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02_580ln_08

# 02_580ln_08 - MIT OpenCourseWare http/ocw.mit.edu 5.80...

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MIT OpenCourseWare http://ocw.mit.edu 5.80 Small-Molecule Spectroscopy and Dynamics Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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5.80 Lecture #2 Fall, 2008 Page 1 Lecture #2: Coupled Harmonic Oscillators: Truncation of an Infinite Matrix For next time, start Bernath, Chapter 5. 1. Approximate separation into subsystems * H ) = h(1) ˆ + h(2) ˆ + (1,2) ( 1,2  H ψ ° (1,2) = φ (1) φ (2) E ° = ε 1 + ε 2 * Matrix elements of H ° H diagonal, non-diagonal 2. Harmonic Oscillator Matrix Elements Simple formulas 3. Organize infinite H matrix in order of E° along diagonal 4. Factor (“BLOCK DIAGONALIZE”) H according to * selection rules for H * permutation symmetry 5. Perturbation Theory * non-degenerate * degenerate * quasi-degenerate (Van Vleck transformation) 6. Correct each block of H for effects of out-of-block terms 7. Secular determinant for each quasi-degenerate block of H . Energy level diagram and fitting formulas. 2 coupled identical harmonic oscillators (like bending vibration of a linear molecule, e.g. CO 2 ) H 1 φ v 1 ( Q 1 ) = ω 1 ( v 1 + 1/ 2 )
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02_580ln_08 - MIT OpenCourseWare http/ocw.mit.edu 5.80...

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