Triatomic Molecule Example

Question 1

What does the KE T equal for the triatomic molecule CO2?

Answer 1

T = 1/2m(O)x1’^2 + 1/2m(C)x2’^2 + 1/2m(O)x3’^2

Question 2

What does M equal for the triatomic molecule?

Answer 2

Same as for diatomic but with 3x3 matrix and masses diagonally down from top left corner.

Question 3

What does V equal for the triatomic molecule?

Answer 3

V = 1/2k(x2-x1)^2 + 1/2k(x3-x2)^2

Question 4

What does K equal for the triatomic molecule?

Answer 4

K = matrix(k, -k, 0, -k, 2k, -k, 0, -k, k)

Question 5

How do we find the eigenvalue frequencies for the triatomic molecule?

Answer 5

Use |K-Mω^2| = 0 again and find the determinant, then rearrange for ω again.

Question 6

How do we get the eigenvector for the triatomic molecule?

Answer 6

Use |K-Mω^2|X = 0 again but have X as matrix(a,b,c) instead of just 2x1 matrix

Question 7

What do we find for mode 1 for the triatomic molecule?

Answer 7

Get free translation, so all the molecules move same direction (to right) when ω1^2 = 0

Question 8

What do we find for mode 2 for the triatomic molecule?

Answer 8

Find symmetric vibrational mode:centre of mass fixed, so left molecules either side move inwards and centre molecule stays still.

Question 9

What do we find for mode 3 for the triatomic molecule?

Answer 9

Same as mode 2 but centre molecule moves to the left to keep C centre of mass fixed.