56
E. KBnig and S.Kremer
ring proton assignments are only a consequence of the results of the calculations and have no other support whatsoever. However, it is worthwhile mentioning that the values of hfs coupling constants for position 6 in o-chloronitrobenzene and 0-nitrotoluene would suggest an inversion of assignments between the 6 and a positions for the corresponding nitrostyrene. In the case of the para isomer a comparison with hfs values for trans-P-Br-p-nitrostyrenel is in favor of the prediction based on the R method of calculation. The Q& values calculated on the basis of these assignments (see Table VI) are in the range of the values found for other molecules.31 In Figure 8 INDO hfs coupling constants for the three isomers, as function of the conformation, are shown. One can see that the variations of ring proton constants are very small in all cases, as expected. The meta isomer is the least sensitive to changes in conformation and the para isomer is the most sensitive one. However, the a proton in the ortho isomer shows the biggest variation, as can be expected, since both through space and through bond effects are large in this case. These results suggest that an investigation of temperature influence on coupling constants might be helpful for the assignments of vinyl proton hfs coupling constants, since temperature changes might affect to a different extent the constants for protons in different positions. Of course, only a deuteration technique can give a conclusive answer. Acknowledgments. The authors are indebted to Dr. W-. J. Hehre, who made a copy of the Program GAUSSIAN 70 available, before submitting it to QCPE. They also thank Dr. V. Malatesta for a sample of o-nitrostyrene.
(2) A. Gamba, G. Morosi, V. Maiatesta, C. Oliva, and M. Simonetta, J. Phys. Chem., 77,2744 (1973). (3) (a) A. Burawoy, J. P. Critchley. Tetrahedron, 5, 340 (1959):(b) J. P. Durand, M. Davidson, M. Hellin, and F. Coussemant, Bull. Chem. SOC.Chim. Fr., 43 (1966). (4) C. Wailing and K. B. Woifstirn, J. Amer. Chem. SOC., 69, 852
(1947). (5) R. H. Wiiey and N. R . Smith, "Organic Syntheses," Coll. Vol. iV, Wiiey, New York, N. Y., 1963,p 731. (6) R . H. Wiley and R. N. Smith, J. Amer. Chem. SOC.,72, 5198 (1950). (7) D. H. Geske and A. H. Maki, J. Amer. Chem. SOC.,82, 2671 (1960);83,1852 (1961). (8) E. R. Talaty and G. A. Russell, J. Amer. Chem. SOC., 87, 4867 11965). (9) I. Bernal and G. K. Fraenkel, J. Amer. Chem. SOC.,86, 1671 (1964). >
(IO) S. F. Nelsen, B. M. Trost, and D. H. Evans, J. Amer. Chem. SOC., 89,3034 (1967). (11) M. Ciaii, Thesis. University of Milan, 1972. (12) S. Wawzonek and M. E. Runner, J. Electrochem. Soc., 99, 457 (1952). (13) D. H. Levy and R. J. Myers, J. Chem. Phys., 41,1062(1964). (14) E. A. Guggenhelrn, Trans. Faraday. SOC., 45,714 (1949). (15) J. W. Smith, Electric Dipole Moments." Butterworths, London, 1955. (16) (a) J. A. Pople, D. L. Beveridge, and P. A. Dobosh, J. Chem. Pbys., 47, 2026,(1967);(b) J . A, Pople, D. L. Beveridge, and P. A. Dobosh, J. Amer. Chem. SOC., 90,4201 (1968). (17) R. Pariser, J. Chem. Phys., 24,250 (1956). (18)J. Del Bene and H. H. Jaffe, J. Chem. Phys., 48,1807 (1968). (19) P. H. Rieger and G. K. Fraenkel, J. Chem. Phys., 39,609 (1963). (20) R. L. Ellis, G. Kuehnlenz, and H. H. Jaffe, Theor. Chim. Acta, 28, 131 (1972). (21) N. Mataga and K. Nishimoto, 2. Phys. Chem. (Frankfurt am Main), 13,140 (1957). (22) C. C. J. Roothaan, Rev. Mod. Phys., 32,179 (1960). (23) R. Zahradnik and P . &sky, J. Phys. Chem., 74, 1235 (1970). (24) W. J. Hehre, R. F. Stewart, and J. A . Pople, J, Chem. Phys., 51, 2657 (1969). (25) E. G. McRae, J. Phys. Chem., 61,562(1957). (26) H. C. Longuet-Higgins and J. A. Pople, J. Chem. Phys., 27, 192 (19571. (27) W l W: Robertson, A. D. King, and 0. E. Weigang, J. Chem. Phys., 35,464(1961). (28) T. Kubota and M. Yamakawa, Bull. Chem. SOC.Jap., 35, 555 (1962);40,1600 (1967);41,1046 (1968). (29) J. Heinzer, "Least-Squares Fitting of isotropic Multiline Esr Spectra," Quantum Chemistry Program Exchange (QCPE) No. 197,Indiana University, Bioomington, Ind.
References and Notes (1) A. Gamba, V. Malatesta, G. Morosi, and M. Simonetta, J. Phys. Chem., 76,3960 (1972).
(30) T. Fujinaga, Y. Deguchi, and K. Umemoto, Bull. Chem. SOC.Jap., 37,822 (1964). (31) A. Carrington and A. D. McLachian, "introduction to Magnetic Resonance," Int. Ed., Harper, New York, N. Y., 1967,p 83.
*
Octahedral d4, d6 Ligand Field Spin-Orbit Energy Level Diagrams E. Konig* and S. Kremer institute of Physical Chemistry il,University of Erlangen-Numberg, 0-8520Erlangen, West Germany (Received March 12, 1973; Revised Manuscript Received August 27, 1973)
The complete ligand field Coulomb-repulsion spin-orbit interaction matrices have been computed for the d4, d6 electron configurations in a field of octahedral symmetry. Correct energy level diagrams are presented.
Introduction Complete octahedral ligand field spin-orbit interaction matrices for the d4, ds electron configurations have been reported both by Schroederl in the strong-field coupling The Journal of Pbysical Chemistry, Vol. 78, No. 1, 1974
scheme and by Dunn and Li2 in the weak-field coupling ~ c h e m e . 3 -In ~ the configurations d2, d8 and d3, d7, "complete" energy level diagrams have been c o n s t r ~ c t e don~ ~ ~ the basis of corresponding matrices and these have been
57
Ligand F i d d Spin-Orbit Energy Level Diagrams
1QOOC-
d‘-octohedral. 6: BOOm-’,
C: L 6.5: 29OVn-‘
Energy level diagram for the d 4 configuration in an octahedral field including spin-orbit coupling B = 800 cm-’, C = 48, { = 290 cm-’. The designation of the terms by the strong-fieid configurational labels and their parent terms corresponds to the largest contribution which is present at a D q value assumed at the right end of the diagram. The labels havsto be supplemented with g subscripts for o h symmetry. Figure 1.
useful in the past to discuss the effect of the interactions and to make certain Dredictions of spectral transition energies. No such efforts seem to have been performed in the d4. d6 problem. In the present study, we have rederived all d< d6 matrix elements in t h e ‘strong-field and weak-field formalisms and, in view of their present and
future utility, we present below complete energy level diagrams for these systems in octahedral symmetry.
Methods of Calculation and Results Racah’s method of irreducible tensor operator^^,^ extended to the coverage of problems in subgroups of SO(3) The Journal of Physical Chemistry, Voi. 78, No. 1, 1974
58
E. K6nig and S. Kremer
500
1000
___-___
1500
Dq lCm-11 &-cchhedrol:B:B06cm-'
C = L B !.:L200n-'
Figure 2. Energy level diagram for the d6 configuration in an octahedral field including spin-orbit coupling: 8 = 806 c m - ' , 420 c m -
l.
Refer to caption of Figure 1 for further details.
The Journal of Physical Chemistry, Vol. 78,
No. 1, 1974
c = 48, =
Ligand Field Spin-Orbit Energy
Level
59
Diagrams
has been applied throughout. Thus, in the weak-field coupling scheme, matrix elements of the spin-orbit interaction may be written as
( v S L D y a 1 2 4 ( r , ) s ,I,. 1 v’S’L’JT’y’a’)
= (-l)J -+
+
s’(30)1/2
1
In eq 1, the expression in braces is a standard 6j symbol, (uSLIIV(lll Ilu’S’L’) is a reduced matrix element of the double tensor operator Vill) of Racah, and {nd is the spinorbit coupling parameter. Matrix elements of the octahedral ligand field potential may be expressed according to
ployed in the strong-field coupling scheme. Both treatments produce, upon numerical substitution, identical results, and these agree with results which were computed on the basis of the published (but corrected) m a t r i ~ e s . l - ~ In Figures 1 and 2 we present the complete energy level diagrams for the d4 and de electron configurations, respectively, subject to a ligand field of octahedral symmetry. The parameter values chosen for B, C, and are approximately suitable for hexahydrated complex ions.ll Since the energies are plotted us. Dq, predictions for complex ions with ligands different from HzO are possible in most cases.12 Hopefully, these diagrams will be useful for the spectroscopist in the same way as the corresponding diagrams in the more common d electron configurations.6.7
Acknowledgments. The authors appreciate financial support by the Deutsche Forschungsgemeinschaft, the Fonds der Chemischen Industrie, and the Stiftung Volkswagenwerk. References and Notes
m,rmr,rws,s w q
(2)
-
In eq 2, (JM1Jr-p) are expansion coefficients defined by Z a r , y being a component of reprethe reduction DJ sentation r and a the multiplicity index.lO In addition, (uSLIIUi4)IIv’S’L’) is a reduced matrix element of the unit tensor operator U(4) of Racah and Dq is the octahedral splitting parameter of the ligand field. Finally, matrix elements of the Coulomb interelectronic repulsion are determined by
w, Jwr,r w y , m
a , a’) (3) where the reduced matrix elements (vSL1 Zi>,(e2/ rL,)llu’S’L’)may be expressed in terms of the (lllC(k)lll) and the Fk(nl,nl). Here, C [ k )are the rationalized spherical harmonic tensor operators and the Fk may be converted, for d electrons, into the Racah parameters A, B, C. The reduced matrix elements appearing in eq 1-3 may be further reduced to coefficients of fractional parentage and other easily calculable numerical quantities like 3j and 6j symbols. Therefore, and in distinction to almost all previous treatments, all matrix elements have been completely computer generated. This procedure provides the
high degree o f reliability hitherto not achieved in ligandfield calculations. An analogous approach has been em-
K. A. Schroeder, J. Chem. fhys., 37,2553 (1962). T. M. Dunn and Wai-Kee Li, J. Chem. fhys., 47, 3783 (1967). Observe that there are numerous errors in the original list of weakfield matrix e i e m e n t ~ .An ~ , ~additional error has been discovered in Table iV of ref 2. It should read
I
(d4,4’ /e I H‘ d4,z1G4) = 20 (22) l Z / l1 Dq E. R. Krausz, J. Chem. fhys., 53,2131 (1970). T. M. Dunn and Wai-Kee Li, J. Chem. Phys., 53,2132 (1970). A. D. Liehr and C. J. Baiihausen, Ann. Phys., 6, 134 (1959). A. D. Liehr, J. Phys. Chem., 67, 1314 (1963). B. R. Judd, “Operator Techniques in Atomic Spectroscopy,” McGraw-Hill, New York, N. Y . , 1963. B. G. Wybourne, “Spectroscopic Properties of Rare Earths,” interscience, New York, N. Y., 1965. E. Konig and S. Kremer, Theor. Chim. Acta, in press. A least2squares fit of the emission spectra of Cr2+ and Mn3+ ions gave B = 699 c m - l , C = 3117 c m - l , and B = 1082 c m - l , C = 3916 c m - l , respectively. The nephelauxetic ratio in M(HzO)Bn+ ions is fl 0.88 if n = 2 and p 0.75 if n = 3. The splitting of the free ion 5D ground term yields [ = 228 c m - l for Cr2+ and { = 348 c m - 1 for Mn3+. The values chosen are approximate averages of these data. Similarly, a least-squares fit of the emission spectrum of the Fez+ ion produces B = 916 cm-’, C = 3867 cm-’. There are no data available for the ion Co3+. The splitting of the 5D ground term of Fez+ gives [ = 416 c m - l . The parameter values were chosen as above. The designation of terms in Figures 1 and 2 does by no means imply pure configurational states. Indeed, some mixing by CI and spin-orbit interaction is often present and the notation corresponds For the purpose of to the largest contribution at Dq = 3000 cm illustration, we give below the percentage content of that eigenfunction which corresponds to the term designation given for the octahedral d6 configuration in Fi ure 2. 45 f 2.5%: 2r#TZ(tz4( 3 T l ) e 2 ( 3 A ~ ) ] ] , r z L ’ E ( t ~ ~ ( * E ))I,.. e r l [ ’ A 1 ( ~ Z ~ - ( ~ A ~ :) ~ ~ ( ~ A ~ ) ) 50 f 2.5%:r 2 , T5,r4[3Tl(t24(3Tl)e2(1E))],~ z [ ’ E ( t z 4 ( ’ A l l e Z ( ’ E ) ) ~ , rz[1E(tz3(2E)e3)]; 55 4z 2.5%: ~ , ~ T l ( t z 4 ( J T l ) e z ( 1 E ) 60 ) ] ; f 2.5%: r4, r5, r ~ [ ~ T i ( t z ~ ( ’ T z (3Az))1, )e rz[lE(t2 (’E)e2(’E))I, ~ 5 [ 1 T z ( t z 3 ( 2 T z ) e65 3 ) ]i ; 2.5%:r5[lT?(t2 (1TZ)e2(1A,))].The listing is always in the order occurring from lower to higher energies, No listing implies 70% or greater content.
-
-
-’.
3
rz.
The Journal of Physical Chemistry, Vol. 78, No. 7, 1974
