There are a variety of models that chemists use to describe bonding in molecules. It is important to remember that models are representations of systems and therefore have strengths but also important limitations. For example, Lewis dot structures, the simplest method for describing how atoms share electrons, do not take into account the geometry of the atoms in the molecule. Valence Shell Electron Pair Repulsion (VSEPR) theory does describe the geometry of atoms, but it does not provide an explanation for the observation that isoelectronic species with the same number of valence electrons can exhibit different geometries. Especially for transition metal complexes, both of these models fall short in describing the bonding of metals. Crystal field theory is a bonding model that is specific to transition metal complexes. This model looks at the effects of a ligand's electric field on the d or f atomic orbitals of a metal center. The interaction results in a break in degeneracy of those atomic orbitals.

In this video, we will focus on MO theory, which is a powerful model that can be used to describe bonding in not only main group molecules, but also is suitable for modeling the bonding in transition metal complexes. Here, we will demonstrate how to generate an MO diagram of metal-containing compounds.

MO Theory:

MO theory describes chemical bonding as the linear combination of the atomic orbitals (LCAO) of each atom in a given compound. The MOs that result from LCAOs describe both the geometry and energy of the electrons shared by a number of atoms in the molecule (i.e., the directionality and strength of the bonds formed by given atoms).

To review the basics of MO theory, first consider the diatomic molecule F₂ (full MO diagram in Figure 1). A fluorine atom has 4 valence atomic orbitals: 2s, 2p_x, 2p_y, and 2p_z. The 2s orbital is lower in energy than the 2p atomic orbitals, which all have the same energy. A linear combination of atomic orbitals will occur between atomic orbitals of similar energy and of matching symmetry. In this case, the 2s orbital on one F atom will interact with the 2s orbital on the other F atom. The addition of these two orbitals results in the formation of a σ bonding MO (Figure 1). Bonding is a stabilizing interaction and, therefore, the resulting σ MO is lower in energy relative to the energy of the 2s atomic orbitals. Subtracting the 2s orbitals results in an anti-bonding interaction (destabilizing), designated as σ*, which is higher in energy relative to the 2s atomic orbitals (Figure 1).

Figure 1. MO diagram of F₂.

Likewise, the 2p atomic orbitals will combine to form bonding and anti-bonding interactions. Like the 2s atomic orbitals, the 2p_z atomic orbitals (which lay along the F-F bond) form σ and σ* interactions. If we consider the 2p_x and 2p_y atomic orbitals, we see that they form different types of bonding and anti-bonding interactions, called π and π*, respectively (Figure 1). It is easy to differentiate between σ and π bonds because σ bonding orbitals are cylindrically symmetrical about the internuclear axis, while π orbitals have a nodal plane along the internuclear axis. The spatial overlap between atomic orbitals that form σ bonds is greater than the spatial overlap between atomic orbitals that form π bonds. Therefore, the resulting π and π* MOs are less stabilized and destabilized, respectively, compared to the σ and σ* MOs formed by the 2p_z atomic orbitals. We can then fill the MOs with the valence electrons for the two F atoms.

Now consider a more complex molecule such as [Co(NH₃)₆]Cl₃ (Figure 2). If we were to use the same process as above (considering the atomic orbital overlap between 2 atoms at a time), generating an MO diagram of this molecule would be extremely challenging. Instead, we can use group theory to first generate a symmetry adapted linear combination (SALC) of the ligands. We can then use symmetry to determine the bonding/anti-bonding interactions that form between the atomic orbitals on the metal and the resulting SALCs.

O­h	E	8C3	6C2	6C4	3C2’	i	6S4	8S6	3σh	3σd
A1g	1	1	1	1	1	1	1	1	1	1		x2+y2+z2
A2g	1	1	-1	-1	1	1	-1	1	1	-1
Eg	2	-1	0	0	2	2	0	-1	2	0		(2z2-x2-y2, x2-y2)
T1g	3	0	-1	1	-1	3	1	0	-1	-1	(Rx, Ry, Rz)
T2g	3	0	1	-1	-1	3	-1	0	-1	1		(xz, yz, xy)
A1u	1	1	1	1	1	-1	-1	-1	-1	-1
A2u	1	1	-1	-1	1	-1	1	-1	-1	1
Eu	2	-1	0	0	2	-2	0	1	-2	0
T1u	3	0	-1	1	-1	-3	-1	0	1	1	(x, y, z)
T2u	3	0	1	-1	-1	-3	1	0	1	-1
Γred	6	0	0	2	2	0	0	0	4	4

Γ_red =A_1g + E_g + T_1u

Figure 2. Linear combination of ligand atomic orbitals of [Co(NH₃)₆]Cl₃.

To generate the SALCs for [Co(NH₃)₆]³⁺, we follow a similar procedure outlined in the "Group Theory" video in the Inorganic Chemistry series:

1. Determine the point group of the molecule.

2. Generate a reducible representation of the ligand atomic orbitals.

3. Reduce the reducible representation to irreducible representations.

[Co(NH₃)₆]³⁺ is in the point group O_h. Since we are only concerned about the bonding at the metal center, we can simply consider the 2s atomic orbitals on each NH₃ ligand. If we follow steps 1-3 for the N 2s, orbitals we find that the reducible representation is Γ_red = A_1g + E_g + T_1u (Figure 2). While the A_1g set represents 1 SALC, the E_g and T_1u sets actually represent 2 and 3 SALCs, respectively, giving a total of 6 SALCs (the same number of ligands in the cation [Co(NH₃)₆]³⁺). The 2 SALCs in the E_g set have the same symmetry and will result in degenerate MOs when they interact with the atomic orbitals of the Co (the same can be said about the 3 SALCs in the T_1u set). Using the character table in Figure 2, we can determine how the atomic orbitals of Co transform in the O_h point group. For example, the d_z² and d_x²–y² orbitals form an E_g set. Since we have 2 ligand SALCs with E_g symmetry, those SALCs will form bonding/anti-bonding interactions with the d_z² and d_x^2–y² Co atomic orbitals. Continuing in the same fashion for all of the valence atomic orbitals of Co, we generate an MO diagram for the transition metal complex, shown in Figure 3. Notice that the remaining d-orbitals (d_xz, d_yz, and d_xy) transform as a set (T_2g) but do not have an appropriate symmetry matched SALC. These atomic orbitals therefore become "non-bonding" MOs. In other words, they do not participate in bonding with the ligands in this transition metal complex.

Figure 3. MO diagram for [Co(NH₃)₆]Cl₃.

Highlighted in Figure 3 are the non-bonding d-orbitals and the σ* orbitals with d-orbital character. When this group of MOs is considered separately from the entire MO diagram it is referred to as the d-orbital splitting diagram of a transition metal complex. Since the d-orbital splitting diagram contains the HOMO and the LUMO, which are typically the most important orbitals to understand the chemistry and spectroscopy of coordination complexes, chemists will often refer to the d-orbital splitting diagram instead of the entire MO diagram. Conveniently, the d-orbital splitting diagram can be filled with the number of de^- on the metal center, since the ligand-based electrons always fill the σ-based MOs in the MO diagram.

Considering the d-orbital Splitting Diagrams for M(dppf)Cl₂:

Consider a simple 4-coordinate metal complex MX₄. MX₄ can exist in two geometries: tetrahedral or square planar. The d-orbital splitting diagrams for the point groups T_d (tetrahedral) and D_4h (square planar) is shown in Figure 4. While the general metal complexes M(dppf)Cl₂ do not have 4 equivalent ligands, and therefore are not in the point groups T_d or D_4h, we can still use these d-orbital splitting diagrams as a model to describe the d-orbital MOs for the two possible geometries.

Figure 4. The d-orbital splitting diagrams for the point groups T_d (tetrahedral) and D_4h (square planar).

Now, consider the d-electron count for M(dppf)Cl₂. Both Ni and Pd are in Group 10 of the periodic table. Therefore, they will both have the same oxidation state (2+) and d-electron count (d⁸). If we fill the two d-orbital splitting diagrams above with 8 electrons, we see that the square planar geometry results in a diamagnetic complex, while the tetrahedral MO diagram is consistent with a paramagnetic species. There are several factors that go into determining which geometry is energetically favored. In the square planar geometry, there are fewer electrons in anti-bonding orbitals, which would indicate that the square planar geometry is more electronically favored. However, we also need to consider the energy required to pair electrons. The electron pairing energy in square planar molecules is higher than that in tetrahedral molecules, which have fewer completely filled orbitals. Finally, we need to consider the amount the σ* d-orbitals are destabilized. Larger metal atoms have greater spatial overlap with ligands, resulting in higher energy σ* d-orbitals.

Finally, we also need to consider the energy contribution from steric repulsions. Tetrahedral geometry is more sterically favored (with angles of 109.5 °) compared to square planar geometry (90 °). Therefore, there are several opposing factors that affect which geometry is more favored, given the identity of M in M(dppf)Cl₂.

We will be able to distinguish between these two geometries using NMR. If the molecule is square planar, we will observe a diagnostic ¹H NMR of a diamagnetic species. If the molecule is tetrahedral, we will observe paramagnetic signals in the ¹H NMR. Finally, we will use the Evans method (see the "Evans Method" video for more details) to determine the solution magnetic moment of the paramagnetic species.

Pd(dppf)Cl₂:
¹H NMR (chloroform-d, 400 MHz, δ, ppm): 4.22 (alpha-H), 4.42 (beta-H), 7.89, 7.44, 7.54 (aromatic)³.

Ni(dppf)Cl_2:
¹H NMR (chloroform-d, 300 MHz, δ, ppm): 20.85, 10.04, 4.23, 3.98, 1.52, -3.31, -7.10.
Evans Method, looking at the ¹⁹F shift of trifluorotoluene:

Observed µ_eff = 3.15 µ_b
Mass of the sample: 9.5 mg
Mass of solution (chloroform-d + trifluorotoluene): 0.8365 g
Temperature of probe: 296.3 K
NMR Field (MHz): 470.06
Reported µ_eff = 3.39 µ_b.⁴
For S = 1 (predicted based on tetrahedral geometry, Figure 4), theoretical µ_eff = 2.83 µ_b.
For S = 3/2, theoretical µ_eff = 3.46 µ_b.

Based on the ¹H NMR data, we see that Pd(dppf)Cl₂ is diamagnetic and therefore exhibits square planar geometry. The ¹H NMR of Ni(dppf)Cl₂ is paramagnetic and therefore is tetrahedral at the Ni center. Evan’s method confirms that Ni(dppf)Cl₂ is paramagnetic, exhibiting a solution magnetic moment of 3.15 µ_b, which is close to the literature reported value for this compound. Since Ni is small, the sterics outweighs any electronic stabilization associated with square planar geometry, making Ni(dppf)Cl₂ tetrahedral. On the other hand, Pd is large and, therefore, has higher energy σ* d-orbitals. In this case, the electronic stabilization greatly outweighs the steric repulsions, resulting in a square planar geometry at Pd in Pd(dppf)Cl₂.

This video demonstrated how MO theory can be used as a model of bonding in transition metal complexes. We synthesized two complexes with the general formula M(dppf)Cl₂. When M = Ni, the 4-coordinate complex exhibits a tetrahedral geometry. Replacing the Ni atom with a larger transition metal (Pd), the molecule takes on square planar geometry.

Previously, we learned about the important role ferrocene plays in the field of organometallic chemistry. Substituted ferrocenes, including dppf, are used as chelating ligands for 1^st, 2^nd, and 3^rd row transition metals. The resulting complexes are used in homogeneous catalysis (i.e., [1,1'-bis(diphenylphosphino)ferrocene]palladium(II) dichloride, Pd(dppf)Cl₂, is a catalyst for C-C and C-heteroatom bond-forming reactions).

Understanding the bonding in transition metal complexes is important for explaining their structure and reactivity. One of the strengths of MO theory is that it provides a good model that can be used to explain the reactivity of transition metal complexes. In many cases, the metal center is the location of any reactivity exhibited by the molecule. Therefore, it is valuable to have a picture of the electron density at the metal center, which is summarized in the d-orbital splitting diagram derived from MO theory (Figure 3). Notice that not only do the MOs in the d-orbital splitting diagram exhibit mostly d-orbital character (the σ* orbitals are closest in energy to the atomic d-orbitals of the metal and therefore most of the electron density in those MOs is centered on the metal atom), but also the splitting diagram contains the HOMO and LUMO of the molecule. Therefore, any chemistry that occurs will directly affect the d-orbital splitting diagram of the molecule.