Question

In the text it was shown that the simple picture of \(\pi\) back-bonding in metal-Cr bonding given by pictures such as Fig. \(2.1\) overestimates the extent of this bonding in \(\mathrm{Cr}(\mathrm{CO})_{6}\). Show that it similarly overestimates it in the tetrahedral molecule \(\mathrm{Ni}(\mathrm{CO})_{4}\). For this, the character table of the \(T_{d}\) point group will be needed. \begin{tabular}{r|rrrrr} \(T_{d}\) & \(E\) & \(8 C_{3}\) & \(3 C_{2}\) & \(6 S_{4}\) & \(6 \sigma_{d}\) \\ \hline\(A_{1}\) & 1 & 1 & 1 & 1 & 1 \\ \(A_{2}\) & 1 & 1 & 1 & \(-1\) & \(-1\) \\ \(E\) & 2 & \(-1\) & 2 & 0 & 0 \\ \(T_{1}\) & 3 & 0 & \(-1\) & 1 & \(-1\) \\ \(T_{2}\) & 3 & 0 & \(-1\) & \(-1\) & 1 \end{tabular}

Short answer

In \(\mathrm{Ni}(\mathrm{CO})_{4}\), all the atomic \(3d\), \(4s\), and \(4p\) orbitals of Ni are filled, leaving no empty orbitals available for back bonding to the \(2\pi^*\) of CO. This suggests that the picture of \(\pi\) back-bonding is indeed overestimated, just as it is in \(\mathrm{Cr}(\mathrm{CO})_{6}\).

Step-by-step solution

Identify the symmetry of molecular orbitals

First, identify the molecular orbitals of \(\mathrm{Ni}(\mathrm{CO})_{4}\). The molecule is tetrahedral and thus its orbitals can be classified using the \(T_{d}\) point group. Ni(II) in \(\mathrm{Ni}(\mathrm{CO})_{4}\) has atomic \(3d\), \(4s\), and \(4p\) orbitals. The carbon monoxide ligands have \(\sigma\) and \(\sigma^*\) orbitals available for interaction. All these orbitals need to be divided into symmetry species of the \(T_{d}\) point group.

Determine the Symmetry of Orbitals

Now, looking at the character table, classify the molecular orbitals into different symmetry species. The atomic \(3d\) orbitals of Ni split into \(E\) and \(T_{2}\) orbitals under \(T_{d}\) symmetry. The \(4s\) orbital remains invariant and thus belongs to \(A_{1}\), and \(4p\) orbitals split into \(T_{1}\) and \(T_{2}\). For each CO, the bonding \(\sigma\) orbital belongs to \(A_{1}\) symmetry and the antibonding \(\sigma^*\) orbital belongs to \(T_{1}\) symmetry, and \(\pi\) bonding orbital and \(\pi^*\) antibonding orbital belong to \(E\) symmetry.

Construct the Molecular Orbital Diagram

Construct the Molecular Orbital (MO) diagram by placing the orbitals at appropriate energy levels and fill electrons. The Ni \(3d\) orbitals are filled with 8 electrons, \(4s\) is filled with 2 electrons and \(4p\) orbitals are empty. Since there are 4 CO ligands, we have 4 bonding \(\sigma\) orbitals filled with 8 electrons, 4 antibonding \(\sigma^*\) orbitals empty and each of 4 CO ligands contribute 2 \(\pi\) electrons, making a total of 8 \(\pi\) electrons in 4 \(\pi\) bonding orbitals, and the \(\pi^*\) antibonding orbitals are empty.

Analyze the Back-bonding

Back bonding can only happen if there are empty orbitals on the metal that can accept the lone pair of electrons from CO. In the MO diagram, one observes that all \(3d\), \(4s\), and \(4p\) orbitals of Ni are filled. There are no empty orbitals available for back bonding to the \(2\pi^*\) of CO. Thus, the picture of \(\pi\) back-bonding in \(\mathrm{Ni}(\mathrm{CO})_{4}\) is indeed overestimated.

Key concepts

Molecular Orbital Theory

The molecular orbital theory (MOT) is a seatbelt for understanding how electrons behave within molecules. It's all about the idea that atomic orbitals combine to create molecular orbitals, which extend over the entire molecule. Here's the game plan: When atoms approach each other, their orbitals mingle and produce new ones, which are either bonding or antibonding. Bonding orbitals are like a cozy hammock, lower in energy, built for electrons to nestle in and stabilize the molecule. Antibonding orbitals, on the flip side, are like a shaky ladder, higher in energy, and they can make the molecule less stable if electrons climb up there.

Now picture this: Electrons fill these orbitals following the aufbau principle, which means they start from the ground floor (lowest energy) and work their way up. Let's not forget about the Pauli exclusion principle, which says two electrons max in each molecular orbital, but only if they are spinning in opposite directions. Lastly, Hund's rule reminds us that electrons are a bit antisocial, preferring empty orbitals in the same energy level before doubling up. These rules guide us in constructing molecular orbital diagrams, which are like floor plans showing where electrons live in a molecule.

Tetrahedral Molecular Geometry

Got four friends? Imagine tying a rope to each one and stepping into the center of a room. Stretch the ropes and have your friends back up until they're equally spaced in 3D. Voila! You've created a tetrahedral geometry just like in molecules such as methane, CH₄, or nickel tetracarbonyl, Ni(CO)₄.

In this geometric hangout, the central atom is the life of the party, surrounded by four atoms or groups at the corners of a pyramid with a square base (minus the base, though). Each friend—or atom—forms an angle of approximately 109.5 degrees with you, which gives you plenty of personal space and creates a pretty symmetrical shape. The beauty of this setup is that it's super stable, keeping everything in a 'chill' low-energy state, and that's what atoms really dig.

Point Group Symmetry

Point group symmetry is like the dance moves of molecules. Each molecule struts its stuff based on its atomic arrangement, and we can classify these groovy moves using point groups. Point groups are basically a set of instructions that tell us how we can rotate or reflect a molecule so it looks exactly the same as it did before.

Remember the character table we mentioned before? That's like the choreography cheat sheet, telling us the moves allowed (symmetry operations) for a particular group. For example, molecules in the T_d point group, like our tetrahedral Ni(CO)₄, are like dance champions—they can do multiple spinning moves (rotations) and stylish flips (reflections) without missing a beat. Why care? Knowing the symmetry helps us predict molecular behaviors such as how they'll interact with light, or how they'll bond - it's crucial for diving deeper into chemistry!

Inorganic Ligand Field Theory

Inorganic ligand field theory is like a VIP event for understanding how central metal ions and surrounding 'fans'—the ligands—interact in coordination compounds. It takes the basics from crystal field theory but cranks it up with molecular orbital theory for a full-blown concert experience. This theory examines the effect these groupies (ligands) have on the energy levels of the d-orbitals of the main metal act.

When ligands approach the metal ion, they don't all come with the same energy—some interact more intimately than others. This can cause the metal's d-orbitals to get all excited and split into different energy levels. Let's just say, not all d-orbitals enjoy the same popularity. In a complex like Ni(CO)₄, the ligands are like backup singers (carbon monoxide) harmonizing with the lead (nickel), affecting the electron arrangement and bonding. This can radically change the color, magnetism, and reactivity of the compound. So, ligand field theory? It's your backstage pass to the quantum mechanics of coordination chemistry.