Partial and Ligand Field Terms E. M. R. Kiremire University of Zambia, P. 0. Box 32379, Lusaka, Zambia Recently, the concept "partial terms" originally introduced by McDaniel (I) was revived in its utilization to generate Russell-Saunders (RS) terms (2, 3). In this article, the derivation of partial terms and their application to yield ligand field (LF) terms' are described. By applying Bethe-Mulliken principle ( 4 , 5 )of descending symmetry, the partial terms are readily derivable. Of ereat interest for the purpose of interpreting electronic spec. . tra of transition metal complexes, are the LF terms aris~ng from d" electron configurations. Consider an octahedral (0,) transition metal complex which descends into a C2h geometry. As we know, under Oh, the d orbitals split up into e, and t2, energy levels. When the Oh changes to the C2h geometry, thee, level also splits up into a, and b,, while the tz, gives two a, plus b, levels. By using these irreducible representations under Cz,, the semimicrostates ( I ) fore," (n= 1,2) and t!, (n = 1-3) confieurations can be constructed. From the semlmlcrosta.tes t h d ~ partial ~, terms are easily obtained. A correlation of the C?h panial terms so obtained with the 0, yields the correspondhg O h partial terms, which can be used to derive LF terms. Where two or more electrons are involved, the application of the direct product rule (6, 7)gives us the appropriate partial term. This procedure is summarized in Table 1. As is evident from the table, the partial terms are obtained from the irreducible representations of the levels occupied by electrons. The resultant partial terms to be
-
utilized to generate LF terms in Oh are E,, AT TT TI,, and AZgfor the respective configurations e:, e i , tZg,tZg,and t&. There is a striking similarity regarding the process of deriving the partial LF terms and the partial RS terms (I, 2). The partial LF terms from the semimicrostates correspond to the MLvalues, and the partial LF terms in Oh correspond to the partial RS terms. Furthermore, just as the RS term of the highest spin multiplicity can be extracted a t the initial stage when a partial term is generated (2,3), so is the LF term of the highest spin multiplicity. Appllcatlon of the Partial Terms To Derlve LF Terms Let us illustrate the method on the t& and t&e: configurations arising from d3 in an octahedral ligand field. The principles developed earlier for deriving RS terms using partial terms (2,3) apply here as well. For the t& configuration with equivalent electrons, its partial term is A% Hence LF term of the highest spin multiplicity corresponding to a (3) spin set (S = 312) is4Aw The LF terms corresponding to the a (2) .8 (1) . . s.~ i nset (S = 112) will be obtained from the direct product of the partial terms T~,andT>laftersubtractingthe A,, term of then (3)spin set. The a (21 and 3 (1, components I Ligand field terms are also known as "spectroscopic terms". "cubic field terms". "group theoretic terms". "terms", or "states".
Table 1. Derivallon 01 OhParllal Terms Using the Semlmlcrostatesot Gh Partial Terms OhLFConfigurations
216
GoSemlmlnostates
Journal of Chemical Education
under Gh
OhPartial Terms
Terms of the Highest Spin Multiplicity
Table 2.
The Derlvatlon of LF Terms ol if,
tion with nonequivalent electrons no subtractionof a term is involved as in the case of configurations with equivalent electrons. However, the product of the spin multiplicities must he evaluated with the help of the newly constructed Table 4. The table' can readilv be expanded depending upon the magnitudes of the spinrnulti&icities iniolved Ynthe multiplication. The result of the direct product is repeated as many times as the number of types of resultant spin multiplicities. For instance T u X Eg = T I , Tz, and 3 X 2 = 4 2 from tables. Hence 3 T ~Xg2ER= 4(Tlg+ Ta) + %(Tlg T a ) The procedure of derivmg the t&ei terms is also summarized in Table 3. The derivation of LF terms for configurations that are more than half filled with electrons can be performed in the same way as in the case of RS terms from similar configurations using partial terms (3). The c o n c e ~ t used s in derivina LF terms of configurations with nonequivalent electrons are equally applical;le to the derivations of RS terms of configurations with nonequivalent electrons. For example the RS terms of the configuration dlP are obtained from the product 2D X 2F.The product D X F (these being the partial terms of dl and P,respectively) yields H , G , F, D, and P (1-3) while 2 X 2 gives 3 1 (Table 4). Hence the RS terms of dlP are 3(HGFDP)and
+
+
Table 3. The Derlvatlon of LF Terms of l b ] ,
+
+
e;
s = 112
+ E*
'(HGFDP). Conclusion '(A,, + E,
+ Gd
A s i m ~ l method e of derivine ~ a r t i aterms l using the ~ r i n ciple of iescending symrnetrylh& been introduce& he-partial terms can be used t o generate L F terms. Although the application of partial terms to obtain LF terms is related to that of McDaniel (I), their derivations are different. Other methods of deriving L F terms are known (4,5,8,10-15). But the current one is much simpler and more systematic. The derivations of RS terms and LF terms are based on similar concepts. Furthermore, the article introduces the derivation of RS terms of confieurations with noneauivalent electrons, which is much faster-than the methods f o h d elsewhere (1621).
ZEg
Table 4. The Mulllpllcatlon of Spln Multlpllcltles 2S+ 1
1
2
3
4
Acknowledgment I would like to extend my gratitude to M. Musopelo for typing the manuscript and the University of Zambia for providing the facilities that enabled me to accomplish this work. Literature Clted
of the u (2) p (1) spin set correspond to the partial terms of t& and t&, which are Tu and Ta (see Table I), respectively. Hence the doublet terms so obtained are 2E,. 2Tjn.and 2T,, The process parallels that used in deriving I% te&s of d3cy the partial terms method (2,3). T o obtain a direct product, you simply read off the result from an appropriate direct (8,9). These operations are summarized in ~ r o d u c table t able 2. The configuration& $t has nonequivalent electrons. However, the component t has equivalent electrons. The terms arising from the t& are%TI,, 'A1,, 'E,, and I T z g (see Table 3). These are multiplied with %Eg term of ei (see Table 1).The multiplication of terms from the components of a configuraThe interested reader may contact the author regarding the construction of Table 4.
1. McDanidD. H. J . Chem.Educ. 1977,54. 147. 2. Kiremire, E. M. R.J. Chem. Educ. 1987.64.951. 3. Kiremire, E. M. R.J. Chem. Educ. 1989.66.419. 4. Cotton, F. A. Chemicai Applications 01Cmup Theory. 2nd ed.; Wiley-Intemcience: New York. 1971; P 260. 5. Fackler. J. P..Jr.. Ed. Symmatryin Chemical Theory:Doaden. Hutchinson andRms: Straudshurg, 1973; p 163. 6. F i g s , B. N. Introduction lo LigondFialda: Inteneiene: London, 1967: p 137. 7. Kett1e.S. F.A. SymmelryondSLruefure; Wiloy: Chicheater. England, l 9 8 a p 193. 8. Kettle, S. F. A. Coordination Compounds; Nelson: London, 1969: p 72. 9. Grifiith. J. S. The Thenry of Transition M e f d Ions. 2nd ed.: Cambridge University: London, 1964: p 386. 10. ReE6, p 149. 11. R e f 9 , p 226. # ~ C. K. ~ Modem ~ ~ Aspects ~ eof Llgand ~ , Field Theory: North~Holland:Amster12. ~ dam, L97L: p 241.
13. Balihausen.C.J.InfroducfionloLi~~ndFieldTheory;McGrav-Hill:NeuYork,1962; p 76.
m. Lever, A. B. P.Inorganic El~ctronicSpectroscopy; Elsevier: Amsterdam. 1968: p 93. 1s. suiton. D. Electronic Spectin of Tronaition M a d Compiexes: McGrau-Hill: London. 1968: p 182. 16. Hsizhorg,G. AtomicSpeclioondAtomicSfruclure,2nded.:Dover: New York, 1944:p 129.
17. Candler, C. Atomic Spedro, 2nd ed.; Arm- Smith: Brisfol. England, 1964: p 1 1 . la. White H. E.1ntmdvcfion to Atomic Spoclm: McGraw-Hill: New York. 1934:pp 185 and 235. 19. Condon, E. U.: Shortley, G. H. Tha Theory of Atomic Sperlm, 2nd ed.: Cambridge University: London. 1964; p 188. 20. R e f 4 . p 246. 1980: p 1356. 21. Cotfon.F.A.: Wi1kinwn.G. Advone~dInorgonicChemistry. 4th d.;
Volume 67
Number 3 March 1990
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