Approximate Molecular Orbital Calculations onMetallo-organic ComplexesBY I. H. HILLIER* AND MRS. R. M. CANADINETReceived 7th January, 1969The results of approximate molecular orbital cdculations on the metallo-organic complexes,palladium bis x-allyl, ferrocene, and dibenzene chromium are presented. The calculations includeelectron interaction and all valence electrons are considered. The bonding involves mainly themetal orbitals and the x orbitals of the hydrocarbon ligand. The results are compared with thoseof other workers and with experimental data.The use of semi-empirical methods in the molecular orbital treatment of inorganiccomplexes has become widespread because the computation of accurate wavefunctionsfor these molecules by the Hartree-Fock self-consistent field method of Roothaanis limited by the large number of molecular integrals which are required.2 Thereis, however, a continuing search for simplsed methods of calculating reliable wave-functions, particularly for those molecules for which a crystal field model is un-satisfactory.One of the earliest such methods, introduced by Wolfsberg andHelmholz (WH),3 and subsequently modified,4 has been used to interpret the bondingin several inorganic molecules. This method, being essentially an extended Hiickelscheme, does not include electron repulsion terms explicitly, and uses empiricalparameters and equations for evaluation of the elements of the secular determinant.For these reasons, many authors have recognized that it may lead to unreliableresult^.^" In particular, the neglect of interatomic coulomb terms leads to too smalla metal-ligand charge separation.By the introduction of systematic and well-defined approximations in Roothaan’s method it is possible to develop other com-putational In this paper the self-consistent charge with electroninteraction (SCCEI) method, which has been developed in previous publications,6* lois used to describe the bonding in some organo-metallic complexes.6* *-loMETHODIn the LCAO-MO approximation, a molecular orbital +i is represented as a linearcombination of atomic orbitalswhere xku is atomic orbital k on centre u, and the coefficients Cik are determined bysolution of the secular equations, yielding the determinantwhere Siu,ju is the overlap intergral and Hi,,,v the matrix element of the FockHamiltonian.Since the approximations introduced to simplify evaluation of thelatter are discussed elsewhere,6* lo they will only be outlined here. Richardson’s* Chemistry Department, The University, Manchester 13.I- I.C.I. Limited, Petrochemical and Polymer Laboratory, The Heath, Runcorn, Cheshire.228 APPROXIMATE MOLECULAR ORBITAL CALCULATIONSseparated Hamiltonian procedure l1 is used to reduce the elements of the seculardeterminant into terms which involve atomic orbital energies and interatomic coulombterms. The former are approximated as the negative of the valency state ionizationenergies (VSIE) and the latter are simplified by using a point charge approximationfor the atomic potentials.The diagonal elements then becomewhere Eiu, the orbital energy associated with atomic orbital xiu, depends upon thecharge qu on atom u obtained via a Mulliken population ana1ysis.lelements similarly becomeThe off-diagonalT i u , j v , (4)where is the two-centre kinetic energy integral. The leading terms in boththese expressions (eqn. (3) and (4)) are similar to those used in the WH method, thedifference being the introduction of an empirical parameter F, usually having avalue near 2, in the off-diagonal terms, viz.,H i u , j v = ( F P P i u , jv(Hiu,iu + H j v , j v ) . ( 5 )The terms in eqn. (3) and (4), which are in addition to those occurring in the WHmethod may be considered as arising from interatomic coulomb or “ ligand field ”effects.The one-electron integrals in these equations are evaluated using the atomicorbitals of the basis set. The calculation is iterated until self-consistency is achieved,when the atomic charges and configurations used to evaluate the elements of thesecular determinant are virtually the same as those obtained by a Mulliken populationanalysis of the resulting molecular orbitals, the criterion being that the differencein the charges is less than 0.01.DESCRIPTION OF THE CALCULATIONSMolecular orbital calculations using this method have been performed on themetallo-organic compounds, Pd(allyl)2, ferrocene, and dibenzene chromium. Previousnon-empirical calculations 3-1 on compounds of the transition metals and un-saturated hydrocarbons have not included the interaction of the a- framework ofthe organic entity with the metal.In the isolated hydrocarbon ligands considered,benzene, cyclopentadiene, and the ally1 radical, the plane of the carbon skeleton isa symmetry plane of the molecule, so that no mixing of the 0- and n-carbon atomicorbitals can occur. However, once the ligand is co-ordinated to the metal, this isno longer the case, so that mixing of Q- and n-orbitals can occur on construction ofthe molecular orbitals. Even if the extent of such mixing is slight, there remainsthe further possibility of significant bonding arising from m.0. constructed of metalorbitals and the ligand orbitals of predominantly a-character with appropriatesymmetry. To assess the importance of such “a-bonding” in these compounds,our calculations have been performed with an a.0.basis set which includes all threecarbon 2p orbitals as well as the 2s and hydrogen 1s orbitals. The metal orbitalsused include the palladium 4d, 5s and 5p atomic orbitals, and the iron and chromiuI . H . HILLIER AND MRS. R . M. CANADINE 293 4 4s and 4p orbitals. The basis orbitals were taken as linear combinations ofSlater type orbitals for the neutral atom in all cases except palladium. Those forpalladium (for metal charge of + 1) were taken from Gray et aZ.,16 for iron andchromium from Richardson et aZ.,17 and for carbon and hydrogen from Clementi?VSIE for the carbon and palladium orbitals were taken from the data of Moore,19the latter from values smoothed over the second row of the transition metals, aftertaking into account the absence of several terms in the experimental data, and aregiven in table 1.VSIE for the other orbitals were taken from the literature.20TABLE 1 .-vSIE FUNCTIONS FOR PALLADIUMVSIE = (Aq2+ Bq+ C ) eV. (1 eV = 16021 x J)orbital4d5s5P2s2Pstartingconfiguration A B0-754.841.1 10.890.8 10.460.490-271.3011.094.2 110.917.327.838-776-556.545.70VSIE FUNCTIONS FOR CARBONs2p” 1 -24 11.75.?pn 1 *66 12-06C8.5412-0213-587.608.809.303.845.215.0819.4310.66Calculations on the Pd(all~1)~ molecule were performed for the two isomericforms postulated from n.m.r. results,21 the “ boat ” and “ chair ” forms of symmetryCz0 and C2, respectively, shown in fig.1. Bond lengths and angles were taken from“ Boat ”C2” “ Chair ” C2hally1 ligandFIG. 1 .-Pd(alIyl)2 isomers30 APPROXIMATE MOLECULAR ORBITAL CALCULATIONSthe X-ray structure of Pd2C12(ally1)2,22 assuming that the xy plane is that definedby the Pd and bridging C1 atoms, and using X-ray data on the substituted methallyls 23to give the orientation of the C-H bonds with respect to the carbon skeleton. Thetransformation properties of the metal orbitals for these two symmetries and theappropriate combinations of ligand orbitals are given in table 2. Ferrocene andTABLE 2.--TRANSFORMATION SCHEMES FOR “ BOAT ” (cz”) AND “ CHAIR ” (c2h)* ISOMERS OFPd(allyl)zbl P xdxz*The transformation properties of the orbitals in C2h are given in parenthesis.t s and p refer to carbon atomic orbitals, h referring to hydrogen 1s orbitals, with the numberingof the atoms as in fig.1.dibenzene chromium were taken to have symmetries DSd and Dsh respectively withbond lengths taken from ref. (24). The z-axis was chosen to be perpendicular tothe aromatic ringsI . H . HILLIER AND MRS. R . M. CANADINE 31RESULTSPd(allyl),The self-consistent atomic charges and configurations for the two isomers aregiven in table 3. In both cases the metal and four equivalent terminal carbon atomshave definite negative charges whilst the remaining two central carbon atoms areelectron deficient. The hydrogen atoms are found to be only slightly charged.TABLE 3.-sELF CONSISTENT CHARGES AND CONFIGURATIONS OF Pd(allyl)2atomPdC1CZH1H2H3charge- 0.3025 - 0.10050-257 10.03300.03 17-0.0345- 0.3 179-0.09560.2574003190.03 14-0.0338s Px0.425 0.5981.048 1.0580.998 0.9400.9670.9681.0350.437 0.5981.047 1.0580.998 0.9400.9680-9691.034cz,, SYMMETRYconfigurationPY Pz dx2-y2 dzz dxy dx, dyz0.235 -0.023 1.869 1.940 1.333 1.999 1.9271-004 0-9900.980 0-825CZJ, SYMMETRY0.226 -0.026 1.880 1.944 1.336 1.998 1.9211.003 0.9880.980 0.824The origin and relative importance of the bonding in these molecules may be assessedby examination of the interatomic overlap populations which give the amount ofelectron density associated with the internuclear region between any two atoms Aand B due to the interaction of orbital i on atom A with orbital j on atom B aswhere cb is the coefficient of the ith atomic orbital on atom A in the kth molecularorbital, Nk the number of electrons in the kth molecular orbital, and Sif the overlapintegral between i and j .When symmetry adapted ligand orbitals are used, a groupoverlap population may be defined, with the group overlap integral G i j replacingSij in eqn. (6). The sum is over all occupied molecular orbitals, and to find thetotal overlap population between A and B, may be summed over all orbitals i on Aand j on B. The major contributions (whose absolute values are greater than 0.1)to the palladium-carbon overlap populations are given in table 4 for each irreducibleTABLE 4.-GROUP OVERLAP POPULATIONS USING THE NORMALIZED SYMMETRY-ADAPTED LIGANDORBITALS OF TABLE 2overlap populationligand terminal centralorbital carbons carbons representation zi;:,SSdXYdXYP xPxPYP zP ZP ZP ZP ZP zP ZPzSS0.232 0 .1 2 50-25 1 0 . 1 1 50.2450-2450.4790-479- 0 . 1 0 9 - 0 . 1 0 40 . 1 80 0 . 1 4 8-0.111 -0.1030.167 0 . 1 5 32 APPROXIMATE MOLECULAR ORBITAL CALCULATIONSrepresentation. Although only one metal-carbon a-interaction appears in this table,there are a number of such interactions having overlap populations of greater than0.05. However, as expected, it is the ligand n-orbitals which are predominantlyinvolved in the bonding. The bond overlap populations between the metal andligand atoms, given in table 5, also show that the bonding is strongest between thePd and terminal carbons atoms.Eigenvalues for the filled orbitals of highest energyand vacant orbitals of lowest energy are given in table 6.TABLE 5.-BOND OVERLAP POPULATIONSsymmetryC," C2hatom 1 atom 2Pd c1 0.3 139 0.3 130c2 0-1 339 0.1333HI - 0.0446 - 0.0466H2 - 0.0460 - 0.0458H3 - 0.0294 - 0.0287TABLE 6.-ELGENVALUES FOR Pd(aIIyI),tboat (CZ"~representation eigenvalue (eV)5.01- 2.08- 4.80- 12.68- 13.38- 13.69- 13.74- 13.98- 16.32- 17.13- 17-31* highest occupied orbital. t The major contributions to the eigenvectors are given in parenthesis.If the free metal atom is assumed to have a filled 4d shell containing ten electrons,a closer analysis of the bond overlap populations in table 4 shows that the majorcontributions to the bonding in the C,, isomer may then be considered to occurvia electron donation : (a) from the ligand a, to metal 5s orbital ; (b) from the metal4dXy to the ligand a2 orbital ; (c) from the ligand b , to the metal 5p, orbital ; (4 fromthe ligand b2 to the metal 5py orbital.Of these, (a) and (d) involve all six carbonatoms, whilst (b) and (c) involve only terminal carbon atoms. These interactionsare reflected in the atomic configurations, table 3, the m a i n features being the relativelysmall metal 4dxy, the high 5s, 5px and 5pV and the small 5p, populations. Thesecomments apply to both isomers equally well as the eigenvalues, atomic charges andconfigurations, and the overlap populations are similar for the two molecules.Inboth cases the individual metal-ligand interactions are identical, any differences inthe bonding arising entirely from differences in the ligand-ligand inteiations. Ourcalculations yield a slightly increased metal-carbon charge separation for the '' chair "isomer (table 3). As these calculations include electron interaction terms, the totalelectronic energy cannot be calculated from the orbital energies as in calculationsof the Hiickel type. However, the sum of the energies of the filled orbitals is morI . H . HILLIER A N D MRS. R . M. CANADINE 33negative by 5.83 kcal for the " chair " form. Since this difference amounts toless than 0.1 % of the total energy of either molecule, and recalling the approximatenature of the calculation, we have not attempted to evaluate accurately the requiredelectron repulsion integrals necessary to obtain a more precise value for the energydifference.Alternatively, we have evaluated the sum of the coulomb interactionenergies arising from the calculated atomic charges, which gives the ionic attractionforces to be weaker in the " chair " form by 0.1 kcal. These estimates may becompared with the experimental energy difference of = 0.5 kcal obtained from variabletemperature n.m.r. studies.21FERROCENE AND DIBENZENE CHROMIUMThe general features of the bonding in these molecules has been previouslydiscussed,13-15* 2 5 9 26 so that we shall only note those points peculiar to the presentcalculation.Computations on these molecules were performed using eqn. (3) and(4) in which the nuclear attraction integrals were evaluated by a point-charge approxi-mation. Initial calculations including the metal 3d, 4s and 4p orbitals gave self-ferrocene and dibenzene chromium respectively. These negative populations ofthe virtual orbitals of the metal atoms arose from the arbitrary partitioning of theoverlap charge density by the Mulliken population analysis. A calculation forferrocene by the WH scheme yielded a similar anomalous result.26 These negativepopulations were taken to indicate that the 4s and 4p orbitals do not participate signifi-cantly in the bonding in these molecules, at least in the ground state. The calculationsconsistent metal configurations of d 6 n a 7 p-1*56 s-0.19 and d5.65 p-1.78 s-0.49 forTABLE 7.-sELF CONSISTENT CHARGES AND CONFIGURATIONS OF FERROCENE AND DIBENZENECHROMIUMconfiguration molecule atom chargeferrocene Fe 1 -02 diiZ2,2, dz2ioo, d;;17C -0.13 s1.03, pA.98, pA.12H 0.03 s O * 9 7di benzene Cr 0.54 d-333,2, dz2ioo, d;i4'sl.OO p 2 * O Q , pk.06 chromium C - 0.06 ¶ H 0.01 so.99TABLE 8.-EIGENVALUES FOR FERROCENE AND DIBENZENE CHROMIUM*ferrocene dibenzene chromiumrepresentation eigenvaluc (eV) representation eigenvalue (ev)a1,(M) - 7.71 a&) - 3.76ez,(M) - 11.03 e2&M) - 8.49elu(L7J - 13.47 el .(Ln) - 15.34a2&) - 18.46 el,(L,) - 17-31el g(M,Ln) - 15-52 el,(M,Ln) - 15-97Ql,(L*) - 19.1 1* The major contributions to the eigenvectors are given in parenthesis ; M = metal ; L = ligand.were therefore repeated using only metal 3d orbitals and gave the atomic populationslisted in table 7.The energies and symmetries of the m.o. containing the 18 mostloosely bound electrons are listed in table 8. In only one of these orbitals, theel, of dibenzene chromium does the 0-framework of the hydrocarbon enter to a34 APPROXIMATE MOLECULAR ORBITAL CALCULATIONSgreater extent than the n-system. The overall effect of the a-framework on themetal carbon bond may be assessed by examining the contributions to the overlappopulations (eqn. (6)). For ferrocene, the interactions involving the p x , p,,, pz and sligand orbitals contribute -0.011, -0.011, 0.115, and -0.001 to the total bondpopulation of 0.092, whilst for dibenzene chromium the corresponding figures are-0.012, -0.012, 0.108 and -0.009, the total bond population being 0.075. Thebonding in the representations elg (dxz, dyz) and e2, (dx2-y2, dxy) contributes 0.073and 0.042 to the n overlap population in ferrocene, and 0.049 and 0-061 in dibenzenechromium.In both molecules, a,, orbitals ( 4 2 ) contribute little to the bonding.The predominant mode of bonding therefore involves the ligand 2pn orbitals, thea-framework contributing a small antibonding effect. Such a result would not havebeen deduced from overlap considerations alone for the overlap matrix contains anumber of terms between the metal d orbitals and carbon a-orbitals (e.g., dyz : 2s)which are greater than 0.1. The various orderings of the molecular orbitals in thesetwo molecules, as calculated by other workers, are summarized in table 9.ArmstrongTABLE 9.-M.O. SCHEMES FOR FERROCENE AND DIBENZENE CHROMIUM"et aZ.,26 who considered the ligand a-orbitals, and F i s ~ h e r , ~ ~ both employed theWH method, whilst the remaining authors used various approximations within theformalism of Roothaan, but considered only the ligand n-orbitals. Ionization data,particularly the photoelectron spectrum of ferrocene (ref. (27), fig. 19) may becompared with the predictions of the various calculations. The 2 : 1 intensity ratiofor the components of the highest energy band at ~ 6 . 8 and 7-2 eV, may be ascribedto the e2, and a,, orbitals respectively. The only calculation to predict this orderis that of Shustorovich and Dyatkina,15 our calculation giving the a,, orbital at-7.71 eV to be of higher energy than the ez,.Further bands in the spectrum occurat ~9 eV and between 12-14 eV, but none of the calculations so far performed canconfidently assign their origin.DISCUSSIONThere are, in general, two approaches by which approximate wavefunctions formolecules as complex as those discussed here may be calculated. The first is theempirical Wolfsberg-Helmholz method with its various modifications, which do notspecify a Hamiltonian, and which utilize experimental data and adjustable parameterssuch as F in eqn. (5) to evaluate the elements of the secular determinant. As th1. H. HILLIER AND MRS. R . M. CANADINE 35approximations in such schemes are not well-defined, it is doubtful if they can beimproved to give mare reliable results.The second approach, to work within theHF formalism of Roothaan, introducing only well-defined approximations to reducethe numerical computation, has resulted in a number of computational schemes,including the one we have used here. These methods have the advantage? apartfrom their more rigorous foundation, that they may be further improved in systematicways, particularly by comparison with results obtained from more accuratecalculations.As already observed, the present method reduces to the WH scheme when onIythe leading terms in eqn. (3) and (4) are used. The present scheme, with the inclusionof the additional terms, has given encouraging agreement with experiment for aseries of transition metal halides,28 and also for SO:- and SF6.6 The m.0.energiesobtained on a series of transition metal carbonyl complexes agree well with thoseobtained from photoelectron spectroscopy. While the agreement is not so goodfor calculations on ferrocene by this or other method^,^^-^^* 2 5 * 26 our value ofthe energy of the a,, orbital of - 7.7 eV is nearer to the experimental value of - 7-2eV than the value of -9.8 eV obtained by a WH calculation.26In this paper we have shown the usefulness of approximate 112.0. schemes inassessing the contributions of the various interactions to the metal-ligand bond.Because of the simplicity of our computational scheme, it is possible to handleproblems with a large number of basis orbitals, so that we have been able to assessthe extent of participation of the a-framework of the hydrocarbon ligands in bondingto the metal.Although we find such bonding (as measured by the overlap population)to be relatively weak, it should be considered in more sophisticated calculations.The results we have given should be more realistic than those calculated by the WHmethod, but the accuracy of any m.0. scheme can only be assessed by agreementwith experiment. The differences between the calculated and experimental ionizationenergies of ferrocene from photoelectron spectroscopy measurements show theneed for further development of approximate computational methods.C. C. J. Roothaan, Rev. Mod. Phys., 1951,23,69.E. Clementi and D.R. Davis, J. Comp. Phys., 1966, 1,223.M. Wolfsberg and L. Helmholz, J. Chem. Phys., 1952, 20, 837.C. J. Ballhausen and H. B. Gray, Molecular Orbital Theory, (W. A. Benjamin Inc., 1965).R. F. Fenske, K. G. Caulton, D. D. Radtke and C. C. Sweeney, Inorg. Chem., 1966,5,951.I. H. Hillier, J, Chem. SOC. A, 1969, 878.C. K. Jsrgensen, S. M. Horner, W. E. Hatfield and S. Y. Tyree, Znt. J. Quantum Chem., 1967,1,191.J. A. Pople and G. A. Segal, J. Chem. Phys., 1965,43, S 136.* M. D. Newton, F. P. Boer and W. N. Lipscomb, J. Amer. Chem. SOC., 1966, 88,2353.lo R. M. Canadine and I. H. Hillier, J. Chem. Phys., 1969, 50,2984.l 1 J. W. Richardson and R. E. 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O'Brien, J. Organometal. Chem., 1967, 9, P 27.22 A. E. Smith, Acta. Cryst., 1965, 18,331.23 R. Mason and A. G. Wheeler, J. Chem. SOC. A, 1968,2543 and 2549.24 Tables of Interatomic Distances and Configurations in Molecules and Ions, Chem. Soc., (Spec.2 5 R. D. Fisher, Theor. chim. Acta, 1963, 1,418.26 A. T. Armstrong, D. G. Carroll and S. P. McGlynn, J. Chem. Phys., 1967, 47,1104.*' D. W. Turner in Physical Methods in Advanced Inor.qa,anic Chemistry, (John Wiley and Sons,28 R. M. Canadine and I. H. Hillier, unpublished results.Publ. no. l l . , 1958).1968), p. 74