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NormalModes

# NormalModes - Classical Normal Mode Analysis Harmonic...

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Colby College Classical Normal Mode Analysis: Harmonic Approximation Each absorption in a vibrational spectrum corresponds to a normal mode. Characteristics of Normal Modes 1. Each normal mode acts like a simple harmonic oscillator. 2. A normal mode is a concerted motion of many atoms. 3. The Center of mass doesn’t move. 4. All atoms pass through their equilibrium positions at the same time. 5. Normal modes are independent; they don’t interact. Carbon Dioxide in Air 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 Single Beam 500 1000 1500 2000 2500 3000 3500 4000 Wavenumbers (cm-1) V = 1 2 k x 2 F = - dV dx = -kx k = d 2 V dx 2 (1) F = ma m d 2 x dt 2 = -kx (2) x = A sin(2 πν t) d 2 x dt 2 = -4 π 2 ν 2 x -4 π 2 ν 2 m x = -kx (3) Coordinates: Atom i: X i , Y i , Z i , Displacements: x i = X i – X i,eq , y i = Y i – Y i,eq Calculate the potential energy V(x 1 , y 1 , z 1 , x 2 , y 2 , z 2 , x 3 , y 3 , z 3 ,…., x N , y N , z N ) 2 V x 1 2 = k 11 xx change of the force on atom 1 in the x-direction when you move atom 1 in the x-direction Asymmetric stretch: CO 2 2349 cm -1 Bend: CO 2 667 cm -1 Asymmetric stretch: H 2 O 3756 cm -1 Symmetric stretch: H 2 O 3652 cm -1 Bend: H 2 O 1595 cm -1 1 2

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NormalModes - Classical Normal Mode Analysis Harmonic...

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