Vibrational levels of water -Tutorial

Vibration frequencies and force constants of water molecule-classical treatmentM. SamiullahSuchi’s Lab Group, 12/01/2009 (?)References:1. Wilson, Decius, and Cross, “Molecular Vibrations: The Theory of Infrared and Raman Vibrational Spectra,” McGraw Hill, NY (1955). rd2. Cotton, “Chemical Applications of Group Theory,” 3 edition, John Wiley & Sons, NY (1990).3. Landau and Lifshitz, “Quantum Mechanics : Non-relativistic Theory,” Addison-Wesley , NY (1958).Classical Mechanics Background• Simple Harmonic Oscillator2 2Kinetic energy, T = ½ m (dx/dt) ; Potential energy, V = ½ k x .2 2Equation of motion: m d x/dt + k x = 0. (1)iωtHarmonic solution: x(t) = A e with A and ω unknown. Plug into 1.2 iωt m(ω - k/m) A e = 0.2 Either A = 0 or ω - k/m= 0. kSince A ≠ 0, we must have . m>>> So, if you know ω, you can learn about force constant k and vice-versa. >>> Frequency ω for a molecular system is normally found by IR and/or Raman vibrational spectrum.Mass-weighted Cartesian Coordinates {q}• Simple Harmonic Oscillator Revisitedq x mLet and let over dot represent d/dt.1 22  2T qT qKinetic energy: , often written as . 2k 2Potential energy: 2V qm k Equation of motion: q q 0m>>> The angular frequency of oscillation:kmFrequency of oscillation of a diatomic molecule (1)2 2Kinetic energy = ½ m (dx /dt) ½ + ½ m (dx /dt) ; 1 1 2 12Potential energy = ½ k (x -x ...