F Orbitals and Metal-Ligand Bonding in Octahedral Complexes Ken Mousseau I. Abstract The independent study will compare metal-ligand bonding in octahedral complexes with rare lanthanide metals. A comparison will be made of f-orbital interactions in hexachloro complexes to that of 3d metal complexes using experimental and theoretical evidence. The f-f ligand to metal charge transfer transistions will be defined using charge transfer spectra data and molecular orbital theory. The extent of the metal 4f-ligand 3p orbital mixing and ineractions and ligand to metal covalency will be estimated. The electrostatic interactions of ligand-ligand repulsion and metal-ligand attraction will be explored as well as bond lengths and bond energies for 4f orbital complexes as compared to that of 3d orbital complexes. With the use of molecular orbital theory, crystal field theory, and the angular overlap model a study of ligand field splitting, f-orbital splitting and the interactions of ligand and metal orbitals will be discussed. It is shown that the ligandligand interactions of f-orbitals differ from that of 3d orbitals in octahedral environment. However both 3d and 4f orbitals display some similar characteristics but have different bonding energies and covalency. II.Introduction The study of f-orbitals and their bonding characteristics were explored. Elements that undergo bonding with f-orbitals are usually located in the Lanthanide series of the periodic table otherwise known as the f-block. F-orbitals have an angular quantum number of 3 and the angular momentum quantum numbers, m l = 0, ±1, ±2, ±3. F-
orbitals can exist in two sets, the cubic set and the general set. A Cubic set is found when the metal is in a cubic environment. A General set is found in all other environments. The two sets have three common orbitals, 4f xyz, 4f z3, and 4f z(x 2 ). The other four orbitals of the f series are 4f x3, 4f y3, 4f x(z 2 ), and 4f y(z 2 -x 2 ). Images of the cubic orbitals can be seen below (Figure 1) 1. In an octahedral environment the f-orbitals will split into two triple degenerate states and one single degenerate state. The irreducible representations for each state are A 2u, T 1u, and T 2u. For octahedral compounds the A 2u represents the f xyz orbital, T 2u represents the f x(z 2 ), f z(x 2 ), and f y(z 2 -x 3 ), and finally the T 1u represents the f x 3, f y3, and f z 3. All f-orbitals are ungerade simply because they are anti-symmetric to the center of inversion. In Figure 1 you cannot see the alternating signs of the wave functions. Lanthanides generally adapt the 3+ oxidation state in compounds. Lanthanide metal bonds are primarily ionic in character, and complexes generally undergo rapid ligand exchange. The 4f electrons in the metal are contracted into the core and unable to participate in bonding. Little π backbonding occurs with lanthanide metals and bonding with ligands. Lanthanide series has an interesting trend called the Lanthanide Contraction. Atomic radii of the metals decrease with increasing atomic numbers. This phenomenon is caused by the poor screening of nuclear charge by 4f electrons which produce s a steady increase in the effective nuclear charge (Z eff ). Relativistic effects influence the shielding characteristics of the inner electrons. The lanthanide contraction will be observed later on. Lanthanides are very strong reducing agents because of the high lattice energies of their compounds or their high hydration energies 2. Bonding interactions of 4f -> 3p have been studied extensively both experimentally and theoretically. One of the most widely used techniques in recent years is the Density Functional Theory (DFT). The central focus of DFT is electron density rather than wave functions. There are several types of DFT type approximations. The four used here to aid in the understanding of the 4f orbital interactions are the Ligand Field (LF), Crystal Field (CF), Ballhausen (BL), and Kohn-Sham (KS). DFT is a very good approximation for 3d metal complexes and with some modifications of the theory it can have a fairly good approximation of 4f orbitals and their bonding interactions. KSDFT has the ability to predict both ground and excited states of lanthanide metals reasonable well by adopting pseudopotentials of 4f electrons. KSDFT model has limitations it can only predict the order of orbitals but not the orbital energies. LFDFT model uses the interelectron repulsion, spin-orbit coupling, and ligand field theory. It is better suited for low symmetry and complex coordination geometries where application of crystal field theory and angular overlap model apply. BLDFT combines both Coulombic and Pauli repulsions, while the CFDFT model only considers the Coulombic contributions 3. You will see that the BLDFT model is one of the more accurate models developed. Using the following DFT models, experimental spectra data, molecular orbital theory, and crystal field theory, a description of metal-ligand bonding in hexachloro octahedral complexes of lanthanide metals will be explored. With the help of molecular 1 http://kitsilano.vsb.bc.ca/teachers/mannandale/oldchemsite/images/4fa.jpg 2 Douglas, B., McDaniel, D., Alexender, J. Concepts and Models of Inorganic Chemistry 3 rd Ed. John Wiley & Sons, Inc.: New York, 1994. 3 Atanasov, M., Daul, C., Gudel, H.U., Wesolowski, T.A., Zbiri, M. Inorg. Chem. 2005, 44, 2954-63.
orbital theory and experimental spectra data, an estimation of ligand to metal and f-f charge transfer can be determined. Orbital overlap and covalency will also be estimated as well as electrostatic interactions such as ligand-ligand repulsion and metal-ligand attraction. The interactions of ligand to metal orbitals will be discussed with the help of molecular orbital theory and a molecular orbital diagram. Finally the ligand field stabilization energies and f orbital splitting will be defined from crystal field theory and angular overlap model. III. Ligand to Metal Charge Transfer Transitions Spectra Data and Molecular Orbital Theory In Table 1 3 the first charge transfer absorption band maxima, CT (cm -1 ), for four hexachloro lanthanide complexes can be seen. The values in the table measure the amount of energy required to transfer an electron from the non-bonding t 1g orbital of Cl - to the 4f orbital (Figure 2) 3. A Co III metal with Chloride Ligands has a charge transfer band at around 37,100 cm -1. The intensities of charge transfer are proportional to the square of the dipole moment 4. From this data the covalency of the Lanthanide to Chloride bond can be estimated. Comparing Figure 2 to the MO diagram (Figure 3) of a sigma and pi donor ligand to a d series metal shows how the orbitals of the ligands mix with the metals. F orbitals are similar in bonding to that of d orbitals in that the symmetry adapted orbitals of the metal must match that of a ligand. Normally the 3p orbitals of Chlorine and the 4f orbitals of the metal are significantly different in energy. However due to a strong ligand-ligand interaction the t 2u and t 1u 3p orbitals of Chlorine 4 Atkins, P., de Paula, J. Physical Chemistry 7 th ed. W.H. Freeman and Company: New York, 2002.
become destabilized and approach closely the 4f orbitals to allow for orbital overlap to occur 3. Bonding orbitals 10a 1g, 7e g, and 10t 1u (Figure 2) are dominated by mostly the ligands but the 3t 2u and 12t 1u is has more metal contribution. Far right columns of Table 1 display the amount of orbital overlap in percentage. The percentages do not exceed more than two which suggests that the covalency of the bonding is low, but there is clearly a greater amount of sigma bonding than pi bonding. Sigma bonds of the ligands are directed to the t 1u and the pi interactions to the t 2u. IV. Electrostatic Interactions of Ligand-Ligand Repulsion and Metal-Ligand Attraction. Polarization of Cations is particularly important because it decreases the screening of the nuclear charge of the cation and decreases the metal-ligand distance 2. As mentioned above, the ligands will form bonds with the t 1u (f x 3, f y3, and f z 3 ) orbitals of the lanthanide series. Like the d orbitals, lanthanide f orbitals also split. In the case of f series we have three sets of degenerate orbitals (Figure 4). A different feature of f orbitals to d orbitals is as Z eff increases, the orbital splitting decreases (Table 2) 3. Table 2 consists of the Ligand Splitting energies (cm -1 ) of Lanthanide to Chloride bonds of the entire Lanthanide series. The data in the middle of the table was calculated theoretically by the methods described in the introduction. The experimental values reside on the column on the right. Most of the Lanthanides splitting decrease as Z eff increases with a few exceptions. Using the Angular Overlap Model (AOM) an estimation of the strength of interaction between the individual ligand orbitals and metal orbitals based on the overlap between them can be made by combining these values for all ligands and orbitals to get a
complete picture. This method adopts both the sigma (e σ ) and pi (e π ) interactions to get an understanding of bond strength and ligand field stabilization energy. In d orbitals the strongest interaction is between a metal d z 2 and ligand p orbital sigma bond. All other sigma bond strengths are referenced to this interaction. Sigma donor pi acceptor ligands, such as carbon monoxide, have empty orbitals and can interact with metal electrons. Because the sigma bond interacts more directly with the metals orbitals then logically e σ will always be greater then e π. Pi donor ligands affect the AOM by dumping electrons into metal vacant orbitals and decreasing the bond strength and giving a positive e π compared to the negative value of a pi acceptor ligand. This effect decreases the ligand field splitting of the metal. In a MX 6 complex a Cr 3+ -Cl - bond, e σ and e π are equal to 5700 cm -1 and 980 cm -1 respectively. From the AOM the LFSE calculation for this complex is 13,180 cm -1 5. Table 3 shows the experimental data of the AOM for hexachloro lanthanide complexes. All parameters of the table are in units of cm -1. As can be seen, the values for the sigma and pi bond are significantly lower than that of a 3d metal orbital. Ligand to metal interactions and bonding of lanthanide metals is considerable less than 3d metal complexes. Bond lengths of Lanthanides in hexachloro complexes are calculated by mimicking the effect of a Na-Cl bond. This is done by placing point charges behind the Ln-Cl bonds at a distance of 2.83 Å. Compared to the bond length of CrCl 6 3+ that is 2.335 Å. In Table 4 the bond lengths of hexachloro lanthanides are greatly longer than that of 3d metal complexes, which is to be expected 3. Table 4 displays a textbook trend of the Lanthanide Contraction, as mentioned above. As Z eff increases the bond lengths decrease. 5 Miesslar, G.L., Tarr, D.A. Inorganic Chemistry 3 rd Ed. Pearson Prentice Hall. New Jersey, 2004.
V. Conclusion Based on the numerical data provided and the theoretical data, f orbitals behave both similar in some ways 3d metal orbitals and different in others. From Molecular orbital theory and AOM, both f and 3d orbitals experience ligand to metal interactions but on different degrees. The bond energies and covalancy of f orbitals are less in energy to that of 3d orbitals. The charge transfer spectra data has shown that no greater than 2% of ligand to f orbital mixing occurs. When compared to the e σ and e π of the AOM and the bond lengths it is concluded that this is to be expected. 3d and 4f orbitals are alike in that they both undergo ligand field splitting in octahedral environments however 3d metal complexes split into two degenerate states while f orbitals split into three. The bonding orbitals of the f series are triple degenerate, and sigma bonding occurs with the f x 3, f y3, and f z 3 orbitals. Pi interactions on the other hand occur with the f x(z 2 ), f z(x 2 ), and f y(z 2 - x 3 ) orbitals and the f xyz orbital is non-bonding. Ligand to metal interactions of f orbitals appears to be of a more covalently nature than interactions of 3d orbitals.