in the table.
Step 4.
Determ ne tne maxlmum value of M ~ f r o mtne values in the MJ colmn. Tne argea va ue of M~represents a value of Jforthat term.
Step 5.
Eliminate the 2J+ 1 values of MJ
Stepl.
~~~
(J,J - I ,
~
~
J+1,-J)
that comprise the J state determined in step 4. Step 6.
Repeat steps 4 and 5 until all Mjvalues are eliminated for the term.
According to our procedure, the fnst step in determining the allowed J levels is to derive the possible terms for the configuration. The jj-coupled terms for an electron configuration in which only one subshell is partially filled will have the form 458
Journal of Chemical Education
where n represents the number of electrons in the partially fdled subshell, and the value of i is constrained such that n Z i Z 0 far suhshells less than half-filled 2e Z i Z 0 far half-filled subshells 2e Z i Z n 2e - 2 for subshells more than half-filled
-
The number of terms will be
+
n 1 for subshelk less than half-filled n for half-filled subshells 4e 3 - n for subshells mare than half-filled
+
Once the terms have been determined, construct a microstate table in which each allowed microstate appears once and only once. Microstates that are simple permutations of the same m, values within each subset of equivalent electrons do not represent different states. Thus, only one of these microstates should be included in the table. We have found a systematic way to set up this table. Const~ctingthe Microstate Table Subsets for Equivalent Electrons Construct a table with n + 2 columns. In the fnst row of the table, label the first n columns with the j states of the electrons in which the smaller values o f j are placed to the left. In the second row, label the first n columns with mj; (i = 1to n) to indicate the allowed magnetic quantum numbers for each electron. Equivalent electrons are grouped. (In our tables, we use double lines t o designate the grouping of equivalent electrons.) within each subset of equivalent electrons, treat the m, values as runnineindices: the values of mi to the rieht varv faster than those to i t s left. ~ i m i l a r l i ,each sibset o"f equivalent electrons is treated as a running index with the subset to the right as the faster changing index. Once an m, value is changed in the group of equivalent electrons to the left, all the possible combinations of mj values for the subset to the right are determined for that one combination. To initiate the construction of the table, set up the first row of numerical values for mj in the table so that the values of mj are maximized. Accomplish this by setting the value of m, a t the far left of each subset of equivalent electrons with the maximum mj possible, that is, the numerical value ofj. The succeeding values of m, to its right decrease by 1 sequentially. Fill out succeeding rows by treating each column of m, values as running indices as described previously. Checks for the Table When this process is carried out correctly, all the values of mj in a particular row for a subset of equivalent electrons are different, and the mj values will decrease from left to right. As a check, if a value of mj in a particular row in a subset of equivalent electrons has a value greater than or equal to another m, value to its left, then the table has been filled out incorrectlv. Once the table is mmpleted, check to be sure that all possible cornhinations have been determined. The nurnher of microstates for a particular term of the form
is given by the product of the binomial coefficientsfor each subset of equivalent electrons.
Table 1. Microstate Tables for a p3Configuration
Table 2. Microstate Table for a p4 Configuration
Tables 1and 2 illustrate the construction of microstate tables for the terms for the p3 and p4 electron conflgurations. In each subset of equivalent electrons, the m, values always decrease from left to right. Furthermore, within each subset of equivalent electrons, the mj value to the right varies the fastest, followed by the one to its left, and so forth.
As in this case, when more than one microstate has the same value of MJ, the question arises Which of the MJ's should be eliminated for a specific value of J?"The short answer is "It doesn't matter as long as the student eliminates one and only one of the microstates (rows)for each of the corresponding (W+ 1)MJ values for the given J." Obviously, this method can become quite laborious for electron configurations with more than a few electrons. For example, the term consists of 90 microstates and 14J levels. Because constructing such a table is time-consuming and tedious, we have included Table 3 which gives all the possible J levels for terms that arise from s", p", d", and f" electron configurations. The electron configurations in Table 3 are .grouped in complementary pairs; the same J levels occur for configurationsof the form 1" and 1"' where n'= 48 + 2 - n. It helps students to construct tables for a few simple cases but not tables consist in^ of hundreds of microstates. Nonetheless, students benefitfr~munderstandin~ how the levels for the more complex cases are determined even if they do not determine the levels themselves.
Using MJ Values To Determine the J Level After constructing the microstate table and determining the MJ values for each microstate, the J levels are determined according to steps 4 and 5. Onee a J level is determined from the maximum MJ value in the table, then the (W+ 1)MJ values associated with that J level are eliminated. For example, in Table 1for the term [(p*)'(p3,&, the maximum MJ value is 312, which results in a J value for this term of 3/2. As step 5 is carried out, all four MJ states 312, 112, -112, and -312 are eliminated. Because there are no MJ states remaining, the only level for this term is the 3/2 level. From the microstate table for the [(pm)'(pm)'1 term in Table 1, the maximum value of MJ is 512. Thus, a J level for this term is 5/2. and the M.7 states 512.. 312.. 112.. -112. -3/2. and 6/2are elkinated, asrndicated with an asterisk next to these values. After eliminating the six microstates associated with the J = 5/2 level, thi maximum MJ value remaining is 3/2, yielding a J level for this term of 3/2. Eliminating the four microstates for this level, indicated with a @ in the table, leaves a maximum MJ value of ID. Thus, the final level for this t e r n is 1/2.
Computer Programming The systematic nature of this method allows easy application to computer programming. Acopy of the FORTRAN program used to generate Table 3 can be obtained by writing M. Campbell. The program consists of less than 100 lines of com~utercode and uses a series of nested DO l o o ~ s and IF stat;rnents to determine the possible microstatks for the inputted term. The values of M., are calculated and tabulated for each microstate, and then steps 4 and 5 of our procedure are applied until all the microstates have been taken into account. jjGoupled States for I" I
" Configurations
Table 3 can also be used to determine jj-coupled states for electron configurations in which more than one subshell is partially filled. In some cases, jj-coupled terms can Volume 71 Number 6 June 1994
459
Table 3. jj-Coupled Energy Levels for sn, p", dn, and f" Electron Configurations
be read directly from the table, but in other cases two or more terms must be combined using the vector model in which each J level of one term combines with every J level of the other term(s). Two Differentj States
When there are only two subsets of equivalent electrons from different subshells and the two j states are different, there are cases in which the J levels can be read directly from Table 3. For example, the s p [(sm)'(pm)'l term has the same J levels as the Kpm)l(pgJ11 term, and the p d [(~m)~(dsn)'l term has the same J levels as the d3[(d3n)2(dm)'lterm.
p2
Nonequivalent Electrons
f4,fi0
For a t e r m such a s s p [ ( S ~ ~ ) ~ ( vector ~ ~ ) ' Icoupling , must be used to determine the J levels because the j values for the nonequivalent electrons are the same. In this case, the twoi = 1/2 vectors couple to yield J [(15n)~'~(hrz)~~'115n,11~912~7/2,512.312 levels of 0 and 1. A term such as p2d [(pvz)'(p3iz)'(dm)'l must 1(f~2)~~'(f712)"~~8,7,6(2),5(3),q31,3(3),2(3),1(2) [(f5i2)4*(h~)0"'~4,2,0 also be determined using the vector model because there are [(fu~~2'4'(hn~2"'~lo,~,e~3~.7(3),7(3),4~8l.3~5~~2~7).1~2~.~~3~ three types of nonequivalent [(fu2)1'~(f7n)36'l1o,~,8(2),7(3~,6(4),5(~).4(5),~(5).2(4).5 electrons. In this case. the J levels of 2 a n d 1 for t h e p2 [(fsn)0'~(hn)4"'la,665,4~2),2~2),~ [(pm)'(p3,)'1 term and the J = [(fs45"'(hn)0"'l~2 [ ( f s 2 ) 4 Q ' ( f 7 ~ ~ ) 1 ' 7 ~ ~ u 2 , t m , ~ i ~ ( 2 ) , 9 ~ 2 ( 2 ) , 7 ~ ( 3 ) , w 2 ( 2 ) ~ 2 ( 2 ) , i ~ 2 512 level of the [(d5n)'l term are combined to yield J levels of 9/2, 712 (twice), 512 (twice), 312 [(f5~)3'b(b~)2"'~21~,19/2,17~(3),15n(4),1~(6),11~(7),912(9),712(8),u2(8),~(6),l~2~3) (twice), and V2; that is, J = 2 [(fw2)2c"(f~n)36'~~,21i2,19~(3),17n(4),1~p),1m(8),lln(tl),9n(ll),7n(12),u2(lo),~(8),ln(4~ combines with J = 512 to give 912, 712, 512, 312, a n d 112, [(fsn)1'5'(f7n)4"'l2t/~,t9~22i7n~2~,i~(3~,i3n~~~,iin~~~,~n~e~,7n~~~,vr~e~,m~~~,~rz~~~ whereas J = 1combines with J = 512 to give 712, 512, and 312. [(fsn)0'"(hrz)5Q'li~n,ii~~~227~12~~/2z2 ~
f5,f9
Pf
[(fsn)6'b(bn)0"'10
[(fsn)5"'(f7n)1'7~6,5,~43322i
[(fs12)4"2(~7~)2"'~10,9,6(3),7(3),~(6),5(5),4(6),3(5)~
~~
~
Ene y Levels, Line Spectra, and$ectral Intensities
Why do we [(fsn~3'3'(f7rz~3~'li~,tt1io~3~,~~5~,8~7~,7~i~),~~3),~~~4~,qi5~,~~i~~,~~ii~,~~e~,o~~~
care what type of coupling scheme describes an atom? Knowing how to deter~~sn)~'"(17~~~"'1i2,11,io~3~,~~4~,~~8~,7~e~,~~~~~,t,i~~,3~1i~,2~13~,1~~~,o(~1 mine jj-wupled levels (or even [ ( f s z ) 1 ' 5 ' ( f 7 n ~ 5 o ' ~ t o , ~ 9 ~ ~ 2 ~ . 7 ~ ~ ~ , e ~ 4 ~ , ~ ~ ~ ~ , 4 ~ ~ [~(,1~s~n5)~~, '2~~(41~7.~t )~~3~~'.1o~ , 4 ~ 2 , 0 LS term symbols) is a useless tool unless the student underf7 [ ( f ~ n ) ~ ' ~ ( b ~ ) " ~ ' 1 7 / ~[(1y~)~"'(f7~)~'~~17~,1~2,1~2(2),1ii2(2),~(3),7~(3),~2(3),312(2),1/2 s t a n d s i t s significance. The type a n, atom [ ( ~ s 1 ~ ) ~ < " ( 1 ~ n ~ ~ ~ ' 1 ~ ~ ~ e ~ ~ , 1 ~ n ~ ~ ~ , i ~ n ~ ~ ~ , i ~ ~of~coupling , ~ m ~ in ~~ ~ ~ ~has ~t~~,gn~ a major effect on i t s most 4 [(f5/~)~(17/2) 1 2 5 ~ . 2 ~ 2 , 2 1 ~ ( 3 ) , 1 9 ~ 2 ( 5 ) , 1 7 1 2 ~ 8 l , 1 5 / 2 ~ 1 0 l , 1 z 2 ( 1 5 ) . 1 1 n ( 1 6 ) , 9 ~ ~ 1 6 ~ , 7 ~ ~ 1 8 ~ . ~ ~ ~prominent 1 6 ) 1 1 ) . 1 ~ ~ ) experimental observable: the line spectrum. The number of lines, their relaIn a term, the lint superscripted number for a set of equivalent electmns refen to the mnfiguration in which the tive positions, and the spectral Subshell is less than half-filled.The numbers in brackets refer to the configurationin which the subshell is more than intensities me all influeneed by half-filled.For half-filled subshells (p3, d5, and f7) some terms exist as mmpiementary pain that have the same J levels. the coupling scheme that preThe same symbolism is used to designate each pair The number in parentheses aner avalue of Jindicates the number dominates in the atom. of levels with that value of Jfor thatterm. 460
Journal of Chemical Education
However, for tellurium and polonium the two higher energy levels of the predicted triplet are very far removed from the ground state, so in these cases jj coupling is a better approximation. The Group 16 case is partieularly interesting due to the order of the lowest energy levels. According to LS coupling, the ground-state triplet should consist of the J = 2 level as the lowest energy level, followed by the J = 1 and then the J = 0 level. This energy ordering is seen in oxygen, sulfur, and selenium. According to jj coupling, the lowest term results in the J = 2 level being the lowest energy level, with the J = 0 level being the next lowest. From the next term. the J = 1level appears as the tdird low4 est enem. This enerm -" orderine is foun$;n tellurium and poloFigure 1. Energy levels for the np4electron configuration of the Group 16 elements. The experimental nium. The fact that the two coupling schemes correctly predict values tor the J levels were taken from ref 10. the experimentally observed order of energy levels for the Group Coupling as the Preferred Scheme: 16 atoms is a rather remarkable accomplishment in the ncreasing Atomic Mass description of atomic structure. Group 16Elements Group 14Elements It is interesting to compare the actual experimentally Another example of how3 coupling becomes more impormeasured properties of atoms with those predict4 theotant as the atomic mass increases is illustrated in Fimrre retically by the different coupling cases. The energy levels 2, This figure shows the energy levels for the ground t&m for the p4 ground-state configurationfor the Gmup 16 eleand first excited terms of the Group 14 elements along ments are shown in F i r e 1along with the predicted levwith the levels predicted assuming LS and jj coupling. LS els for LS and jj coupling. coupling is a very accurate description for carbon, silicon, From Figure 1we see that the LS prediction is very good germanium, and tin, although the energy spacings infor oxygen, sulfur, and selenium although the energy spaccrease as the atomic mass increases. However, for lead the ings in the triplet increase as the atomic mass increases. two higher energy levels of the triplet are very far removed from the ground state, so jj coupling is a hetter approximation. In the case of the n'snp excited state, LS coupling describes carbon and silicon very well. However, for germanium, tin, and lead the energy levels appear more as two double-energy levels, a s predicted by jj coupling. These observations for the Group 14 elements are true in general. For ground-state electron configurations very heavy elements follow jj coupling, whereas light to moderately heavy elements follow LS coupling. However, in excited-state configurations, jj coupling is often observed in lighter atoms.
-
4
Genera2 Appearance of the Spectrum
Another indication of which coupling scheme best describes an atom is the general appearance of the s~ectrumand the inFigure 2. Energy levels for the tp2and n ' s ~electron configurations of Ihe Group 14 elements. The tensity relations for the lines in experimental values forthe J levels were taken from ref 10. the spectrum. As a n example, Volume 71 Number 6 June 1994
461
Figure 3 shows the LS- andjj-allowed transitions from the n'snp excited state to the np2 ground state for the Group 14 elements. The spectrum for carbon is not shown because it has the same general appearance as the silicon spectrum, except shifted to shorter wavelengths. Transitions 1 through 8 are allowed when LS coupling applies; all eleven transitions are allowed in ji coupling. In an LS-coupled atom, these transitions should consist of six closely spaced lines with two other lines widely separated. Carbon, silicon, and germanium show this pattern very clearly, whereas in tin and lead this pattern is obscured. Table 4 shows the experimental relative line strengths (11)for the eleven transitions in Figure 3 along with the theoretically calculated values based on LS andjj coupling (12).Neither coupling scheme predicts the experimentally obsewed line strengths exactly, although from the table it is apparent that LS coupling gradually changes to jj coupling in going from light to heavy atoms. Lande Interval Rule
250
300
350
450
400
500
Wavelength (nm) Figure 3. Dipole-allowed transitionsfrom the rlsnp to np2configurationsin the Group 14 elements. The transitions labelled 1 throuoh 8 are allowed in LS-couoled atoms. All eleven transat ons are al owed in .lauplea-atoms The spectrum forcamon is not shown oecaJse it has essenlla ly tne same appearance as !hat of sl lmn excepl the I nes are shlheo to shoner wave engths Table 4. Theoretical and Experimental* Relative Line Strengths Sfor the np2 n'snp DipoleAllowed Transitions for the Group 14 Elements
-
rans sit ion^ 2
np (1)
- n'snp
1 ~ - ' ~ 9
SLS*
Sc
Ssi
So.
Ssn
S P ~Sjjt
20
20
16
19
24
34
20
Another way to verify the type of coupling in an atom is to compare the obsewed energies in a multiplet with that predicted by the Land6 interval rule. If LS coupling applies in the atom, then the ratio of the intervals between two successive components (Jand J + 1) in a multiplet should be proportional to J + 1. This relationship was first formulated by Land4 and is called the Land$ interval rule. Thus, for a 3Pterm, the Land6 interval rule predicts that the energy difference between the J = 2 and J = 1levels should be twice as large as the energy difference between the J = 1and J = 0 levels. Areview of Figures 1and 2 indicates that the Land6 interval rule applies for light and moderately heavy atoms in their ground states, but deviations are seen for ground states of heavy atoms and excited states of moderately heavy atoms.
Group 5 Elements
Another illustration of how doupling changes from light to heavy atoms is reflected by the extent to which the energy differences within a multiplet follow the Land6 interval rule for the transition elements. For example, the d3 configuration in the Group 5 elements results in a 4Fterm with J levels of 912, 712, 512, and 312. According to the Land6 interval rule, the energy differences should be in the ratio 9:7:5. The Group 5 elements V, Nb, and Ta are found to have experimentally measured ratios (10)of 8.4:6.8:5. 7.3:6.4:5. and 4.1:4.9:5. The liehter vanadium atom follows the ~ h d interval 6 rule fa& well, whrrras the heavier tantalum atom deviates markedlv. This t w n of behavior is seen in essentially every group in the table. Conclusion
. . 'Elgwrimental line strength values were calculated hom the Einstein transition probab~litiesin ref 11. The line strengths are normalized such that ZSx= 300 for each element.
%anransition numbers refer to the numbered transitions in Figure 2. *Theheoretical line strength assuming LSeaupling. tThwretical line strength assuming jj mupling.
462
Journal of Chemical Education
In this paper we have described a simple and systematic method for determining jj-coupled levels. We have found that students are able to use this method aRer about a half-hour of class time devoted to its description and the working of examples. Once the student knows how to determine atomic terms, it is important to illustrate the significance of the different coupling schemes through real examples. Diagrams of energy levels and atomic spectra are important ways of showing the correlation between ex-
perimentally determined energy states and theoretically derived states described by LS and jj coupling. It is also useful for the student to understand that other relationships, such as spectral intensities and the Land6 interval rule, can be used to verify the type of coupling that occurs in an atom. Literature Cited 1. 1alAtkins.P WPhysimlChemisby, 5thed.;Freeman: NewYork, 1994: pp447-455. ( b ! A l W , R. A.:Silky, R.J.Physlm1 Chemistry; W~ley:New Yark, 1992; pp37G 375. lcl Naggle, J. H. Physiml Chamisfry, 2nd 4.:Sldt,Foreman: Glenview, IL, 1989;pp 712728.1d!Lwine,I.N.PhyaicolChemishy:3dd.;McCraw-EI11: New York 1g88; pp 63M40, Berry, R, s,: S, A: Rosr, Physial mley:NewYork, 1980, pp 168.199. (0Barmar, G. M. physieol chamisby, 5th ~d.; MGraw-Hill: New York, 1988: pp4M65.1g)Dybtra. C. E. Quonhlm Chamlafry and M d a u l a r S p a t m o p y : Rentiee Hall: Englewood Cliffs, NJ, 1992; pp 2& 270. 2. I a ) S h r i ~ e r , D . ~ ; A t h a , W.;Langford,C.H.InowanieChsmislry:Freeman:New T! Yo* 1930; pp 434-41. lbl Butler I. 8.;H a d , J. F Inawanb Chemistry Pnnciph8 andApplbztlons; BenjaminiCumminga:New York, 1999: pp 4245. IdLee, J.
D. coneisp ~~~~~~b chamisby, 4th 4.:chapman and H ~ IN: ~ W york, 1991: pp 940-950 (dl Huheey. J. E. Inorganic ChemiatryPrinc;ph ofStructuroandRmc. B V ~ ~znd J . d.; H W ~ Iand ow: N ~ W york, 1978; pp 810.814. (el meader. G. L.; TW, D. A. ~norgonicchamisby; ~ r e n t i c e ~ d~l n: g ~ e w o oGIG. d NJ. 1991; pp 4248. lfl Owen. S. M.: Brooker, A. T A Gui& Lo M d m Inorganic Chemistry; W,ley:NewYork. 1991:pp 156160. (g! Shsrpe,A. G.InorganbChemidry.3rd ed.: wlley: NCW ymk. 1992; pp 7275. 3. Rubio. J.: Perel, J.J. J. Cham. Edue. 1986, 63,476. 4. lirtUe,R. R . A m r J.Phys. 1967.35.26. 5. Cornan. M. J. Cham Ed-. 1973.50. 189. 6. Hyde, K.E. J C h r m E d u c 1915,52,87. 7. Vi-te,J. J. Cham. Educ. 1989,60,561. 8. Richtmyer, F. K.; Kennard, E. H.; Cmper, J. N. Inlmduelion foM&rn Physics, 6th ed.: Mdjraw-Hill: New York, 1969: p 457. 9. Leighton, R. B. fin=ipl= d M o d D m P h ~ i c sMcCraw-Ell: ; New York, 1959: P 271. 10. Moore, C. E. Ammie Emrsy h o d s 1 ; NSRDS-NBS 35; U.S. Government %ting %ce Washinpton. DC. 1971;Vols. 1-3. 11. Wieae, W.lr:Martin, 0. A. Womlengfha a n d h s i t i o n Robabilities forAfrmsond Afomlclons, Part 1I: NSRDS-NBS 68; U S Government Printing Ofice: Washington. DC, 1980. 12. Condon, E.U.; Shortleyi G. H. The TheoryofAttmiiSppef~;C c b r i d g e U ~ i i i i i I t y : Cambridge, 1935: pp 247.285.
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