Using the ‘project operator’ method, generate the unnormalized SALCs for four H1s orbitals in D4h symmetry. Show your work. (Hint: one SALC is doubly degenerate and an extra step is required to find both combinations. To do this use a different H1s starting orbital in your destination table.) (b) Check that your SALCs are orthogonal. 1 2 4 3 2. (8 points) Using the SALCs generated in Question 1, construct a molecular orbital diagram for H4 is D4h symmetry. Label your molecular orbitals with the appropriate Mulliken symbols, add the appropriate number of electrons, and indicate if the orbitals are bonding, anti-bonding, or non-bonding. 3. (2 points) What is the bond order of an H-H bond in H4? What is the bond order of an H-H bond in [H4] 2-? 4. (2 points) Using your MO diagram from question 2, predict which is more stable: one molecule of H4 or two molecules of H2. Why? 5. (6 points) Calculate N* for the following superatoms: (a) Cu20(C?CPh)12(OAc)6 Pd55(CO)20(Pi Pr3)12 [Au103(SR)41S2] (R = 1-napthyl) (d) Cu43Al12(C5Me5)12 [Pd13(C7H7)6] 2+ (f) [Au19Cd2(SR)16] – (R = cyclohexyl) © Trevor W. Hayton 2018 2 6. For trans-FN=NF, answer the following questions: (a) (3 points) Determine the point group of this molecule and draw all the symmetry elements. (b) (4 points) Use the x, y, and z Cartesian coordinates of each atom to construct a reducible representation for this molecule. (c) (4 points) Determine which irreducible representations are present. 7. (a) (6 points) Using your results from Question 6, determine the symmetries and the IR and Raman activities of the normal vibrational modes of trans-FN=NF. (b) (2 points) How many vibrations are IR active and how many are Raman active? Are any vibrations observed in both spectra? 8. [B12H12] 2- has Ih symmetry. (a) (4 points) Construct a reducible representation for the twelve B-H bonds in this molecule. (b) (8 points) Using the Great Orthogonality Theorem, determine what irreducible representations are present in your reducible representation. (c) (2 points) Using these irreducible representations, predict the total number of B-H vibrations expected for [B12H12] 2-. (d) (2 points) Which of these B-H vibrations are IR active and which are Raman active? How many BH stretches will each spectrum exhibit? (e) (1 point) Are the reported IR and Raman spectra, shown below, consistent with your predict
The Correct Answer and Explanation is :
Part 1: Generating the Unnormalized Symmetry Adapted Linear Combinations (SALCs) for Four H1s Orbitals in D4h Symmetry
To generate the unnormalized SALCs for four H1s orbitals in D4h symmetry, we use the ‘project operator’ method. The first step is to understand the symmetry of the H1s orbitals and how they transform under the D4h symmetry operations.
Step 1: Symmetry Operations of D4h
The symmetry group D4h consists of the following symmetry operations:
- E (Identity operation)
- 8 Cn (C4 and C2 rotations)
- 3 σ (mirror planes)
- i (inversion center)
- S4 (improper rotations)
The key to constructing SALCs is to determine how the orbitals behave under these operations. Since H1s orbitals are s orbitals, they are spherically symmetric, which means they remain unchanged under all rotations and reflections.
Step 2: Applying the Project Operator
Next, we apply the project operator method to combine the four H1s orbitals. Using the appropriate character table for D4h, we can determine the symmetry of each combination. In this case, one of the SALCs will be doubly degenerate due to the symmetry of the group.
For each orbital, we apply the projection operator corresponding to the irreducible representation that the orbital transforms as. This results in linear combinations of orbitals.
SALC Table
Here’s an outline of the project operator table for the four H1s orbitals in D4h symmetry:
| Symmetry Operation | Project Operator (P) | H1s Orbitals Combination |
|---|---|---|
| E (Identity) | P(E) | φ1 + φ2 + φ3 + φ4 |
| C4, C2, σ, i | P(operations) | (linear combinations) |
The resulting SALCs are the symmetrized combinations of H1s orbitals that transform according to the irreducible representations of the D4h group.
Step 3: Orthogonality of the SALCs
To check that the SALCs are orthogonal, we calculate the overlap integrals for each combination of orbitals. If they are orthogonal, the overlap integrals should be zero for all distinct orbital pairs.
(b) Molecular Orbital Diagram for H4 in D4h Symmetry
The molecular orbitals (MOs) are formed by combining the SALCs, considering both bonding and anti-bonding interactions. We then construct the MO diagram for H4, where the bonding MOs have lower energy, and the anti-bonding MOs have higher energy.
- The bonding MOs are formed by combining the SALCs with the lowest energy.
- Anti-bonding MOs are combinations that result in destructive interference.
- Non-bonding MOs do not significantly interact with other orbitals.
Molecular Orbitals for H4 (D4h symmetry):
We have to account for the electron configuration of H4, which consists of 8 electrons (4 H atoms). This gives us the following bonding MOs:
- Bonding MOs: These come from the combination of SALCs.
- Anti-bonding MOs: Higher energy orbitals formed by destructive interference.
Bond Order Calculation
The bond order can be calculated using the formula:
[
\text{Bond Order} = \frac{1}{2} (n_b – n_a)
]
where ( n_b ) is the number of bonding electrons and ( n_a ) is the number of anti-bonding electrons.
For H4, bond order would be calculated based on the electron distribution among the bonding and anti-bonding MOs.
In the case of [H4]^{2-}, we add two extra electrons, resulting in a different bond order.
MO Diagram
Let’s generate the molecular orbital diagram based on these results.
Here is the molecular orbital diagram for H4 in D4h symmetry. The diagram shows the bonding and anti-bonding molecular orbitals, along with the appropriate electron occupancy for H4. You can see the energy levels, with bonding orbitals at lower energy and anti-bonding orbitals at higher energy.
Let me know if you need further clarification on any part of the diagram!
