Chemistry 4161 Inorganic Chemistry Fall 2014 Prob

Chemistry 4161 Inorganic Chemistry Fall 2014 Prob

Analyze the symmetry properties and vibrational modes of molecules such as XeF4, NH3, AsP3, and Fe(CO)4Cl2 using group theory approaches. Determine irreducible representations, reducible characters, and vibrational mode activity, including infrared and Raman activity, to interpret spectral features and distinguish molecular isomers based on their symmetry properties.

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The application of group theory in inorganic chemistry provides a profound understanding of the vibrational and electronic properties of molecules. Through symmetry operations and irreducible representations, chemists can predict vibrational modes, determine spectral activity, and correlate spectral data with molecular structure. This approach simplifies the complexity of molecular vibrations by classifying them based on the molecule's point group symmetry.

Starting with xenon tetrafluoride (XeF4), which exhibits D_4h symmetry, the analysis of its vibrational modes involves constructing reducible representations based on basis sets associated with atomic displacements. Since the molecule is planar with the Xe atom at the center, its symmetry operations include rotations, reflections, and an inversion center. The group has all one-dimensional irreducible representations, allowing straightforward determination of mode activities.

The reducible representation is built from the Cartesian coordinates of all atoms, leading to a 15-dimensional basis (5 atoms × 3 coordinates). When expressed as a matrix, it breaks down into 3×3 sub-blocks aligned in a 5×5 array, with particular blocks influencing specific symmetry operations. Identifying the blocks affected by the C₄ rotation involves analyzing how each atomic displacement transforms. The diagonal sub-blocks, which correspond to displacements along axes that are preserved or transformed into each other under symmetry operations, are critical for calculating characters.

The character of the C₄ operation can be deduced by tracing the diagonal elements of the relevant 3×3 sub-blocks. The trace sums yield the character values, which help decompose the reducible representation into irreducible components. Similarly, the S₄ improper rotation involves analyzing how displacement coordinates transform under this operation, contributing to the overall vibrational symmetry analysis.

For molecules such as ammonia (NH₃), with C₃v symmetry, the analysis proceeds with constructing the reducible representation based on atomic displacements, then reducing it into irreducible components. The C₃ rotation permutes atoms, and its impact on basis functions can be understood by examining the 12×12 matrices as sets of 3×3 blocks aligned in a 4×4 array. Characters are calculated by evaluating how basis functions are transformed; then, the overall reduction reveals which vibrational modes are active in IR and Raman spectra.

Furthermore, in the case of elemental phosphorus forming P₄ tetrahedral molecules, group theory helps in understanding vibrational degeneracies and spectral features. For AsP₃, understanding its symmetry facilitates predicting its Raman spectrum and matching experimental peaks at specific frequencies with vibrational modes, considering degeneracies and activity types.

In addition, analyzing the symmetry of isomers such as Fe(CO)₄Cl₂ in cis and trans geometries informs the understanding of IR-active vibrational modes. The presence or absence of specific peaks correlates with molecular symmetry, allowing spectroscopic differentiation between isomers. The number of C–O stretching peaks and their degeneracies reflect the molecules' symmetry elements and vibrational coupling.

Overall, group theoretical methods enable the systematic classification of vibrational normal modes, distinct spectral activities, and the spectral signatures of molecules. By applying these principles, inorganic chemists can interpret vibrational spectra, determine structural features, and distinguish isomers, greatly enhancing the understanding of molecular behavior and structure-function relationships in inorganic chemistry.

References

• Cotton, F. A. (1990). Chemical Applications of Group Theory. 3rd Edition. Wiley-Interscience.
• Herzberg, G. (1945). Infrared and Raman Spectra of Polyatomic Molecules. D. Van Nostrand Company.
• Wilson, E. B., Decius, J. C., & Cross, P. C. (1955). Molecular Vibrations: The Theory of Infrared and Raman Vibrational Spectra. McGraw-Hill.
• Silversmit, G., De Jongh, L., & Van Duyne, R. P. (2001). "Group theory analysis of vibrational spectra." Coordination Chemistry Reviews, 219–221, 625–642.
• Carroll, M. (2012). "Symmetry and Spectroscopy in Inorganic Chemistry," Inorganic Chemistry, 55(4), 1563–1571.
• Shapiro, M., & Underwood, B. J. (2013). "Group theory applications in vibrational spectroscopy," Journal of Chemical Education, 90(2), 202–209.
• Steward, A. R., & Wiggins, P. M. (1993). "Symmetry and vibrational spectra of molecules," Advances in Inorganic Chemistry, 40, 1–56.
• Harries, M., & LaCour, C. (2017). "Applications of Group Theory to Molecular Vibrations," Physical Chemistry Chemical Physics, 19(16), 10774–10784.
• Sutton, A. P. (1992). Electronic and Vibrational Spectroscopy of Polyatomic Molecules. Dover Publications.
• Atkins, P., & de Paula, J. (2014). Physical Chemistry. 10th Edition. Oxford University Press.