THE VIBRATIONS OF MOLECULES II

THE CARBON DIOXIDE MOLECULE

Student Instructions

These instructions accompany the MATHCAD diskfile co2run.mcd.

The student should first complete the exercise for the simple harmonic oscillator in horun.mcd with its instructions.

This program illustrates the vibrations of CO2. We will assume that the force constant for each C-O bond is k. The molecule is arranged along the x axis with coordinates of the three atoms x1, x2, and x3. The kinetic energy is the sum of the kinetic energy of the atoms:

The extent Q1 that the left bond is stretched or compressed compared to its equilibrium bond length of re is

Similarly for Q2	.

Taking the time derivatives we get and .

If we assume that the center of mass does not move we get

We can set up the potential energy in terms of the stretching of each bond.

Unfortunately we have described the kinetic energy T in terms of x coordinates, and the potential energy V in terms of Q coordinates. In order to obtain the equations of motion we must describe T and V in the same coordinates. We choose to use the Q coordinates, so we must obtain both T and V in terms of Q's. The Q's in terms of x's are

To obtain the x coordinates in terms of the Q's we invert the matrix:

Where .

The kinetic energy can be calculated substituting the 's above into

and after considerable algebra we get

Collecting the constants in B11 and B12 we can rewrite the kinetic energy

which with the potential energy

can be substituted into Lagrange's differential equation first for Q1 then for Q2

to give the resulting equations of motion.

Unfortunately these equations cannot be solved by MATHCAD in this form. We must solve them for each of the second derivatives:

The values of K and the masses of the atoms are now included in the computational document where the differential equations are solved.

EXERCISE 1

The input to the differential equations routine has some adjustable constants. We must set values of the masses of the atoms m1 and m2 which can be taken as 16 and 12 for O and C. The value of the force constant should be set to 200. The column matrix Q gives the initial values of the Q's and their initial velocities:

Set each initial velocity equal to 3 and each initial position equal to zero. Calculate all four pages of the document. Which vibration does this simulate? For the animation part place the cursor (red cross) just to the right of the top of the animation plot and click on 'Graphics' and 'Animation' and 'create.' Then with the mouse block out the portion of the picture you wish to animate. Then press the lightbulb button to activate the automatic mode. The automation will now be recorded. A picture will appear at the left for the playback feature. Clicking on the lower left button will activate the playback.

Note the Fourier transform of the motion of Q1 on the last page. This is a plot in frequency space of the vibrational frequencies contained in this motion.

EXERCISE 2

Go to the Q matrix and change one of the derivatives from 3 to -3. Repeat the calculations. Has the frequency become larger or smaller?

Which vibration does this represent?

EXERCISE 3

Go again to the Q matrix and change the initial POSITION Q1 to 2 and change the initial VELICITY to 3. Repeat the calculation. Note that the motion becomes a mixture of the two modes and appears to be haphazard. But the Fourier transform can display two frequencies (even though the phasing of the maxima is not great).