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 asthe "Hyde method" (3), "spin factor method" (41, "short-cut calculation method" (5), and "numerical algorithm techni- que" (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 the- ory. Without knowing the origin of these generating func- tions it is difficult to follow and be convinced by their dem- onstration. 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 advanced group-theoretical methods are not generally un- derstood by chemists. Both the group-theoretical methods and the non-group- theoretical methods mentioned above have been a ~ ~ l i e d to . . 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. It 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- Table 1. 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 irreducible Spln Multlpliclty representation Young diagarn Zms (9 (2S+ 1) of S(4 7 (heptet) [el 5 (quintet) 15.11 I (singlet) 13-31 8 (octet) 171 6 (hextet) W.11 Volume 66 Number 11 November 1989 893
Transcript
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 as the "Hyde method" (3), "spin factor method" (41, "short-cut calculation method" (5), and "numerical algorithm techni- que" (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 the- ory. Without knowing the origin of these generating func- tions i t is difficult to follow and be convinced by their dem- onstration. 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
advanced group-theoretical methods are not generally un- derstood by chemists.
Both the group-theoretical methods and the non-group- theoretical methods mentioned above have been a ~ ~ l i e d to . . 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-
Table 1. The Relatlonshlp among Total Spln (S), Multlpllclty (25 + I), the Spin-State Partltlon (I.%, Young Diagram), and the lrreduclble Representation of S(n) for the SIX and Seven Electrons
irreducible Spln Multlpliclty representation
Young diagarn Zms (9 (2S+ 1) of S(4
7 (heptet) [el
5 (quintet) 15.11
I (singlet) 13-31
8 (octet) 171
6 (hextet) W.11
Volume 66 Number 11 November 1989 893
functions of E2 and 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
G = Q,, x G.,i. (1)
where GsPatid is the spatial group for the n equivalent elec- trons; GSpi, is the symmetric group S(n) describing the per- mutation properties of the spin function.
Since every group is isomorphic to a symmetric group S(n) (lo), GSpati.l can he depicted in terms of the symmetric group S(n).
Specifying the Spin Function of G,,,, by S(n) 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 correla- tion 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 elec- trons 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.
Specifying the Spatial Functions of G,,,, 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 elec- trons (n) and the total spin (S).
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 prop- erty of conjugate, we know that a direct product representa- tion between conjugate pairs contains the totally antisym- metric 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.
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 prop- erty 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
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 resul- tant 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 621 .
Two electrons, n = 2.
Three electrons, n = 3.
x(G: doublet) = 1/3f[x(G)I3 - x(G3))
Four electrons, n = 4.
894 Journal of Chemical Education
Table 2. Character Tables of the Symmetric Groups S(n)
2 0 2 -1 0 Singlet [2,12] 3 I -1 0 1 [i'] 1 -1 1 1 -1
D"2 2 2 COS 1/29 0 3 " 4 2 cos 1/2$ + 2 cos 3/29
The spatial group, G,,.ti.l, of an atom is the three-dimen- sional rotation group R(3). The Young Diagram (YD), sym- metric 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 arbi- trary rotation, C(+) , are required. This character table is shown in Table 3.
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A few examples of application of eqs. 4 2 1 t o the atomic term symbols are given in the following:
1. P(P2): 1 = 3, Dl = D? 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 p3 configurations are2P+ ZD + &S. d3(d7): 1 = 2, Dl = D2, the d orbitals transform as D2 within R(3) (see Table 6). The terns for d3 configurations are2P + 2D(2) +2F +20+2H+4P+W . - . - - . - . - . d4(d6): I = 2, D' = D2, the d orbitals transform as D2 within R(3) (see Table I ) . The terms for d4 configurations are 'S(2) + 'D(2) + 'F + 'G(2) + Y + 3P(2) + 3D + 3F(2) + 3G + 3H + &D.
Table 4. Examole Awllcatlon for fW21
x(G) in D3 7 1 + 2 c w $ + 2 ~ 0 s 2 $ + 2 c o s 3 $ x(G2) in D3 7 1 + 2 c o s 2 $ + 2 ~ 0 s 4 ~ + 2 c o s 6 $
x(G: singlet) 28 4+6cos$+6cos2$+4cos3$ = 1/2( [~(0) ]~ + +4cos4$+2cos5$+2cos6$ x(G2)i - D O + D ~ + D ~ + D ~
5. d5: 1 = 2, Dl = 0 2 , the d orbitals transform as DZ within R(3) (see Table 8). The terms for d5 confmations are + 2P + 2D(3) + 2F(2)+2G(2) +2H+21+4P+4D+4F+4G+6S.
Tabla 6. Examole Aoollcatlon for Bid7)
x(G) in D2 s 1+2cos$+2cos2$ x(GZ) in DZ 5 1+2cos2$+2cos4$ x(G3) in D2 5 1+2cos3$ f 2cos64
x(G: doublet) 40 6+12ms$+1Ocos2$+6cos3$ = 1/3([~(G)]~ - + 4 c o s 4 $ + 2 ~ 0 ~ 5 $ x(GS)I -D1+D2(2)+D3+D'+DS
=2P+2D121+2F+2G+2H
Table 7. Example Appllcatlon for d'(d6)
G' E C(44
x(G) in D2 5 1 + 2 c o ~ $ + 2 c o s 2 $ x(G2) in D2 5 1 + 2 c o s 2 4 + 2 c o s 4 ~ x(GS) In D2 5 1+2cos34+2cos6$ x(G4) in D2 5 1+2cos4$+2cos8$
x(G: singlet) SO 8+12cos$+12cos2$+8cos3$ = 1/12([x(G)]'- + 6 c o s 4 $ + 2 ~ 0 ~ 5 $ + 2 c o s 6 $ 4x(G)x(G3) + - Do(2) + D2(2) + D3 + D1(2) + D6 3[x (@)I2! = 'S(2) + 'D(2) + 'F+ 'G(2) + '1
x(G: hextet). 46 6+12cos#+ 1 0 ~ 2 4 + 6 c o s 3 0 + eq 20 scos4#+4cos54+2~os6$
- D ' + D ~ + D S + D * + D S + D ~
= 'P+~D+V+~G+%+~I
X ( G : octet). 1 1 eq 21 - Do
= 8.5
Volume 66 Number 11 November 1989 897
6. @(PJ): 1 = 3, D l = D3, the f orbitals transform asD3 within 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) + 3F(9) + 3G(7) + 3H(9) + 31(6) + 3K(6) +
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 2S(2) + 2P(5) + 'D(7) + zF(lO) + 2G(10) + 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 H + B I + 8 S .
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).
Acknowledgment Valuable suggestions by the referee are gratefully appreci-
ated.
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 l w y M L J i ' n r m h ~ c . 19h?.M;-I. 6 Klrmurr K 31 I1 .I Chrm h o w !lh7,ti<.95l - Curl H F : K . . D ~ ~ ~ # . L . J K m .I I h $ a I96O.X.B-
~ ) ~ ~ ~ ~ ~ ~ ~ . ~ . n K J . C ~ ~ ~ P ~ < . I ~ , ~ . ; . I I ( ~ 9. Jud6.B. R. Phys.Rau. 1367,162,28.
la. Fraleigh, J. B. A F i ~ l t Cmme in A b ~ t m f Algebra, 3rd ed.;Addison-Wesley: Reading, MA, 1982: p 72.
11. Flurry. R. L. Symmetry Groups: Theory and Chsmieol Applirotionr: Prentia-Hall: Englewoad Cliffs. NJ, 1980; pp 151.165.
12. Flurry. R. L. Quantum Chemultry: Pnntice-Ha1l:Englewmd Cliffs.NJ, 1983:pp 112- ,9Q
13. Littlewood, D. E. The Theory of Croup Chorocfrrs, 2nd 4.; Oxford University: New York, 1950: pp 266271.
