08_580ln_576

08_580ln_576 - 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 #8 Fall, 2008 Page 1 Lecture #8: The Born-Oppenheimer Approximation For atoms we use SCF to define 1e orbitals. Get V eff (r) for each e in field of e ’s in all other occupied orbitals. ψ (r) = φ 1 (r 1 ) φ N (r N ) single antisymmetrized product function. This is a way of defining our zero-order complete basis set. It is a bad approximation and accurate ab initio electronic wavefunctions are CI — linear combination of many configurations (product functions). For molecules, we separate Ψ r;R, θ , φ ) into a product of electronic, vibrational, and rotational ( functions Φ i (r;R) χ iv J | JM . This is the Born-Oppenheimer approximation. It is based on a good approximation (e move much faster than nuclei) and most molecular eigenstates can be well described by single electronic*vibrational*rotational product. BUT WHAT DO WE HAVE TO SLIP UNDER THE RUG? How to separate H ( r,R, θ , φ ) ? some subtle stuff — return to this for polyatomic molecules 1. CLAMPED NUCLEI T N 0 get electronic Φ i ( r ; R) and nuclear V i (R) by neglecting and Φ i 2 Φ j χ (R) . Φ i Φ j 2. For the i-th electronic state, H ROT VIB ( R, θ , φ ) separated into H ROT ( θ , φ ) + H VIB define | JM basis set neglect part of H ROT Define V iJ (R) = V i (R) + B i (R)[J(J + 1) – 2 ] effective potential Define χ ivJ (R) vibrational basis set. 3. EXACT ψ — use BO ψ ° to go beyond BO approximation, then put the neglected terms back into H spectroscopic perturbations adiabatic vs. diabatic limits (neglect of either 2 or electrostatic terms) Potential Energy Surfaces are the central organizing concept of molecular spectroscopy. Recipe: 1. write exact H 2. neglect inconvenient terms 3. solve the simplified equation to define a complete basis set 4. put the neglected terms back in.
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5.80 Lecture #8 Fall, 2008 Page 2 H = T e + T N + V eN + V NN + V ee Defined with respect to center of mass. See [Bunker, J. Mol. Spect. 28 , 422 (1968)] for T e = p i 2 = 2 i 2 neglected e induced center of mass wobble. 2m e e i i internuclear distance T N = p ˆ 2 A + p ˆ B 2 T N ( R, θ , φ ) = T N (R) + H ROT ( R; orientation of with respect to lab XYZ θ , φ ) A B N 2 ⎡ ∂ ⎛ ∂ ⎞ ⎤ radial only KE T (R) = 2µR 2 R R 2 R H ROT ( R, θ , φ ) = 2 ( 2 R 2 ) 2  rotational constant hcB(R) nuclear angular momentum R J L S µ = m A m B m A + m B Z A e 2 Z B e 2 = V NN = + Z A Z B e 2 R V ee = + V eN + r i R A  r i R B 
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08_580ln_576 - MIT OpenCourseWare http:/ocw.mit.edu 5.80...

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