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2 1350 CAN. J. CHEM. VOL makes it a good candidate for theoretical study and the extensive number of nmr studies of it and its derivatives gives us something with which to compare our calculations. What is the ground state of uranocene? U(IV) has anf'configuration with the lowest state being 'HA. The point symmetry group of U(C,H,)> is either D,,, or D,,, and since it will make no difference in the calculations we will assume the former for convenience. The eight-fold symmetry results in a crystal field with no spherical harmonics with M > 0 and, therefore, the 'HA state is split by the crystal field into five states which are: IJ,MJ) = 14,+4); 14,?3); 14,+2); 14,k I); (4,0) There is no agreement in the literature as to which of these five states is the ground state. The 14,?4), 14,+3), and 14,O) states have all been suggested as the ground state in uranocene. Karracker (13) predicted (4,+4) ground state based on theoretical considerations. This was confirmed by Hayes and Edelstein (14) in their analysis of the magnetic susceptibility of powders of uranocene, neptunocene, and plutonocene. Amber- ger et al. (15) fitted their susceptibility data to a model in which the ground state was 14,O) with the 14,? 1) state only 17 cm-i above it. Dallinger et al. ( 16) found with Raman studies that the first excited state was at 466 cm-' and differed from the ground state by AMJ =? I. They proposed a ground state of 14,+4) with the first excited state being 14,k3) by fitting the theoretical susceptibility against the experimental susceptibility versus temperature plot. They obtained a slightly poorer fit when the two states were reversed. Fischer (1 1) has estimated (pii2 - p12) to be 8.8 Bohr magnetons from pseudo-contact shift studies and claims this requires a 14,-+3) ground state with the first excited state being /4,+2). Luke and Streitweiser (12) have also estimated the pseudo-contact shift and obtain a value of (p'- p12) = 12.5 Bohr magnetons which we estimate would best fit a ground state of 14,?3) with the first excited state being 14,?4). Hayes and Edelstein (14) also predicted a )4,+3) ground state on the basis of Wolfsberg-Helmholtz MO computations. The present evidence suggests three possibilities: 14,?4) is ground state with 14,+3) at 466 cm-' or 14,?3) is ground state with either 14,?4) or 14,?2) at 466 cm-'. We will do our calculations for all three possibilities. Molecular orbitals of U(C8H8), For D,,, the MO's for the p, orbitals of COT (cyclooctatetraene) will be given using the numbering system for individual carbon atoms as shown below. The lower ring has atom number 9 under 1, 10 under 2, etc. Since the f orbitals belong to the representations, we give in eq. [I] only the n. MO'S belonging to those representations. -1 X El,, Ezu, and E,, irreducible 'The appropriate atomic orbitals for the 14,4), 14,3), and 14,2) states of the f' configuration are given in eq. [2] below.

3 +L - McGARVEY The f orbitals are combinations of real functions that match the above IT orbitals in symmetry. We therefore combine the IT orbitals in an analogous fashion to give the complex functions listed in eq. [3] below. C3' fo = lz(5z2-3r2)) $0 = +(Azu) 1 - I f- - -[+lx(5z2 - r2)) - ily(5z2 - r2))] ++I = --{~+(E~,)[x(5z'- r')] - i+(e1,,)[y(sz2 - r2)]) + I - v5 1 ff-2 = --[lz(x2 - y',)? ilxyz)] v5 I - f-. - [ +lx(x2-3 )- ily(3x2 - y2))l We now combine the metal f orbital and the IT ligand orbitals to get the complex MO's listed in eq v5 v5 I - v5 +' --{+(E2,)[z(x2 - y')l L i+(e2,0[xyzl) 4 = { E 3 [ x x i$(e3,)[y(3x2 - y2)]} v5 We now replace the f orbitals in eq. [2] with the F functions of eq. [4]. It is these functions that will be used to calculate the contribution of covalent bonding between f orbitals and the IT orbitfls of COT to the paramagnetic shift. In practice we will calculate the shift for atom 1 and will need to worry only about the pi, pi, and p; terms in the above MO's. Calculation of paramagnetic shift for I3C The calculation of the paramagnetic shift is a straightforward but tedious procedure using the theory of Kurland and McGarvey (17). If there is only one excited state to include in the calculation, the equation for the shift takes on the following form: where we have assumed the ground state to be 14,*3) and the excited state to be 14,t4), A is the energy difference between the two states, and the other symbols have their usual meanings. To obtain the result when 14,t4) is the ground state and 14,t3) the excited state, we simply make A negative in value and to obtain the shift when the excited state is 14,+2), we simply replace 14,f 4) by (4, t2) in the above equation. The matrix elements of eq. [5] are given in the Appendix along with the steps needed in their computation. The calculation was done for carbon nucleus I and contributions from unpaired spin on neighboring atoms 2 and 8 have been included. The A?,, term comes from pi and the R-' term comes from pf and p;. The Fermi contact term Q has contributions from all three atoms and is estimated by standard methods (18). Using the matrix elements in the Appendix and eq. 1.51, the shift is calculated for the x, y, and z principal directions and the results are given in eq. [6]. Please note that the shift for both x and must be calculated because the local symmetry around the carbon nucleus is not axial, even though the molecule itself has axial symmetry. The equations given in eq. [6] are for the case when the ground state is 14,t3) and the nearest excited state is 14,t4). AH [61 (K),,?,,,( 1 - e-aikt + C,, ,626: + 46: + 586: + 8b:] + D,,,[--- + ) A 2P ~CPC Q A I,, -- = 6600A(l + e-""'){[ pnpn R' RNPN RNPN ~NPN The average value for the shift is given in eq. [7]. The corresponding equation when the excited state is 14,t2) is as follows:

4 1352 CAN. 1. CHEM. VOL Please note that because the orbital momentum and spin-orbit coupling are not quenched, the dipolar terms of A2,, and R-"re not averaged out and therefore make a major contribution to what many people refer to as the "contact shift". Calculation of paramagnetic shift for 'H The 'H shift resulting from spin in thep: orbital of the neighboring carbon atom is readily calculated from the present equations. We simply drop out the A?,, and R-' terms and replace Q/gNpN by (Q/gN~N- R,..L~R-~) in the z shift equation, by (Q/gNkN + 2gC4R-') in thex term, and by (Q/gNFN- gcp,,r-') in they term. The Q now is the appropriate hyperfine term for 'H in aromatic radicals and the R-"erm is the dipolar contribution from the spin in the carbonp, orbital which does not average out to zero when the angular momentum is not quenched. The resulting shift equation for the case of a 14,?3) ground state with the first excited state of 14,?4) is as follows: and for the excited state of 14,?2) the equation becomes Comparison with experiment We can now attempt to calculate numerical values to compare with experimental values. The following values were used for the parameters in the equations: A2,('.'C) = 90.8 MHz Q('") = MHz gcp,cgnp,nr-3('") = 7.39 MHz Q(lH) = MHz To obtain estimates of the magnitude and sign of the shifts predicted by these equations, we need to make assumptions concerning the values of the four b parameters. One reasonable assumption would be to assume that only the topmost filled n orbital in the COT ion would be involved in covalent bonding with the f orbitals. This would assume that 6, = 6, = b, = 0, since the E,, is the topmost filled level. Another possible assumption is to make all n orbitals equally involved in bonding, which would assume that b = 6, = b, = bz = b3. The results of making these two assumptions for three different arrangements of states are given in Table 1. The values in Table 1 should be compared to what most experimentalists have called the contact shift, that is, the isotropic shift from which the pseudo-contact shift has been subtracted. Fischer (1 1) reports the following values for these shifts: (AH/H),,,("C) = ppm (AH/H),,,('H) = 38.1 ppm Luke and Streitweiser (12) report for 'H a value of 34.2 ppm. A comparison of the experimental values with the table entries quickly reveals that the theory will not account for the experi- ~ental results, since it would require in most cases that b' or b; be negative or, in one case where the signs agree, it would require an unreasonably large value of b;. We must therefore conclude that the major contribution to the contact shift in uranocene comes from spin transfer mechanisms other than the one investigated here. It must not be concluded from the above arguments that f orbital covalency is unimportant in uranocene, because in the most likely cases examined, that is for a 14,*3) ground state with only 6, different from zero, the predicted contribution to the shift is quite small for reasonable values of bf and would be difficult to detect. Thus we have shown the existence of a sizeable contact shift in uranocene and its derivatives is not an indication of covalency, at least in the f orbitals, and any changes in the shift do not necessarily indicate changes in covalency. Polarization mechanism of spin transfer The polarization mechanism of spin transfer is more difficult to handle in a theoretical or semi-theoretical manner. In principle, exchange interactions between the unpaired spins in the f orbitals induce a small uncoupling (or polarization) of paired spins in filled orbitals of the system. This is in general small,

McGARVEY TABLE I. Paramagnetic shifts (pprn) Ground state 14,+3) 14,*3) 14-24) Excited state )4,+4) 14,*2) 14-23) b,, = b, b, = 0 (AH/H)avc('3C) 67.5b; (AHIH 'HI - 16.

5 McGARVEY TABLE I. Paramagnetic shifts (pprn) Ground state 14,+3) 14,*3) 14-24) Excited state )4,+4) 14,*2) 14-23) b,, = b, b, = 0 (AH/H)avc('3C) 67.5b; (AHIH 'HI bf b = b,,= b, = b2 (AH/H)2,vc('3C) 1210b2 (AH/H)a,c('H) b2 but if the filled orbital involves an s orbital of the atom in question, the Fermi contact term will be large enough to produce a measureable effect. Quite a few filled orbitals could make a contribution to this effect and this makes interpretation of such shifts very difficult. A simple model was used to interpret and predict the polarization component of the nmr shift in lanthanide fluorides (2-9) which can be applied to uranocene. 'This will at least allow us to compare the magnitude of the effect between the lanthanides and the actinides. In this model, it was assumed that the matrix elements of A,; were proportional to the matrix elements of S,. That is I Using this formalism and assuming a ground state of 14.+3) and first excited state of 14,+4), we obtain for the average shift l If we assume Kli = K, = K and if we assume that the entire "C shift in uranocene is due to the polarization mechanism, we calculate a K value of MHz from eq. [12]. This is, surprisingly, close to the value estimated by Edelstein er a/. (19) who assumed a 14,?4) ground state with no contributing excited states. K,,, = MHz for "F-Gd" and MHz for "F-Eu'~. Since K is a hyperfine constant and is proportional to g,, we must scale these numbers by the ratio of gn's before comparing them. K = MHz would become MHz for a "F nucleus. Thus the polarization of spin in the carbon atom adjacent to the U" is only slightly larger than that of the fluorine ion adjacent to Gd" or Eu". Can we determine which filled orbitals in the molecule make the major contributions to spin transfer by polarization? Streitweiser et al. (20) have proposed that it is the filled bonding orbitals which involve f and d orbitals of uranium combined with TT orbitals of the COT anion. We consider this to be unlikely for two reasons. First, these orbitals will have only a small admixture of the f and d orbitals so that the exchange integrals with the f electrons will be small (and therefore the spin polarization will be small) and this coupled with the fact that these MO's do not have any direct admixture of carbon s orbitals makes it highly unlikely that polarization of these orbitals can account for the measured nmr shift. The magnitude of K estimated earlier would require a polarization of the spin of better than 10% in the TT orbitals if this were the only source of the nmr shift. Second, these orbitals would contribute only by a double polarization which would yield a shift of opposite sign to that observed. A more likely source is the polarization of the outer 6s and 6p electrons in the uranium. Overlap of these electrons with the u bonding electrons of the COT anion will result in a renormalization of the two sets of orbitals which will (lo), in turn, result in a polarization of the 2s and 2p orbitals of the carbon atom. If this is the case, we would expect the nmr shift to be related more to overlap of the outer electrons in the metal and ligand atoms (which is a function of the bond distance) than to the nature of the bonding interactions. If this is the case, we would expect the nmr shift of "C to be similar for any carbon attached directly to the uranium atom. This appears to be borne out in the limited data available on "C shifts. In eq. [12], K is divided by g, so that we can infer from the fact that the magnitude of the 'H shift in uranocene is less than one third that of the "C shift that the polarization of the spin near the 'H nucleus is less than one third of that near the "C nucleus. This is reasonable since the polarization must be transmitted through the C-H bond which also causes the reversal in sign of the polarization. Again we would expect the 'H shift to be influenced only slightly by bonding changes, if the spin polarization is transmitted primarily by overlap between the outer electrons of the uranium and carbon atoms. It has been commented on several times (1 1, 12) that the "contact" shift in (CsHs),UX compounds varies only slightly with X. Further, the "contact" shift in U(C,Hx)2 is not much different than that observed in U(C5Hs),. Conclusions There are several conclusions to be drawn from this work. The theoretical approach used in the work has been successfully applied to the lanthanide system, where it showed that covalency involving the f orbitals played a major role in determining the paramagnetic shift for the ions in the second half of the series. Thus we have every expectation that the present calculations should give a good estimate of the sign and magnitude of the shift from f orbital covalency in uranocene. The failure of the calculations to explain the sign and.magnitude of the observed shifts must, therefore, indicate that either there is little covalency or (more likely) the ground state is 14,+3) with covalency involving primarily the El, TT MO of COT, since this was shown to lead to very small paramagnetic shifts. The paramagnetic shift must be determined mostly by the polarization mechanism of spin transfer. It appears that this mechanism operates mainly through the overlap of the outer 6s and 6p orbitals of uranium with the u bonding orbitals of the carbon atom. Therefore, the nmr shift in these complexes is not very sensitive to the nature of the bonding between the ligand and the uranium atom. Another important, but less obvious, conclusion is the importance of terms other than the Fermi contact term in determining the sign and magnitude of the isotropic shift in actinide complexes. The A2, term was larger than the Q term in the "C

6 1354 CAN. J. CHEM. \ computations and the R-.' term was just as large as the Q term in the 'H case. Thus these calculations point out that it is unreasonable to characterize the isotropic shift as a contact shift in the actinide systems and even more unreasonable to consider the shift as a measure of the esr Q parameter for unpaired electrons in the IT system. Acknowledgement This work was supported by an operating grant from the Natural Sciences and Engineering Research Council of Canada. MM' A(MM1) a(mm'0) a(mmii) n(mm'2) n(mm'3) It is quickly established that all the (+,,,IA;'I+,,,) are identical, so that calculation of one will suffice. Since our nucleus of interest is carbon nucleus 1, we assume only it and its nearest neighbours, nuclei 2 and 8, will make a contribution. Thus we can write Appendix Calculation of the (4Mlp;/4M1) matrix elements in eq. [5] is straightforward. We take g, = 415 and Ji = Li + Si. If we further assume that only the central orbital part of the MO contributes to L;, we obtain the matrix elements The final values in the above equations are obtained by assuming N, = N2 = N, = N, - 1, which should be a reasonable assumption for f orbitals. The calculation of the ( ~ M I A ~, I ~ matrix ~ ~ ~ elements ~ ~ M ' ) is more laborious. We first consider the AN; terms. Since these terms are centered on the ligand nucleus, we consider only the +,, terms in the MO's. Since these have only p, orbitals, we can ignore the orbital momentum part of AN; in eq In fact, the only portion of the AN, operator that makes a contribution is the sr terms, so that we write The dipolar (second term in eq. [A2]) term in A;' gives contributions froni all three nuclei. For nucleus 1 we obtain For nuclei 2 and 8 we use the point dipole form of A,', which is gcpc(3 cos2 R - I)R-'. R is the distance between nucleus 1 and nucleus 2 or 8 and R is the angle between this direction vector and the z axis. Since R = 90, we get The Fermi contact term (first term in eq. [A2]) would give zero for the MO's in eq. [I]. However, we know from free radical studies of such aromatic rings that polarization of filled bonding a orbitals produces a hyperfine interaction proportional to the density of spin in the p; orbital on the nucleus in question plus the density in the nearest neighbours. Since the hyperfine interaction of a spin in +,,, is calculated for the z direction by (+,,,IAZ1I+,,,) and the fraction of spin on any nucleus is 16-' for each +,,, orbital, we can write for eq. [A41 the following: where Q/16 is the Fermi contact part of the hyperfine interaction constant for a free radical with unpaired spin in a +,,, orbital. Q can be estimated by procedures given in ref. 18. A similar procedure is adopted for the AN.,/gNpN and ANy/gNpN terms. The general form for the matrix elements is the coefficients in eq. [A81 are as follows: The ( 4 ~ N;/I:NpN14M') 1 ~ matrix elements then take the general form [A31 (~M]AN~/~N~N~~M') = A(MMr) a(mm1m)b,?,, 3 r,,=o x (+,,,IA:'l+,,,) where the coefficients for appropriate MM' values are as follows: MM' B(MM1) b(mmro) b(mmii) b(mm'2) b(mm'3) 43 fi/ fi/

7 McGARVEY 1355 In an analogous fashion to that used for the AN;/gNpN terms we find The cross terms are: These cross terms disappear when we compute the average nmr shift. 1. B. R. MCGARVEY. J. Chem. Phys. 65, 955 (1976); 65, 962 (1976). 2. M. R. MUSTAFA, B. R. MCGARVEY, and E. BANKS. J. Magn. Reson. 25, 341 (1977). 3. A. REUvEN1 and B. R. MCGARVEY. J. Magn. Reson. 29, 21 (1978). 4. R. J. BOOTH, M. R. MUSTAFA, and B. R. MCGARVEY. Phys. Rev (1978). 5. A. REUVENI and B. R. MCGARVEY. J. Magn. Reson. 34, 181 (1979). 6. B. R. MCGARVEY. J. Chem. Phys (1979). 7. A. REUVENI and B. R. MCGARVEY. J. Magn. Reson. 36, 7 ( 1979). 8. R. J. BOOTH and B. R. MCGARVEY. Phys. Rev. 821, 1627 (1980). 9. L. C. STUBBS and B. R. MCGARVEY. J. Magn. Reson. 50, 249 (1982). 10. R. E. WATSON and A. J. FREEMAN. Phys. Rev. 156, 251 (1967). I I. R. D. FISCHER. In Organometallics of the,f-elements. Edited by T. D. Marks and R. D. Fischer. D, Reidell Pub. Co., Boston, MA p W. D. LUKE and A. STREITWEISER, JR. In Lanthanide and actinide chemistry and spectroscopy. Am. Chem. Soc. Symp. Ser Edited by N. M. Edelstein. Am. Chem. Soc. Washington, D.C p D. G. KARRACKER. Inorg. Chem. 12, 1105 (1973). 14. R. G. HAYES and N. EDELSTEIN. J. Am. Chem. Soc. 94, 8688 ( 1 972). 15. H. D. AMBERGER, R. D. FISCHER, and B. KANELLAKOPULUS. Theoret. Chim. Acta, 37, 105 (1975). 16. R. F. DALLINGER, P. STEIN, and T. G. SPIRO. J. Am. Chem. Soc. 100, 7865 (1978). 17. R. J. KURLAND and B. R. MCGARVEY. J. Magn. Reson. 2,286 (1970). 18. J. E. WERTZ and J. R. BOLTON. Electron spin resonance elementary theory and practical applications. McGraw-Hill, Inc. New York N. EDELSTEIN, G. N. LA MAR, F. MARES, and A. STREITWEISER, JR. Chem. Phys. Lett. 8, 399 (1971). 20. A. STREITWEISER, JR., D. DEMPF, G. N. LA MAR, D. G. KAR- RAKER, and N. EDELSTEIN. J. Am. Chem. Soc. 93, 7343 (1971).

Electronic Spectra of Complexes

Electronic Spectra of Complexes Interpret electronic spectra of coordination compounds Correlate with bonding Orbital filling and electronic transitions Electron-electron repulsion Application of MO theory