Hole Formalism and Its Effect on Partial and RS Terms E. M. R. Kiremire University of Zambia, P. 0. Box 32379. Lusaka. Zambia A numerical method of generating partial terms and using them to derive RS (Russell-Saunders) terms has been advanced? On examining the partial terms so generated, an eouivalence relationshin is discerned. For instance the Dart i h term of p' is the same as that of p2 and those of dithe same as those of d3. This equivalence relationship is given in the table for p", dn, f", and gn configurations. The relationship can he regarded as "hole formalism" based on a halffilled shell. Thus the partial terms of O are identical to those of P,namely H, F, and P. Also the corresponding RSterms of the highest spin multiplicities can directly be obtained from the partial terms. As a consequence of this relationship, it is only necessary to know the partial terms for the configurations with electrons ranging from zero (n = 0)t o n = 1 where 1 is the aneular momentum of the orbital under consideration. In the case of dn ( 1 = 2), all we need to know are the partial terms for do..d'. .and d2in order to derive the RS terms of anv dn configuration. Let us consider the derivation of RS terms of d4configuration using the partial terms. The quintet term (S =>) is obtained directly from the table. The triplet terms arise . ~ partial terms for 4 3 ) are F and from the spin set c ~ ( 3 ) pThe P and that of j3 is D as read from the table. These are multiplied out and a redundant term subtracted as explained earlier' to leave genuine triplets. The singlets are similarly derived. This procedure is summarized in Figure 1. The equivalence relationship of RS terms by the so-called "hole formalism" can also readily be verified using the partial terms. Take the case of d6.A microstate with the highest spin multiplicity (S = 2) will have the spin set a(5)P. The component 4 5 ) has an S partial term while B has a D partial 8 set is given by the term. Hence the RS term of ~ ~ ( 5 1spin product S X D = D. Therefore the corresponding RS term of
the highest spin multiplicity is 5D. Similarly the triplet and singlet terms are generated by the operation of theappropriate partial terms and removal of redundant terms. This is summed up in Figure 2. As expected the RS terms of d6 are identical to those of d4. By this procedure, the equivalence relationship of RS terms of configurations comes out naturally without using the usual concept of "positrons" occupying the holes.
' Kiremire. E. M. R. J. Chem. Educ. 1987, 64. 951, and the references therein. a(3)= aaa.
Figure 1. The derivation of d4 RS terms using parlial terms
m e Equivalence Relatlonshlp of Partial Terms of p", d", t", and gn and the Correylondlng RS Terms ol the Highest Spln MuHlpllcllies Elemon Configurations
d"
f"
8
CwrespondingRS Terms of the Highest Spin Muiiipliclties
dl d2
dS d4 d3
S D FP
'S 2O '(FP)
10
P P
S F HFP IGFOS
'S 2F WFP) "IGFDS)
g9 g8 9' 9" g5
S G KHFP MKIHGF(2)P NLKi(2)HGIZ)FDW
'S 2G TKHFP) '[MKiHGF(?)P] "NLKl(2)W2)FO(2)S]
f'
P t9 8"
Panial Terms
so 9' g2 g3 9'
r fS
'OS sG O(KHFP) '[MKIHGF(P)P] B[NLKl(2)HG12)FD(2)S]
Volume 66 Number 6 June 1989
479
Conclusion Hole formalism hased on a half-filled shell gives the equivalence relationship of partial terms. Just as log tables are used in arithmetical operations, the partial terms can he used to derive RS terms of a given electron configuration. The partial terms may also he used to verify the equivalence relationship of RS terms according to the hole formalism hased on a filled shell. The selective derivation of RS terms of the highest spin multiplicities is very useful for dnconfigurations in the interpretation of electronic spectra of octahedral weak ligand field complexes.
Figwe 2. The derlvatim of dBRSVmns using partial terms.
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Journal of Chemical Education
Acknowledgment I would like to thank Miss M. Musopelo who contributed her valuable time to type the manuscript and my lecturers, namely John Thorley and Frank Bottomley, who introduced me to the concepts of spectroscopy and group theory.
