Atomic Term Symbols by Group Representation Methods Jyh-Horung Chen National Chunghsing University, Taichung, Taiwan, R.O.C.
Ford's paper (I) in a past issue of this Journal is an excellent account of obtaining molecular term svmbols bv group theory. Presented in this paper isan improved version of Ford's method for ohtaininr termsvmbols of atoms (1.2). A number of non-group-thioreticai methods such a s t h e "Hyde method" (3), "spin factor method" (41, "short-cut calculation method" (5), and "numerical algorithm technique" (6) apply only for the atomic term symbols. Curl and Kilpatrick (7)determine the term symbols for atoms by using generating functions derived via group theory. Without knowing the origin of these generating functions i t is difficult to follow and be convinced by their demonstration. Wybourne (8) and Judd (9) use a much more sophisticated group-theoretical method to find the atomic terms for the f" and gn electronic configurations, but these
Table 1.
advanced group-theoretical methods are not generally understood by chemists. Both the group-theoretical methods and the non-grouptheoretical methods mentioned above have been a .~. ~ l i to ed fiolve for the atomic term symbols only. The improved Ford's method described in this DaDer is useful for ohtaininr either atomic or molecular term &bols. I t will he descrged for atomic term symbols with illustrative examples. Method
Since the electron is a fermion, the total wave function of a system of electrons must be antisymmetric with respect to interchange nf any two electrons. Furthermore, hecake the angular momentum operators L' and 5'? commuce with the Hamiltonian, the zeroth-order functions should be eigen-
+
The Relatlonshlp among Total Spln (S),Multlpllclty ( 2 5 I), the Spin-State Partltlon (I.%, Young Diagram), and the lrreduclble Representation of S(n) for the SIX and Seven Electrons
Spln Young diagarn
Zms
Volume 66
(9
Multlpliclty (2S+ 1)
irreducible representation of S(4
7 (heptet)
[el
5 (quintet)
15.11
I (singlet)
13-31
8 (octet)
171
6 (hextet)
W.11
Number 11 November 1989
893
functions of E2and S2. TO get a zeroth-order function, we sometimes have to take a linear combination of the Slater determinants of electronic configurations. The total group (G) for the n equivalent electrons of an atom described above is given by (1) x G.,i. where GsPatidis the spatial group for the n equivalent electrons; GSpi,is the symmetric group S(n) describing the permutation properties of the spin function. Since every group is isomorphic to a symmetric group S(n) (lo), GSpati.lcan he depicted in terms of the symmetric group S(n). G = Q,
Specifying the Spin Function of G,,,,
by S(n)
Three electrons, n = 3.
The allowed permutational symmetries of spin function in eq 1 can be obtained from a two-rowed Young diagram directly. A Young diagram is a group of hoxes arranged to illustrate the cyclicstructure of theclass. The largest cycle of the partition is written first, with succeeding shorter cycles on succeeding rows, all the rows being aligned to the left ( I , 11,12). The total spin (S) can be determined by filling the hoxes of the first row with electrons of spin m, = 112 and those in the second row a value of m, = -112 and then summing the m, of all the boxes of the Young diagram. The partition of n electrons into the Young diagram corresponds to a unique spin state. Further, there is a one-to-one correlation between the partition of n electrons and the irreducible representation of S(n). Thus for agiventotal spin S, the spin part must belong to an irreducihle representation of the symmetric group S(n) characterized by partition of n electrons through the Young diagram. As an example, for n = 6,7, the relationship among total spin (S), multiplicity ( 2 s I ) , the spin-state partition (i.e., Young diagram) and the irreducible representation of S(n) is shown in Table 1.
+
by S(n) The spatial part of the n-electrons wave function in an atom must belong to a certain irreducihle representation of the symmetric group S(n), specified by the number of electrons (n) and the total spin (S). Specifying the Spatial Functions of G,,,,
Specitying the Total Functions of G by S(n)
In each S(n), the conjugate of the Young diagram results from the interchange of rows with columns. From the property of conjugate, we know that a direct product representation between conjugate pairs contains the totally antisymmetric irreducihle representation [I"] once and only once. For electrons, the spatial and spin functions must transform as conjugate irreducihle representation of the appropriate S(n) so that their product will he totally antisymmetric. Therefore, a r spin function requires a r spatial function, where r is the irreducihle representation of S(n) conjugate to r (I). The partition of n numbers, which relates to any class of S(n), can be expressed in the form of cyclicstructure.
(P)= (lb',
.. ,ibi, . ..nb")
i = l , 2 , . ..n
(2)
where bi is the number of cycles of length i. The character xdG: F,) of G in the a representation of the spatial group adapted to the representation F, of the symmetric group S(n) is shown in eq 3. This assures the antisymmetric property of the fermion.
The sum is over all classes, p, of S(n);n is the degree of the S(n); h, and x(p:F,) are the order and_character of the pth class in the irreducihle representation rjof S(n) conjugate to that of the spin (i.e., rj); the product is over the cycle length, i, of the partition for the class ( P )shown in eq 2; G' is the ith 894
Journal of Chemical Education
power of the operator G of the spatial group; and x(G) is the character of Gapatial. The numerical calculation of eq 3 requires the knowledge of characters of irreducible representations of S(n). Tahle 2 lists S(2) through S(6) along with classification of the resultant spin states (11,13). In Tahle 2, we have connected the conjugate representations by lines. Using these results the explicit formulae, eq 3, for all irreducihle representations up t o n = 7 are rendered in the eqs 6 2 1 . Two electrons, n = 2.
x(G: doublet) = 1/3f[x(G)I3 - x(G3))
Four electrons, n = 4.
Character Tables of the Symmetric Groups S(n)
Table 2.
Desr-
4
S(4)
(1')
6(2.12)
1
[2,12] [i']
3(Z2) e(3.1)
6(4)
1 -1 2 -1 1
1 -1 0 1 -1
1 1 0
2 3 1
I -1
1 0 -1 0 1
(I6)
30(4. 1)
1 4 5 6 5 4 1
1 0 -1 0 1 0 -1
Spin8asis Quintet Triplet Singlet
20(3. 2)
20(3, 12)
15(Z2,1)
10(2,13)
2q5)
1
1 1
1 0 -1 1 -1
-1 0 -1 1 1
1 0 1 -2 1 0 1
1 2 1 0 -1 -2 -1
1 -1 0 1 0 -1 1
-1
(1')
1 4 4 6 1)
gO(4.2)
90(4. 1')
40P2)
120(3,2,1)
40(3, 13)
1 5 9 10 5 16 5 10 9 5 1
1 0 -1 0 0 1 0 0 -1 0 1
1 -1 1 0 -1 0 -1 0 1 -1 1
1 1 -1 0 -1 0 1 0 1 -1 -1
1 -1 0 1 2 -2 2 1 0 -1 1
1 0 0 -1 1 0 -1 1 0 0 -1
1 2 0 1 -1 -2 -1 1 0 2 1
15(Z3) 45(Z2,12) 1 -1 3 -2 -3 0 3 2 -3 1 -1
1 1 1 -2 1 0 1 -2 1 1 1
1 3 3
2 1 0 -1 -2 -3 -3 -1
Hextet Quartet Daublet
120(6)
Spln Basis
1 -1 0 1 0 0 0 -1 0 1 -1
Heptet Quintet Triplet Singlet
char act^ Table for R(3)
Table 3. R131
E
D"2
2 4
03"
15(2. 1')
Spin Bask
Cldl
2 COS 1/29 2 cos 1/2$
+ 2 cos 3/29
The spatial group, G,,.ti.l, of an atom is the three-dimensional rotation group R(3).The Young Diagram (YD), symmetric group (S(n)), and rotation group R(3) are used to yield the atomic term symbols. To derive the R(3) group, only the character of the identity operation, E, and an arbitrary rotation, C(+), are required. This character table is shown in Table 3. Volume 66
Number 11
November 1989
895
A few examples of application of eqs. 4 2 1 t o the atomic t e r m symbols are given in the following: 1.
P(P2):1 = 3, Dl = D?
5. d5: 1 = 2, Dl = 0 2 , the d orbitals transform as DZ within R(3) (see 2P 2D(3) Table 8). The terms for d5 confmations are 2F(2)+2G(2)+2H+21+4P+4D+4F+4G+6S.
+ +
the f orbitals transform as D3 within R(3) (see Table 4). The terms for fZ configurations are 3P 3F 3H + 'S+'D+'G+% 2. p3: I = 1,Dl = Dl, the p orhitab transform as D1 within R(3) (see Table 5). In summary, the terms for p3configurationsare2P+ ZD
+
+ +
+ &S.
Tabla 6.
Examole Aoollcatlon for Bid7)
d3(d7): 1 = 2, Dl = D2,the d orbitals transform as D2 within R(3) (see Table 6). The terns for d3configurations are2P 2D(2)+ 2 F +. 2 0 +.2 -H- +. 4 -P +. W- . d4(d6):I = 2, D' = D2, the d orbitals transform as D2 within R(3) (see Table I ) . The terms for d4configurations are 'S(2) 'D(2) 'F 'G(2) Y 3P(2) 3D 3F(2) 3G 3H &D.
+
-
+ +
+
Table 4.
+ +
+ + +
+
+
Examole Awllcatlon for f W 2 1
s
x(G)in D2 x(GZ)in DZ x(G3) in D2
5 5
x(G: doublet) 40 = 1 / 3 ( [ ~ ( G)]~ x(GS)I
x(G)in D3 x(G2)in D3
7 7
x(G: singlet) = 1/2([~(0)]~ x(G2)i
+
28
1+2cos$+2cos2$ 1+2cos2$+2cos4$ 1 + 2 c o s 3 $ f 2cos64 6+12ms$+1Ocos2$+6cos3$ +4cos4$+2~0~5$ -D1+D2(2)+D3+D'+DS =2P+2D121+2F+2G+2H
1+2cw$+2~0s2$+2cos3$ 1+2cos2$+2~0s4~+2cos6$ 4+6cos$+6cos2$+4cos3$ +4cos4$+2cos5$+2cos6$
Table 7.
Example Appllcatlon for d'(d6)
- D O + D ~ + D ~ + D ~
='S+'D+'G+'I
x (G: bipM)
= l~211x(O12 x(G2)I
Table 5.
21
3+6cw$+4cos2$+4cos36 +2cos4$+2cos5$ -D'+D3+Ds =3P+3F+SH
Example Appllcatlon for p' G'
E
C(44
x(G)in D2 x(G2) in D2 x(GS)In D2 x(G4)in D2
5 5 5 5
1+2co~$+2cos2$ 1+2cos24+2cos4~ 1+2cos34+2cos6$ 1+2cos4$+2cos8$
x(G: singlet) = 1/12([x(G)]'4x(G)x(G3)+ 3[x (@)I2!
SO
x(G: triplet) = l16([x(G)14+ 2[x(G)I2x(G2) 2x(G4)- [x(G211?
45
x(G: quintet) = 1/24([x(G)]' 6rlGl12rlGI
5
+
896
Journal of Chemical Education
8+12cos$+12cos2$+8cos3$ +6cos4$+2~0~5$+2cos6$ Do(2) D2(2) D3 D1(2) D6 = 'S(2) 'D(2) 'F+ 'G(2) '1
-
+ +
+ +
+
+ +
7+14cos$+10cos2$+8cos3$ +4cos4$+2cos5$ D1(2) D2 D3(2) D4 D" = V(2) 3D ZF(2) =G
+ + + + + + + +
1+2cos$+2a%Z$ D2 = SD
Tabla 8. R(3)
C(4)
E
Do D' D2 D3 D4
1 3 5 7 9
D5
11
D'
13
Table 9.
Example Applbatlon tor
d R(3)
5 5 5 5 5
1 x(G) In D2 1 + 2 ~ ~ x(G2)In D2 x(GS)In D2 l+2ws4+2cw29 1 + 2 ~ 0 ~ ~ + 2 m s 2 ~ ~ 2 c o s 3 ~x(G4)In D2 x(GS)In D2 1+2cos4+2cos2~+2cos34 +2ms4# x(G: doublet). 1+2wsd+2ws24+2ms34 eq 11 +2~0~4m+2~0~54 1+2cos4+2ms29+2ws34 +2~0~4r$+2~0~5#+2m~6#
Example Application tor
C(+)
E
75
11+20wr4+18ms24+12ms34 +8ws44+4ms54+2ws64 D' D2(3) DS(2) D4(2) -Do + D 5 + DB
+ +
x(G: werlet). eq 12
24
x(G: hextet). eq 13
1
P(P)
1+2cos4+2oos24 1+2COSZb+2COS4# 1+2cos3~+2cos64 1+2cos4$+2cos84 1+2c0s5~+2cos10~
+
+
4+8ms$+6ws20+4cos34 2 cos 44 -D1+D2+D3+D4 =4P+4D+++'G
+
1
-
Do
= 8.5
Table 10.
Examole Aoolicatlon tor f'
. . ~ ( 0 in 3 0%
7
x(G: singlet), eq 14
490
1+2cos64+2cos12~+2ws184 46+84ws4+82rns2++7Oms34 +62~~4Q+46ws54+38ws 6~+24ms7~+18ws84+10 cos9~+6coslo4+2cos11~+2 ~ o 126 s Dq4) D' D1(6) D3(4) D4(8) 014) DB(7) D7(3) D8(4) D9(2) Di0(2) = 1a4) 'P 'D(6) 'F(4) 'G(8) 'H(4) 'l(7) 'K(3) 'L(4) 'M(2) 'N(2) ' 0
-
+ +
+
x(G: dwblet). eq 18
784
+
+ + + + + + + + ++ ++ + + + + + +
x(G: triplet), eq 15
588
56+112cos~+100ms2~+90 cos3Qf 72co64@+58ms54+ 4Ows64+28ws7#+ l6cos8$ l o cos 94 4 cos 104 2 cos 114 D'(6) 0%) D3(9) D4(7) DS(9) D6(6) D7(6) D8(3) D0(3) Dlo D" = 'P(6) 'D(5) =F(9) 3G(7) 'H(9) %6) =K(6) 'L(3) %(3) ' N '0
+ +
x(G: quintet). eq 16
140
-+ + +
+
+ +
+
+
+
+ +
-
+
+
+ +
1+2cos4+2ws2++2ms34 D3
+
+
+
+ + + + + + + + + + + + + +
x(G: ~ w t e t ) . eq 19
392
+
+ + + +
+
+
+
+
+ +
+ + +
x(G: hextet). eq 20
46
+
40 76cos 4 72 cos 24 6Ocos 34 +5oms4++36cos54+26ms 6++16ws7$+10cos84+4ms 94 2 cos 104 D0(2) D'(2) D2(6) D9((5 047) DS(5) De(5) 073) D8(3) D9 DO ' = 'S(2) 'P(2) 'D(6) 'F(5) 'G(7) 'H(5) 'l(5) 'K(3) 'L(3) 'M 'N.
+
+ +
+ + + + + + + +
7
+ + + +
+ +
16+30cos~+28ws2~+22ws30 +18cos4d+12ms50+8cos64 +4cas74+ 2cos64 -Do D' D2(3) D3(2) D4(3) D5(2) DB(2) D7 D' = 5P %(3) SF(2) $G(3) 5H(2) $42) 6K IL
+
x(G: heptet). eq 17
+
+ +
72+140ws~+130~2~+116 cos3++96cos44+76~os54+ 580osB~p+40ws7~$+26cos84 16 cos 94 8 cos 104 4 cos 114+2ws12#-D0(2)+D'(5)+ D2(7) Ds(lO) 0110) D5(9) De(9) D'(7) D8(5) D8(4) D'O(2) D" Dl2 = a2) + >P(5) 2q7) 2F(10) *G(10) 2H(9) 21(9) 2K(7) 9(5) 'MI4) 2N(2) 20 2O
-
+ +
+
+
+
+
+
+
+ +
+
+
+
+
+
6+12cos#+ 1 0 ~ 2 4 + 6 c o s 3 0 + scos4#+4cos54+2~os6$ -D'+D~+DS+D*+DS+D~
='P+~D+V+~G+%+~I X ( G : octet).
eq 21
= 7F
Volume 66
1
1
-
Do
= 8.5
Number 11 November 1989
897
1 = 3, D l = D3, the f orbitals transform asD3within R(3) ( s e e Table 9). The terms for P c o n f i g u r a t i o n s are ' S ( 4 ) + 'P ' D ( 6 ) + ' F ( 4 ) + 'G(8) 'H(4) 'I(1) 'K(3) + 'L(4) ' M ( 2 ) + 'N(2) 'Q + 3P(6) + 3D(5) 3 F ( 9 ) 3G(7) 3 H ( 9 ) 31(6) + 3 K ( 6 ) 3L(3) 3M(3) 3 N 30 &S SP 5 D ( 3 ) SF@) 5 G ( 3 ) SH(2) &K S L 7F. 7. f?1 = 3, D l = D3, the f orbitals transform as D3 within R(3) (see Table 10). The terms for 'f configurations are 2 S ( 2 ) 2 P ( 5 ) 'D(7) zF(lO)+ 2 G ( 1 0 ) + 2H(9) 21(9)+ ZK(1) XL(5) + 2M(4) + 'N(2) + '0 + 2Q + 'S(2) + 4P(Z)+ *D(6) I F ( 5 ) + l G ( 7 ) + "H(5)+ 41(5) 4K(3) &L(3) 4M 4N+ 6 P + 6 D + S F + 6 G + 6.
@(PJ):
+ + + + + + + + + + + + + + + +
+
+
+
+ +
+
+ + +
+
+
+
+ + +
+
+
6H+BI+8S.
The method presented here is more or less consistent with the results mentioned in Flumy's books (11, 12) but is an improvedprocedure. The use of the eqs 4-21 to find the term symbols is neater and more straightforward than Flurry's demonstrations (11,12).
898
Journal of Chemical Education
Acknowledgment
Valuable suggestionsby the referee are gratefully appreciated. 1. Ford, D. 1. J. Chsm.Educ. 1972.49.336. 2. Chen.J. Y.;Chcn.J. H. Chemistry (TheChineaeChem. Soc.:Tahan,China), 1987.45, A229.
3. Hyde, K. E. J.Chpm.Educ. 1975,52,87. 4. MeDaniel. D. H.J. Chem. Educ. 1977.54.147. 5 D ~ I ~ t ~ . l . . , E l lM w yL J i ' n r m h ~ c .19h?.M;-I. 6 Klrmurr K 31 I1 .I Chrm h o w !lh7,ti
