05_lecnotes_rwf

05_lecnotes_rwf - MIT Department of Chemistry 5.74, Spring...

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MIT Department of Chemistry 5.74, Spring 2004: Introductory Quantum Mechanics II Instructor: Prof. Robert Field 5.74 RWF Lecture #5 5 – 1 When we create Ψ ( Q , t = 0) that is not an eigenstate of the isolated molecule, time-independent H , that | Ψ ( Q , t )| 2 will evolve. Usually a nonstationary “coherent superposition of eigenstates” state is produced by a sudden perturbation, such as a short pulse of electromagnetic radiation. We have looked at the evolution of a large number of “wavepackets” in 5.73 and 5.74. But there is a lack of a simple picture of how Ψ ( Q ,0) is produced, what are the simple forces that cause it to evolve, and how all of this is related to the frequency domain spectrum. We need to understand the nature of “the pluck”. Heller’s formulation of the relationship between the absorption spectrum, I ( ω ), and the Fourier transform of the ⟨Ψ ( t )* Ψ (0) autocorrelation function provides a unified conceptual and computational framework. It was revolutionary! ⟨Ψ Ψ Electronic Absorption Spectrum FT of ( t )| (0) traditional frequency domain formulation restated as Fourier Transform of the autocorrelation function wavepacket evolution as responsible for features in autocorrelation function what are the features of the wavepacket as determined by the upper and lower electronic potential surfaces local view of V( Q ) from wavepacket/autocorrelation picture vs. global view from Franck- Condon picture. see Heller JCP 68 , 3891 (1978) I k () c k j δ ω − ω jk ) = c k j δ 1 ( E ( E j E k )) ω j h initial transition transition± 1 ω jk = ( E j E k ) state intensity frequency± h notation e single prime, g double prime 2 I v ′′ eg ω q ′′ ω − ω vv vv ) g eg ( v e Franck-Condon factor 2 q = v v e vibrational overlap squared g
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5.74 RWF Lecture #5 5 – 2 It is necessary to know complete V and V g up to at least energy of v and v g ′′ and to integrate over all e e 3N–6 normal coordinate displacements. Usually do not have this information except for diatomic molecules. Use displaced harmonic oscillators and qualitative F-C factors derived from diatomic molecules.
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05_lecnotes_rwf - MIT Department of Chemistry 5.74, Spring...

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