Question

1 Normal modes of CO2 molecule In the linear CO2 molecule with the equilibrium configuration symmetric with respect to the oxygen atoms, OC=CO=a, the potential energy depends on dis- tances OC, CO, and angle OCO. (a) Apply the conditions separating vibrational modes from translations and rotations of the molecule as a whole. Check the correct number of remain- ing physical modes. (b) Characterize the normal modes qualitatively and draw pictures of corre- sponding atomic displacements. (c) Argue that for the modes along the line the normal coordinates are simply the sum and the difference of the displacements of the oxygen atoms. Write down the corresponding part of the Lagrange function introducing the elastic force coefficient k for the atomic displacements OC and Co along the line and find the normal frequencies. (d) For the transversal modes assume that the bending potential energy can be written as U= 1k_a?A? where ki is the corresponding elastic force coefficient, and _y(O(1)) – y(C) y(O(2)) – y(C) (2) is the deviation of the angle OCO from 180°. Find the normal frequencies for the bending modes.

Answer 1

total no. of atom N= 3 total degoree of freedom =3N = Fril + Frot = 3 & because molecule can more Foot = 3 for Non-linear molecule = 2 For linear molecule Frot & traws alor q x,y & 2 t axis) Frit= 3N- Fot- ftram = 3N-2-3= 30-5 = 3x3-5 - 955 = 4 I Frib= 4 = Normal modes (vibration mode I Normal mode = Fil = 4. toocto symmetrical stretching Es o unsymmetrical stretching bend bend 2019/10/27 18:26