The snectroscooic terms arisine from the electron occu-

Darl H. MeDaniel University of Cincinnati Cincinnati, Ohio 45221

pancy 2 nonequ&alent orbitals -&I be obtained from an analvsis of all possible microstates for the configuration in

Spin Factoring as an Aid in the Determination of Spectroscopic Terms

question but may he obtained more easily from <he product of all terms arising from the occupation of the equivalent or- bitals alone (I).' A similar simplification is possible for terms arising from electron occupancy of equivalent orbitals by factoring into a and B spin sets. This powerful method was developed and applied to atomic spectroscopic terms by Shudeman in 1937 (3). I t has recently been rediscovered using a group theoretic approach by Judd (4). Chemists have not availed themselves of this method due, perhaps, to the limi- tation of the literature treatments to terms for monatomic soecies. The ournose of this Daner is to aonlv the method of . . . . .. . spin factoring in a direct, simple fashion to illustrate how the Runsell-Saunders term svmhols for molecules and for ions in various crystal fields &ay he obtained. The development presented will not rely on familiarity with references (3) or (4).

Spectroscopic Terms for Atoms As is k n o q , the number of microstates corresponding to

any electronic configuration is given hy the coefficient of the appropriate term in the expansion of (e + h)" where the ex- ponent of e represents the electron occupancy, the exponent of h represents the hole occupancy, and n is the total number of electrons that may occupy the set of orbitals. For thep or- bitals we have (e + h)6 = e6 + 6e5h + 15e4h2 + 20e3h3 + 15e2h4 + 6eh5 + he; i.e only one microstate is possible for a p6 configuration, six microstates forp5 orp', 15 forp4 etc., and in general n!/n.!nh! microstates for a configuration of n. electrons and nh holes distributed among n spin orbitals. Factoring the polynomial into a and B spin sets we have (e, + h,)"/2(es + hs)"I2. For the p orbitals we have

(ee3 + 3eR2k, + 3enkn2 + hU3)(ep3 + 3ep2kp + 3ephp2 + hd3)

Partial terms2 mav he associated with the semimicrostates2 of the spin factored polynomial by inspection of the ML values arisine from a articular orbital occupancy (see Fig. 1). The . . ML v&es for each semimicrostate are simply the sum of the ml values of the occupied orbitals and a partial term of a given L value arises from a set of semimicrostates running from ML = -L through Mr. = +L. In the above example the e3 and h3 semimicrostates are clearly associated with S terms while the e2h and eh2 semimicrostates are associated witb P terms, as mav he shown bv an examination of the Mr values of the se&iiicrostates Gee Fig. 1). The product of p&ial t e r n from an a soin set and a B soin set eives terms havine L values which run idinteger stepi3 from IL: - Lsl through TL, + LpI. Each

Presented at the 169th National Meeting of the American Chemical Society in Philadelphia (1975). ' For the systematic examination of microstates to obtain term designations see Reference (2).

The designation semimicrostates ia used to emphasize that only electrons of one spin type are permitted in contrast to the usual mi- crostates where hoth a and 8 soins are allowed. Each horizontal line in Figure 1 contains one aemimicrostaW. Partial terms arise from semimirrostntes and are thus nswciated with n single spin set.

"or examole the ~roductof oartial terms haviwL values of 3 and 5 would have.^ values of 2,3,4,5,6,7, and 8.

Figure 1. Some partial terms from semimicrostates

partial tern has a well defined M s value, and each product of partial terms has a well defined Ms value. The product of the partial terms associated with particular sets of semimi- crostates may thus be associated with single columns in the usual Mr Venus MQ arravs used to obtain suectroscooic terms. The product of the a and 8 semimicros&tes automatically vields Pauli allowed microstates. and the match between the Dumber of microstates and the tbtal degeneracy of the terms produced serves as a check on the mechanics of the procedure. Thus from the partial terms obtained for thep orbitals we are in a position to obtain the spectroscopic terms arising from the entire manifold of microstates possible for the occupancy of a set of equivalent p-orbitals.

A p6 configuration can arise only from (e,3)(eg3) yielding an S X S = S. The Ms value is zero since the a and p spin sets are equally populated and the total degeneracy is 1 X 1 = 1, so the onlv s~ectroscodc term is a 'S.

A p 5 c~nkguration-can arise only from (e,3)(3e02) and (3em2)(en3) vieldine S X P = P with M,T = f H and a total de- - - generacy of6. Theonly term arising f iompris thus a 2P.

A p4 configuration contains the microstates (e,3)(3ea) and (3e,)(es3) as well as (3em2)(3eP2). The former yield an S X P = P term with M s = i1 and adegeneracyof 6. Thesemust be six of the nine microstates of a 3P term. The (3e,2)(3es2) - P X P = S, P, and D with Ms = 0 and a degeneracy of 9. The P components witb Ms = 0 complete the 3P term, and the 'S

Volume 54, Number 3, March 1977 1 147

and 'D terms account for the remaining microstates with M s = 0.

A p3 configuration contains the microstates (e,3)(ha3) and (ep3)(h,3) which give two of the four microstates of a 4S term. The other two components of the 4S as well as 2D and 2P terms come from P X P = S, P, and D resulting from (3ea2)(3ep) and (3e,)(3ee2).

The p2, pl , and pO configurations give rise to the same terms as the D ~ . D ~ . and p6 configurations due to the symmetry . of e and h terms in the polyno&ial expansion.

Table 1 gives the partial terms arising from the occupancy of a single spin set b; a given numher of electrons, or holes, for the orbitals through g. The number in italics following each set of partial terms is the appropriate coefficient from the ex~ansion of (e, + h,)n/2and represents the number of sem- imkrostates for the particular electron (or hole) occupancy, which is equal to the sum of the orbital degeneracy of the nnrtial terms eiven. The nartial terms were obtained from r-- ~ - - ~ - ~ ~ ~ - arrays of Mr, values for semimicrostates of the type shown in ~ i ~ u r e I.

An exampleof the use of spin factoring to simplify the de- termination of soectrosco~ic terms for an f4 confiwration f~llows.4 An f 4 c&iguration will include microstates arising from e,4, e,3eg, and e,2es2. All terms of high spin multiplicity will he replicated in those arising from microstates of lower M s values.

t (35e"')(hs7) - S[(S)(I G F D S)] total degeneracy = 70 (he7)135e#) MS = *2

t (35e"3)(7ep) - 3[(F)(I G F D S ) ] total degeneracy = 490 (7e,.)(35ep3) Ms = *l (includes

quintet states above)

(21e,2)(21es2) - I[(HPF)(HPF)] total degeneracy = 441 Ms = 0 (includes quintet and triplet states above)

The above yields 5(1 G F D S), 3[P(3) D(2) F(4) G(3) H(4) I(2) K(2) L MI, and '[S(2) D(4) F G(4) H ( 2 ) I(3) K L(2) Nl Spectroscopic Terms lor Molecular Specles and for Atomic SDecles In Nonspherlcal Cryslal Fleldss

The spectroscopic terms into which the free atom states are split by a reduction in symmetry may be determined by the methods of group theory and are independent of the spin multiplicity of the states involved (6). Accordingly, we can use the correlation of free atom states and states in lower sym- metry to assign the partial terms arising from the occupancy of a single spin set of orbitals (either a or 8) in various sym- metries. In a cubic field (i.e. under symmetry group 0) an s orbitalgoestoanal,pgoestotl ,dtoe+tp,ftoa2+t~+tz, etc.; and an S state goes to an A1 state, a P state to a TI state, etc. From this it follows immediately that a zero population of an orbital set yields an A1 (or its equivalent) state, and that an occupancy of an orbital set by one electron yields the same designation for the term as for the orbital. Since the p orbitals retain their degeneracy under 0 symmetry, being designated as t under 0, the terms arising from occupancy of the t t orbitals may he obtained directly from those of the p or- bitals; i.e.

P.,O yields S t 10 yields A t p,,' yieldsP t l l yields TI p,: yields P tl2 yields TI p,: yields S t13 yields At

Gwup theory has heen used lo develop the terms for the full/" manifold ( 50 ) and thefin manifold (5bJ.

;Tables of direct ~nrductr. and correlation tables mav be found in reference (12). The methods bywhichsuch results ma; be obtained are detailed in group theory texts, such as reference (9).

An fm7 yields an S term; hence azl t13 t23 must yield an AI term. Since we know that a21 yields an Az, and t 13 yields an A1 term, then tz3 must yield an A2 term. In like manner a dm5 configuration yields an S term; hence e2t23 must yield an A1 term; since t23 yields an A* term, e2 must also yield an A2 term. Finally, from the hole formalism we know that tz2 ( ~ r ) t 2 ~ (8) must give a Tp term, and since tz3 gives an Az, tz2 must give a T I term. These results, along with those for several other svmmetries, are given in Table 2. (It may be noted that the partial terms ari& from the two electron occupancy of any degenerate set of orbitals are given by the antisymmetric di- rect product of thesymmetry species to which the orbital set belongs, as required by the Pauli exclusion principle.)

Althoueh the oartial terms of Table 2 were arrived at bv a - - - ~ - ~ ~ ~~ ~. consideration of the spectroscopic states for atoms in various

Table 1. Partial Terms Arising from t h e Occ~pancy of a Single Sriin Set [either u or 81

occLpancv (elecflon. or holed 0 1 2 3 4

Orbital Set S S111 S ( 1 ) P Sill PI31 d 8 1 1 ) D ( 5 ) PF(101 f S ( 1 ) F ( 7 ) PFH1211 S D F

G I ( 3 5 ) g S ( 1 ) G ( 9 ) P F H K l 3 6 ) PFI21G SDI21FG[21

H I K M H I I 2 1 K L N

r h e orbital degeneracy is given i n parenthesis: t h e number of par- tial terms, i n square brackets.

Table 2. .Partla1 Terms Arwng from the Occupancy of o (or 81 Orbital Sets on Vartour Field Svrnmetrlesa

.- -- Elec t ron Partial Electron Partial

Orbital Occupancy Term Orbital Occupancy Term

Under 0 and Td

1 0 1 0 1 2 0 1 2 3 0 1 2 3

Under D, 0 1 0 1 0 1 2

Under D- o 1 0 1 0 1 2 0 1 2 0 1 2

Under D , 0 1 0 1 0 1 0 1 0 1 2

Under D, 0 1 0 1 0 1 0 1 0 1 2 0 1 2

Under I 0 1 0 1 2 3

' 0 1 2 3 0 1 2 3 4 0 1

2 3 4

a In symmetries for Which g and u or ' and " are appropriate labels Odd electron occupancy of u a t " orbitals re ru l t r in u o r " partial terms; all others will be labeled g or '.

symmetries, the use of the partial terms for generating spec- troscopic terms does not have this restriction. A few examples of applications follow.

The spectroscopic terms for a tz3 configuration in 0 sym- metry would be IAg (from tZa3 X ha3 + ha3 X t q 3 ) + 2A2 + 2E + 2T1 + 2T2 (from 3tzm2 X 3tzO1 + 3tza1 X 3hO2 - T I X Tz, note that the A2 arising from this product contains the Ms = f $ needed to complete tbe 4Az already found.)

The spectroscopic terms of an e2 configuration under D s Da, Ds etc. or 0 symmetry would be a 3A2 (from ee2 X hp2 + h,2 X ea2) + 'A1 + ' E (from 2e,' X 2eS1 - E X E which contains the Ms = 0 component of the 3A2 term).

A r 2 configuration is quite similar to the e2 configuration vieldine a 32- + ' Z + + 'A . In this case the 'A arises from the . . . . . - -. . . . -.

There are, of course, other methods for obtaining the results of the last few examples (7).The primary difficulty lies in the assignment of spin multiplicity to the individual terms. The eas; of makingsuch assignments appears to be the prime virtue of this method.

Using the spin factoring approach one may readily convert one-electron energy level diagrams into qualitative spectro- scopic term diagrams for any electron population. One-elec- tron energy level diagrams for the crystal field splitting of d-orbitals in most eeometries mav be readilv obtained bv the methods of ~ r i s h i a m u r t h ~ a n d ~ c h a a p (6 For example, a trieonal hinvramidal eeometrv (Dl*) oroduces the followine - . -. . . . s t A g fi& knergy oAer

from which the order of energies of the electron configurations

I 3-> ,2

3 1 A2

Orbital Terms Occupancy

Figure 2. correlation diagram f w me d2 configuration in a %bong Oh field.

B % + + *ttf * * + %lt+ ,+-% 5

4~1g 3~2,, 2

T2s Es '"r. Figure 3. Some ground state terms in a weak 00 field.

for dZ (ignoring spin pairing energy and configuration inter- actions) would be

d"' < e"'dl < e'2 < ale" <ale' <at'%

The terms for nonequivalent electrons may be ohtained in the normal fashion as the sinelet and triolet s t a w arisine from the direct product of the r&resentatibns designated i y the orbital labels; i.e.

The doubly occupied alf orbital would give only the 'Al' term. The terms for which spin factoring offers simplification are those arising from equivalent electrons-the e'2 and e"2 configurations. Remembering that ' X ' = 'and " X " = ', while

X " = " all of the doubly occupied equivalent orbitals must give terms labeled '. The e2 configuration under D3 has been worked out above, thus

and

3A2', lA{ and lE'

The strong field limit term diagram is illustrated in Figure 2. The term diagram in the weak field limit may be obtained in a straiehtforward fashion from the enerw levels of the d2 terms and correlation tables for the ~ 3 a n d ~ J g r o u p s . Wood has ouhlished a comdete enerw level diaeram for d q n D t r "- .., . symmetry (10).

Althouehthe number of microstates vieldine a oarticular term is e&al to the total degeneracy ofthe te;m, it is some- times useful to consider a sinele microstate obevine Hund's rules of maximum mukiplicit; as the "ground state-configu- ration" and to associate with it the ground state spectroscopic term. This has been done for the d electrons in a crystal field of Oh symmetry in Figure 3. Since a completely filled a or 13 d-shell gives no contribution to the orbital term, the orbital desienations for d6 throueh dl0 are a repetition of those for dl &rough d , and indeed'the manifold (;fterms of maximum multiolicitv will he identical. leading to the same spin allowed trans~ilons.in the crystal field spectia. The grounditate terms mav he written by inspection from a knowledne of the pan~al terms and remen&ri;lg that the product of A Z and T; or T2 changes the T subscript, but the product of A2 and E is just E. The hole formalism relates the spectra of states of similar spin multiplicities in the well-known manner (9).

Calculations have recently been made on the electronic structure of a number of boranes (11). For the unknown BmHlo4- with a pentagonal dodecahedra1 arrangement (point group l h l it was predicted that the ground state electronic configuration would he (h,)"(fis. The determination of the full set of spectroscopic terms for all permitted multiplicities of this one confiewation would he a formidable task bv con- ventional metho&. The terms arising fromg4 and h6 config-

Volume 54, Number 3, March 1977 1 149

urations are developed helow.6 The full manifold of terms from h6g4 would of course he the direct product under Ih of all of the terms fromg4 with all of the terms from h6, a total of some 1144 terms.

Total Degener-

acy

Thanks are due Professor John J. Alexander for useful suggestions regarding this manuscript.

Literature Cited I11 Henber&G.,"Atomi~SpedraandAtomicStrueture."Do~~rPYbli~~ti~n~.NewYork,

19U.l I21 Douglan.8. E..and McDaniel, D. H., "ConeepUandModelsofInorganicChemistry."

John Wiley and Sons, Ine.. NewYork. 1966; Hyde, K. E.. J. CHEM. EDUC., 52.87 (19751.

I31 Shudeman, C. L. B., J. ~ m n k l i n Inaritute, 224,501 (19371. I41 Judd, B. R., Phya. Re"., 162.28 119671. 151 la1 Curl, JI., R. F.,and Kilpstriek. J. E., Amer. J. Phys., 28.357 (1960); lbl Wybournc,

B. G.,J Ch~m. Phys., 45. LlW (19661. 16) Beshe, H., dnn. Phjsib, 4 135 (1929). I71 Goscinski. 0.,sndOhrn,Y..lnr. J. QunnlumChem., 2.815 (1968);Fard,D.l.,J. CHEM.

EDUC., 49,336 (1972); Ellis, R L., and J@e:H. H., J. CHEM. EDUC.,48,97 (1971): Mu1liken.R. S.,Reu. Mod. Phys.. 4, i (19321.

I81 Krishnarnurthy. R.,sndSehaap, W. 8.. J.CHEM.EDUC.,46,799 (19691. I91 Cotton. F. A,, "Chemical ApplicationsofGmupThoory:'2nd Ed.. Wilcy-Intarscience,

Now York, 1971. 1101 Wmd. J.S..lnorg. C h m . 7.852 11968). 1111 Armstmng.D. perkins ins. P.G.,andSt~urart.J.J.P., J. Chem. Sor., DaltonTrana..

"his example is given solely to illustrate the ~implifiration of a formidable prohkm hy spin fanoring. The reader should note theeaw of assignment of spin multiplicities instead of laboring over theme- chanics of obtaining direct products.

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150 I Jowml of Chemical Education