Question

VSEPR (valence state electron pair repulsion) theory was formulated to anticipate the local geometry about an atom in a molecule (see discussion in Section 25.1). All that is required is the number of electron pairs surrounding the atom, broken down into bonded pairs and nonbonded (lone) pairs. For example, the carbon in carbon tetrafluoride is surrounded by four electron pairs, all of them tied up in \(\mathrm{CF}\) bonds, whereas the sulfur in sulfur tetrafluoride is surrounded by five electron pairs, four of which are tied up in SF bonds with the fifth being a lone pair. VSEPR theory is based on two simple rules. The first is that electron pairs (either lone pairs or bonds) will seek to avoid each other as much as possible. Thus, two electron pairs will lead to a linear geometry, three pairs to a trigonal planar geometry, four pairs to a tetrahedral geometry, five pairs to a trigonal bipyramidal geometry, and six pairs to an octahedral geometry. Although this knowledge is sufficient to assign a geometry for a molecule such as carbon tetrafluoride (tetrahedral), it is not sufficient to specify the geometry of a molecule such as sulfur tetrafluoride. Does the lone pair assume an equatorial position on the trigonal bipyramid leading to a seesaw geometry, or an axial position leading to a trigonal pyramidal geometry? The second rule, that lone pairs take up more space than bonds, clarifies the situation. The seesaw geometry in which the lone pair is \(90^{\circ}\) to two of the SF bonds and \(120^{\circ}\) to the other two bonds is preferable to the trigonal pyramidal geometry in which three bonds are \(90^{\circ}\) to the lone pair. Although VSEPR theory is easy to apply, its results are strictly qualitative and often of limited value. For example, although the model tells us that sulfur tetrafluoride adopts a seesaw geometry, it does not reveal whether the trigonal pyramidal structure (or any other structure) is an energy minimum, and if it is, what its energy is relative to the seesaw form. Also it has little to say when more than six electron pairs are present. For example, VSEPR theory tells us that xenon hexafluoride is not octahedral, but it does not tell us what geometry the molecule actually assumes. Hartree-Fock molecular orbital calculations provide an alternative. a. Optimize the structure of \(\mathrm{SF}_{4}\) in a seesaw geometry \(\left(C_{2 v} \text { symmetry }\right)\) using the HF/3-21G model and calculate vibrational frequencies (the infrared spectrum). This calculation is necessary to verify that the energy is at a minimum. Next, optimize the geometry of \(\mathrm{SF}_{4}\) in a trigonal pyramidal geometry and calculate its vibrational frequencies. Is the seesaw structure an energy minimum? What leads you to your conclusion? Is it lower in energy than the corresponding trigonal pyramidal structure in accordance with VSEPR theory? What is the energy difference between the two forms? Is it small enough that both might actually be observed at room temperature? Is the trigonal pyramidal structure an energy minimum? b. Optimize the geometry of \(\mathrm{XeF}_{6}\) in an octahedral geometry \(\left(\mathrm{O}_{\mathrm{h}} \text { symmetry }\right)\) using the HF/3-21G model and calculate vibrational frequencies. Next, optimize \(\mathrm{XeF}_{6}\) in a geometry that is distorted from octahedral (preferably a geometry with \(\left.C_{1} \text { symmetry }\right)\) and calculate its vibrational frequencies. Is the octahedral form of \(\mathrm{XeF}_{6}\) an energy minimum? What leads you to your conclusion? Does distortion lead to a stable structure of lower energy?

Short answer

To solve this problem, first optimize the structures of SF₄ and XeF₆ in their different geometries using the Hartree-Fock model and the 3-21G basis set. Next, calculate the vibrational frequencies for these optimized structures. Analyze the energies and vibrational frequencies to determine the energy minimum for each molecule and compare the results with the predictions from VSEPR theory. For SF₄, check if the seesaw structure has lower energy than the trigonal pyramidal structure, and for XeF₆, examine if the octahedral form is an energy minimum or if distortion leads to a more stable structure.

Step-by-step solution

Optimize the structures

Using appropriate software that can handle Hartree-Fock calculations, optimize the structures for each of the given molecules and their geometries. - For SF4: optimize in seesaw (C2v symmetry) and trigonal pyramidal geometries. - For XeF6: optimize in octahedral (Oh symmetry) and distorted octahedral geometries.

Calculate vibrational frequencies

Calculate vibrational frequencies for the optimized structures of SF4 and XeF6.

Analyze the results

Analyze the energies and vibrational frequencies of the optimized structures to answer the following questions: - For SF4: - Is the seesaw structure an energy minimum? (Look for the structure with the lowest energy) - Is the seesaw structure lower in energy than the trigonal pyramidal structure, as VSEPR theory suggests? - Calculate the energy difference between the two structures. - Is the energy difference small enough to observe both structures at room temperature? - Is the trigonal pyramidal structure an energy minimum? - For XeF6: - Is the octahedral structure an energy minimum? - Does distortion lead to a more stable structure of lower energy? Remember that the results provided by the Hartree-Fock model are only qualitative and they should be compared with VSEPR theory predictions to draw conclusions about the stability and relative energies of the different molecular geometries.

Key concepts

Molecular Geometry Optimization

The process of molecular geometry optimization is akin to finding the most comfortable position for a group of people on a sofa; each individual tries to find a spot where they have the most personal space. Similarly, atoms within a molecule adjust their positions to minimize repulsive forces and maximize stability, resulting in the most energetically favorable arrangement.

Using computational methods, such as the Hartree-Fock calculations mentioned in the exercise, scientists can simulate this 'comfort-seeking' behavior by adjusting atomic positions and calculating the energy at each step. Optimization continues until the lowest possible energy configuration is achieved, which is considered an energy minimum. If further adjustments increase the energy, the structure is said to have reached its optimal geometry.

Applying VSEPR Theory and Hartree-Fock Calculations

Using the example of sulfur tetrafluoride (\textsf{SF}\(_4\)), VSEPR theory predicts a seesaw geometry based on the electron pair repulsion. However, to confirm this shape is the energy minimum, calculations using the Hartree-Fock method are necessary. When a molecule like \textsf{XeF}\(_6\) defies simple VSEPR predictions, more advanced methods like Hartree-Fock are particularly useful in exploring potential geometries and finding the conformation with the lowest energy, whether it be octahedral or a distorted shape.

Hartree-Fock Molecular Orbital Calculations

Imagine trying to solve a jigsaw puzzle where each piece represents the behavior of an electron in a molecule. Hartree-Fock molecular orbital calculations are similar to this task; they attempt to describe the positions and energies of electrons within molecules. This quantum chemistry approach considers electrons to move independently in an average field created by all other electrons.

The calculations are based on a set of mathematical equations called the Hartree-Fock equations, which must be solved iteratively to find the distribution of electrons that minimizes the energy of the molecule. This approach generates a molecular orbital diagram depicting the energy levels and occupancy of the electrons.

Connecting Theory and Computational Models

In the context of the exercise, after performing Hartree-Fock calculations for the \textsf{SF}\(_4\) and \textsf{XeF}\(_6\) molecules, the optimized geometries can be compared to the predictions made by VSEPR theory to evaluate their accuracy and gain insights into the molecular shapes that could not be explained by VSEPR alone. Complementing qualitative models with quantitative calculations allows us to verify which molecular geometries are true energy minima.

Vibrational Frequency Analysis

Vibrational frequency analysis is a technique that could be likened to playing various music notes by stretching guitar strings. Each vibrational mode of a molecule is a 'note' that depends on the shape and mass of the molecule, much like the sound depends on the length and tension of the guitar string.

This type of analysis provides information about the stability of a molecule and the nature of its chemical bonds. By calculating the vibrational frequencies, chemists can determine if a predicted structure corresponds to an energy minimum. Frequencies corresponding to real vibrations (positive values) indicate a stable structure, whereas imaginary frequencies (negative values or written with an 'i') suggest that the structure is not at an energy minimum and is, therefore, unstable.

Practical Implications for \textsf{SF}\(_4\) and \textsf{XeF}\(_6\)

Applying vibrational frequency analysis to \textsf{SF}\(_4\) and \textsf{XeF}\(_6\), as outlined in the exercise, helps confirm whether the seesaw and distorted octahedral geometries, respectively, are stable. It checks if the molecules in these conformations would vibrate naturally without leading to a structural change, which is critical for validating both the VSEPR theory and the results from Hartree-Fock calculations.