Problem 1: XeF4 is a square planar molecule with D4h symmetry. 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.

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 coordinate is transformed to another coordinate.

Problem 2: NH3 is a trigonal pyramidal molecule with C3v symmetry. The molecule is oriented 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 individual matrix elements for only the 3 × 3 sub-block(s) that lie on the diagonal. Determine the reducible character of the C3 operator.

2b: Determine the reducible representation

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 obtain the irreducible representations of the normal modes. Indicate which modes are infrared active.

Problem 3: White phosphorus is a form of elemental phosphorus that is important for synthetic 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. Diawara, 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 centered on each atom i. Determine the characters of this reducible representation.

3c: Carry out the reduction of Γ in Part 3b to its irreducible representations. Do the total degrees of freedom equal 3N?

3d: Subtract out the translational and rotational degrees of freedom from Part 3c to obtain 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.

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.