Page 164 - Numerical Methods for Chemical Engineering

Problems 153

where
[m] [n] [m] [n]
U ˆq ± εe + εe − U ˆq ± εe − εe
|(∂U/∂q n )| ˆ q±εe [m] = (3.284)
2ε
Thus, we have the finite difference approximation, accurate for small ε,
[U mn (ε, ε) − U mn (ε, −ε)] − [U mn (−ε, ε) − U mn (−ε, −ε)]
[H(ˆq)] mn =
4ε 2
[m] [n] (3.285)
U mn (ε 1 ,ε 2 ) ≡ U ˆq + ε 1 e + ε 2 e
HINT: You only need to perform the finite difference calculations for n ≤ m, and then
T
generate the other elements through symmetry, H = H .
(d) Compute the angular frequencies of each vibrational mode. Note that you will have
some zero-frequency modes corresponding to net translation and rotation of the
molecule. Neglect these. Also compute the eigenvectors, and save all results in a .mat
file.
(e) Then, using animate 2D vib.m as a guide, write a MATLAB routine that makes a movie
of the vibrations associated with a selected mode. Using this visualization, list the
modes in order of increasing frequency and describe the nature of the corresponding
oscillations. These modes are the peaks that may be observed in IR or Raman spec-
troscopy, depending upon the symmetry properties of the corresponding oscillations.