lecture35 - 5.61 Physical Chemistry Lecture#35 1...

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5.61 Physical Chemistry Lecture #3 5 ± 1 VIBRATIONAL SPECTROSCOPY R R 0 A + B separated atoms V(R) Harmonic Approximation As we’ve emphasized many times in this course, within the Born- Oppenheimer approximation, the nuclei move on a potential energy surface (PES) determined by the electrons. For example, the potential felt by the nuclei in a diatomic molecule is shown in cartoon form at right. At low energies, the molecule will sit near the bottom of this potential energy surface. In equilibrium bond length this case, no matter what the detailed structure of the potential is, locally the nuclei will “feel” a nearly harmonic potential. Generally, the motion of the nuclei along the PES is called vibrational motion, and clearly at low energies a good model for the nuclear motion is a Harmonic oscillator. Simple Example: Vibrational Spectroscopy of a Diatomic If we just have a diatomic molecule, there is only one degree of freedom (the bond length), and so it is reasonable to model diatomic vibrations using a 1D harmonic oscillator: 2 2 P ˆ 1 ˆ 2 P ˆ 1 2 ˆ 2 H ˆ = + k R = + m ω R 2 µ 2 o 2 µ 2 o where k o is a force constant that measures how stiff the bond is and can be approximately related to the second derivative to the true (anharmonic) PES near equilibrium: 2 V k o 2 R R 0 Applying Fermi’s Golden Rule, we find that when we irradiate the molecule, the probability of a transition between the i th and f th Harmonic oscialltor states is: 2 V W fi 4 fi 2 δ ( E i E f ω ) + δ ( E i E f + ω )
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2 5.61 Physical Chemistry Lecture #34± where ω is the frequency of the light (not to be confused with the frequency of the oscillator, ω o ). Because the vibrational eigenstates involve spatial degrees of freedom and not spin, we immediately recognize that it is the electric field (and not magnetic) that is important here. Thus, we can write the transition matrix element as: 2 2 2 V = φ f * µ ˆ i E 0 φ i d τ = E 0 i φ f * µ ˆ φ i d τ = E 0 i φ f * eR ˆ φ i d τ 2 fi Now, we define the component of the electric field, E R , that is along the bond axis which gives 2 2 2 V = E R φ f * eR ˆ φ i d τ 2 = e E φ f * R ˆ φ i d τ 2 fi R So the rate of transitions is proportional to the square of the strength of the electric field (first two terms) as well as the square of the transition dipole matrix element (third term). Now, because of what we know about the Harmonic oscillator eigenfunctions, we can simplify this. First, we re- write the position operator, R , in terms of raising and lowering operators: ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 * * 2 2 2 , 1 , 1 ˆ ˆ ˆ ˆ 2 2 1 2 fi R f i R f i fi R f i f i e V e E a a d E a a d e V E i i φ φ τ φ φ τ µω µω δ δ µω + + + = + = + = + + 1 1 i i φ + + 1 i i φ where above it should be clarified that in this expression i " never refers to √-1 it always refers to the initial quantum number of the system. Thus, we immediately see that a transition will be forbidden unless the initial and final states differ by one quantum
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