Finding the Terms of Configurations of Equivalent Electrons by Partitioning Total Spins Liu Guofan Liaoning University, Shenyang. Liaoning, China
M. L. Ellzey, Jr. The University of Texas at El Paso, El Paso, TX 79968
Atomic spectral terms are of fundamental importance to the investigation of atomic structure in atomic . physics, . quantum ch(mis~ry,and srruct~lr~lrhemistry. Finding these tcrms for ronfieurstionj of euui\dent electruns can he difiicult, however. The usual methods such as "electronic arran( Z ) , and "determinantal wave cement" ( I ) , "Slater araohic" . . runctim" rJ,, are so rediui~sthat thev are scarcely pracrical for larger cunfj~.uratimi. \2'hile the "Hvde method" ( 4 ) and the "spin factor method" (5)represent improvements they are still laborious for the difficult cases. The method presented here, without proof, is a refinement of the spin factor method. It is simpler and easier to apply than any of the foregoing and gives the correct terms for pN, dN, and fN configurations. It could, indeed, be called the "short-cut calculation method". T h e rules for this method are given below followed by illustrative examples. Rules 1. For a configuration of N equivalent electrons, nlN, the possible total spins are listed in decreasing order: S = N12, (NI2) - 1 , . . . ,112 or0 2. N, and Na are determined for each S (aand P represent the two m, values of 112 and -112, respectively). M-L,., and M~L,, are obtained from the formulas,
+
3. If N, = 0 or N, = 21 1, then L, = 0 only, since M-L,., = 0. If = 1. N, = 1or N. = 21, then L, = 1 only, since If 1< N, < 2 1, then L, = M " L , ~M ~ " L , ~~ 2,. , 1 for M=L,~, odd, and L, = M"L,~~.M"L,~. - 2, . . . , 0 and Mn~,.J2 for M*L,,,.~even. The same process is used for Lb.
..
4. Far S = NI2, the values of L are given by the vector sum of L,
and La: L = L a + L g , L o + L a - 1 ,..., (La-LgI For all other S, L is the differences of the vector sums of two
successive values of S: L = (L, + L ~L,,
+ La - I, . . .
.
IL, - L& - (L,
+ Lo,L, + LB
-1,.
Volume 64
Number 9
.. , IL,
- LglJs+,
September 1987
771
Therefore, L, = 6,4,3,2,0;LB= 3, and
Examples 1. np2 (np4):1 = 1, N = 2, S = 1, 0 a. S = l , N , = 2 , N g = 0
9,
8, 7, 7,
L=
6, 6, 6,
5, 5, 5, 5,
4, 4, 4, 4,
3 3, 3,
3,
2, 2, 2,
1
1, 0 - ( 5 , 3 , 1 ) 1
3 MOLmai= 0
ThereforeL, = 1;Lp = 0; and L = L, SP.
+ Lg = 1 + 0. The term is
Therefore, L, = 6,4,3,2,0; Lo = 5,3,1; and 1 1 , 1 0 , 9 , 8 , 7 , 6 , 5 , 4 , 3 , 2 , 1 9 , 8 , 7 , 6 , 5 , 4 , 3 , 2 , 1 8, 7, 6, 5, 4, 3, 2, 7, 6, 5, 4, 3,
Therefore L, = 1,La = 1, and L = (2,1,0) - 1 = 2, 0 for the terms ID and IS. In summary, the npz (np4) configuration terms are 3P, ID, and IS.
ML ' ,.
=0
ThereforeL, = 3 , l ; Ls = 0; andL = 3 , l . The terms a r e v a n d
4P. b. S = 112, N,
= 2, No = 1
9 , 8 , 7 , 6 , 5 , 7, 6, 5, 6, 5, 5,
Therefore L,
= 3 , l ; LB= 2; and
4 , 4, 4, 4,
3 , 3, 3, 3, 3
2, 2, 2,
1, 1, 0 1
= ll,lO, 9(2), 8(3), 7(5), 6(5), 5(7), 4(6), 3(7), 2(5), l(4)
= 5,4, 3,2(2), 1
The terms are 2H,%G,2F, %D(2),and 2P.In summary, the nd3 (nd') configuration terms are bF, 4P. 2H, W, 2F,2D(2),and ZP. 3. nfVnf9): 1 = 3, N = 5, S = 512,312,112 a. S = 512, N, = 5, No = 0 M",,,
=3
+ (3 - 1) + (3 - 2) + (3 - 3) + (3 - 4) = 5
Therefore, L, = 5,3,1; Lg = 0; and L 6H, 6F, and 6P.
= 5,3,1.
The terms are
The terms are 20, 2N, ZM(2),lL(3), 2K(5),21(5),2H(7),%(6), ZF(7),zD(5),and 2P(4). In summary, the nf5 (np)configuration terms are =H, 6F, 6P, 4M, 4L, 4K(2), 41(3),4H(3),4G(4), &F(4),4D(3),4P(2),4S,20,2N,2M(2),2L(3),2K(5), 21(5),2H(7), ZG(6),2F(7),2D(5),2P(4). T h e terms are the same for complementary configurations, t h a t is, configurations with 2(21+ 1)- N electrons ( N holes). T h e results of these rules agree with those of Slater for p, d, and f orbitals (6). Llteralure Cited 1. Murrell,J. N.:Kettle, 8. F.; Tedder. J.M. The Chemiroi Bond;Wilcy: New York, 1978. 2. 8hter.J.C.ouontum TheomofAfomicSt~urcure:McCraw-Hill:NewYork, 1960;Vol.
l,P302.
..
'
8. Slstec,J.C. Quantum TheoryolAfomieStructur~;MeCraw-Hill:New York, 1960; Vol. 4. Hyde, K. E. J Cham. Educ. 1975,52,87. 6. MeDanie1.D. H. J. Chrm.Educ. 1917,51,147 6. Re12, p 304.
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Journal of Chemical Education
