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Ligand Field Theory of Metal Sandwich Complexes The parameter P " / s through its significance as intermolecular energy/molecular surface may be related to the critical data and hence to e and 4. We have
H ~ / x , x=~ (2(1
-
f)
+
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- e @/3 + (P2/9)(-
u + TC,) (10)
Here third powers of e and 4 and also products of 0 and 4 with 1 - $. have been neglected. Also, Rowlinsonll has pointed out that a similar expression may be obtained from the solubility parameter theory. For HE
HE1x,xz = (e2/4 - e @/z + cp 2/4)(- u +. TC,) (11) Equations 10 and 11 are similar to eq 6a except for the in-
clusion of a term in @. It is of interest that the Monte Carlo calculationssa agree with the presence of a small (62 ferm. However, eq 10 and 11 gave poor correlations of the data in Table I. The van der Waals result, Le., the simple eq 6b seems more promising but further tests are desirable, particularly with mixtures containing a homologous series, e.g., cycloalkanes OMCTS.
+
Acknowledgment. We gratefully acknowledge the support of the National Research Council of Canada, (11)J. S. Rowlinson, "Liquids and Liquid Mixtures," 2nd ed. Butterworths, London, 1969,p 339.
Ligand Field Theory of Metal Sandwich Complexes. Axial Field Spin-Orbit Perturbation Calculations for d1(d9), d2(d8), and d3(d7) Configurations Keith
D. Warren
Department of Chemistry, University College, Cardiff, Wales, United Kingdom
(Received October 12, 7972)
Complete ligand field perturbation calculations, including spin-orbit coupling, have been carried out for dl(dQ),d2(d8), and d3(d7) configurations in axial, Cm*, symmetry, using the strong field formalism. The application of the results to the interpretation of the d-d electronic spectra of metal sandwich complexes is discussed, with particular reference to the metallocenes of vanadium, cobalt, and nickel.
I. Introduction The utility of the ligand field model for the assignment of the low-energy d-d transitions of metal sandwich complexes is now generally accepted,l and the approach has been used with some success for a number of the metallocenes. Thus, following the initial use of the method by Scott and Becker2 for ferrocene, the spectra of both vanadocene and nickelocene were similarly interpreted by Prins and van Voorst.3 More recently Sohn, Hendrickson, and Gray4 have treated the spectra of ferrocene and ruthenocene, and of the ferrocinium and cobaltocinium ions on a ligand field basis, while nickelocene has been further treated by Scott and Matsen5 and by Pavlik, Cerny, and Maxova.6 It was shown by Scott and Matsen5 that metallocenes, bisarene metal compounds, and a number of related complexes, may all be treated on the basis of a n effective axial (Cmu) symmetry, as long as only dn configurations are involved, and currently full perturbation treatments, excluding spin-orbit effects, are available for dZ(d8) configurations in both the weak and the strong field schemes.s,6 Similar partial treatments for the d3 configurations have been given by Prins and van Voorst3 in the strong field basis, and by PerumareddP and by DeKock and Gruen,s in the weak field scheme, and for the d6 and d5 systems of ferrocene and the ferrocinium cation partial strong field matrices have also been calculated.4 However, only for
d2(ds) systems is an axial field calculation including spinorbit coupling availab1e;g this though is expressed in the weak field basis which is not very suitable for metallocenes and related species since the low nephelauxetic ratios found (@ 0.5) indicate that the eigenstates correspond to fairly well-defined strong field configurations. Thus, in order to develop further the ligand field model for metal sandwich complexes the complete axial field energy matrices, including spin-orbit coupling, have been calculated in the strong field scheme for dl, d2, and d3 (d9, d8, and d7) configurations. Using these results the available data for the d-d spectra of the corresponding metallocenes have been analyzed, and general predictions made in those cases for which experimental spectra are lacking. In particular the present results permit a more detailed interpretation of the spin-forbidden transitions recently reported for nickelocene. In addition, relatively
-
C. J. Ballhausen and H. B. Gray, "Coordination Chemistry," Vol. I. ACS Monograph No. 168,American Chemical Society Publications, Washington, D.C., 1971. D. R. Scott and R. S. Becker, J. Organometai. Chem., 4, 409 (1965). k. Prins and J. D. W. van Voorst, J. Chem. Phys., 49,4665 (1968). Y. S. Sohn. D. N. Hendrickson, and H. B. Gray, J. Amer. Chem. SOC.,93,3603 (1971). D. R. Scott and F. A. Matsen, J. Phys. Chem., 72,16 (1968). I. Pavlik, V. Cerny, and E. Maxova, Collect. Czech. Chem. Commun., 35,3045 (1970). J. R. Perumareddi, J. Phys. Chem., 71,3144 (1971). C. W. DeKock and D. M. Gruen, J. Chem. Phys., 46,1096 (1967). C.W. DeKock and D. M. Gruen, J. Chem. Phys., 44,4387 (1966). The Journaiof Physical Chemistry, Vol. 77, No. 13, 1973
Keith D. Warren
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little work has been done on the magnetic susceptibilities and magnetic resonance phenomena of the above systems and their treatment by a complete spin-orbit perturbation method should serve to facilitate further progress.
11. Theory and Calculations X-Ray diffraction studies have shown that the crystalline metallocenes may occur either with the staggered, conformation of the cyclopentadienyl rings, as in ferrocene and the other 3d complexes, or with the eclipsed, Dsh, arrangement found in ruthenocene and osmocene.10 For bisbenzenechromium, and probabiy for other bisbenzene complexes, a D6h point group obtains,lO but it has been shown by Scott and Matsen5 that when only dX configurations are considered all systems of Cn, C n h , C,,, Dnd, and Dnh symmetries may be treated in terms of a purely axial (C,,) ligand field, as long as n 2 5. The one-electron ligand field potential is then VLF = A1VoO A2V02 A3V04 where the A, are expansion coefficients and the V$ spherical harmonics. The magnitudes of the A, are determined by the nature and geometrical arrangement of the ligands, but may be taken up into the axial parameters, Ds and Dt, thereby giving the same potential for all the symmetries considered. In such a field the d orbitals split into three sets, ddz2:do), d d xz, d yz: d +I), and 6(dx2 - y 2 , dry:d A, with one-electron energies E ( u ) = 2Ds - 6Dt, E(*) = Ds 4 Dt, and E(6) = -2Ds - Dt, following the definitions of Piper and Carlin.11 Usually these one-electron core energies follow the order 6 < a
