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Chemistry 4161: Inorganic Chemistry, Fall 2014 Problem set 4
Due date: Tues November 25 In class, we derived the reducible representation for the 3N degrees of freedom of the wa-‐ ter molecule, which has C2v symmetry. C2v is a particularly simple group as all the irreduci-‐ ble representations are one-‐dimensional (also known as an Abelian group), and all the irre-‐ ducible characters are +1 or –1. In the first two problems, we will work through examples involving groups where the irreducible characters have values other than +1 or –1. Problem 1: XeF4 is a square planar molecule with D4h sym-‐ metry. The molecule is oriented such that it lies flat on the xy-‐ plane, with the Xe at the origin and each Xe–F bond parallel to either the x-‐ or y-‐axis. The principal C4 axis is aligned along the z-‐axis. The atoms are numbered as indicated on the figure to the right. Consider a reducible representation involving the set of (xi, yi, zi) coordinates centered on each atom i, for a total of 5 × 3 = 15 basis functions. We order these as: (x1, y1, z1, x2, y2, z2, x3, y3, z3, x4, y4, z4, x5, y5, z5). In this representation, each operator of the D4h group is a 15 × 15 matrix. However, we can view the 15 × 15 matrix as a set of 3 × 3 sub-‐blocks arranged in a 5 × 5 array.
1a: For the C4 operator (counterclockwise 90° rotation about the z-‐axis), mark with an “x” to indicate the sub-‐blocks that contain non-‐zero elements in the 5 × 5 array below.
1b: Write the individual matrix elements for only the 3 × 3 sub-‐block(s) that lie on the diagonal in the above array. 1c: Determine the reducible character of the C4 operator. 1d: For the S4 operator (4-‐fold improper rotation about the z-‐axis), write the individual matrix elements for only the 3 × 3 sub-‐block(s) that lie on the diagonal. Determine the reducible character of the S4 operator. 1e: State a simple rule for determining the contributions to the character when one co-‐ ordinate is transformed to another coordinate.
CHEM 4161 Fall 2014 Problem Set 4 2
Problem 2: NH3 is a trigonal pyramidal molecule with C3v symmetry. The molecule is ori-‐ ented such that the principal C3 axis is aligned along the z-‐axis. Consider a reducible representation involving the set of (xi, yi, zi) coordinates centered on each atom i, for a total of 4 × 3 = 12 basis functions. In this representation, each operator of the C3v group is a 12 × 12 matrix. Similar to Problem 1, we can view each 12 × 12 matrix as a set of 3 × 3 sub-‐blocks arranged in a 4 × 4 array.
2a: For the C3 operator (counterclockwise 120° rotation about the z-‐axis), write the in-‐ dividual matrix elements for only the 3 × 3 sub-‐block(s) that lie on the diagonal. Deter-‐ mine the reducible character of the C3 operator. 2b: Determine the reducible representation Γ. 2c: Reduce Γ to its irreducible representations. Do the total degrees of freedom equal 3N? Note that each two-‐dimensional irreducible representation E counts as two degrees of freedom. 2d: Subtract out the translational and rotational degrees of freedom from Part 2c to ob-‐ tain the irreducible representations of the normal modes. Indicate which modes are in-‐ frared active.
Problem 3: White phosphorus is a form of elemental phosphorus that is important for syn-‐ thetic applications. It consists of tetrahedral P4 molecules, where each P atom is arranged on the vertices of a tetrahedron and bonded to three other P atoms. In contrast, tetrahedral As4 is unstable, even though As is only one element below P on the periodic table. AsP3 has been isolated at ambient temperature, and a tetrahedral structure where an As replaces a P at one vertex of the tetrahedron has been proposed [B.M. Cossairt, M.-‐C. Di-‐ awara, and C.C. Cummins, Science 323, 602 (2009)]. The Raman spectrum of AsP3 exhibits four peaks at 313 cm–1, 345 cm–1, 428 cm–1, and 557 cm–1.
3a: Sketch the proposed AsP3 structure. What is its molecular point group? 3b: Consider a reducible representation involving the set of (xi, yi, zi) coordinates cen-‐ tered on each atom i. Determine the characters of this reducible representation Γ. 3c: Carry out the reduction of Γ in Part 3b to its irreducible representations. Do the to-‐ tal degrees of freedom equal 3N? 3d: Subtract out the translational and rotational degrees of freedom from Part 3c to ob-‐ tain the irreducible representations of the normal modes. Indicate which modes are Raman active. Is the observed Raman spectrum consistent with the proposed structure? Note that a mode with E symmetry is doubly degenerate, meaning that it is actually two vibrations with the same energy, thus appearing as a single peak in a spectrum.
CHEM 4161 Fall 2014 Problem Set 4 3
Problem 4: Consider the cis and trans isomers of Fe(CO)4Cl2, whose structures are given below:
Determine the number of C–O stretching peaks in the infrared spectrum of each isomer. Can the infrared spectra distinguish between the two isomers? Note that a mode with E symmetry is doubly degenerate, meaning that it is actually two vibrations with the same energy, thus appearing as a single peak in a spectrum. For the purpose of determining the infrared-‐active modes, it is sufficient to consider only the irreducible representations that are infrared-‐active.
cis-‐Fe(CO)4Cl2 trans-‐Fe(CO)4Cl2
