Boron trifluoride, B F 3 B F 3 , is a planar molecule with F B

Boron trifluoride, ${BF}_3$, is a planar molecule with $F-B-F$ bond angles of ${120}^{\circ }$. It has the same symmetry operations as the mass and spring system in Problems 8 and 9, hence has the same character table. Its nontrivial normal modes are basis functions for the irreducible representations $A_1+B_2+2E_1$.

(a) Which of the modes are IR active?

(b) Which of the modes are Raman active?

(c) Which mode is Raman active but not IR active?

Data from Problem 8

Consider a planar system of three identical masses and three identical springs. At equilibrium each mass is at the apex of an equilateral triangle. A spring links each pair of masses.

How many displacement coordinates are there?

How many normal modes are there?

One of the normal modes is a basis function for the identity irreducible representation. Sketch this normal mode.

Data from Problem 9

Consider a planar system of three identical masses and three identical springs. At equilibrium each mass is at the apex of an equilateral triangle. A spring links each pair of masses.

Here is its character table (inversions omitted).

Use coordinate displacements as the basis for a reducible representation. In this representation what are the characters of each of the symmetry operations listed in the table?

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