04_580ln_fa08

04_580ln_fa08 - 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 #4 Fall, 2008 Page 1 of 9 pages Lecture #4: Atoms: 1e and Alkali 1e– Atoms: H, He + , Li 2+ , etc. coupled and uncoupled basis sets: | j s m j or | ± s ² ± ˆ 2 ± ( ± +1) centrifugal term ³ 2μr 2 ³ V ± (r) effective potential 2μr 2 radial Schrödinger Equation ˆ ± ·s ˆ spin-orbit 3 r n-scaling (also μ and Z) exact, integer n and integer Z inter-relationships notation Self Consistent Field to define 1e orbitals: Alkali atoms (one e outside closed shells) extension of scaling semi-empirical, non-integer n* and Z eff 2 ( Z eff ) IP E n ± ± 2 seems like we have 2 different kinds of corrections for the ( n µ ± ) same thing. Effective core potential. Quantum defect theory — a scattering based model constant phase shifts along Rydberg series properties that probe inner vs. outer parts of orbital penetrating vs. non-penetrating orbitals Qualitative differences between 1e and alkali-like electronic structures Patterns: Assignment Prediction and Extrapolation Information about complicated part of from “fudge factors” 1-e Atoms “Hydrogenic” H ² ( r, ± , ² ,s ) = H ² = 2 ³ ² 2 Ze 2 + Ze 2 2 1 3 ˆ · s ˆ r 2 c r kinetic potential H ² spin-orbit heavily energy energy weighted at includes integrate over ± , ² 2 ( + 1 ) nucleus · (r, ± , ² )= + ¸ V (r) effective radial potential 2 R n (r)Y m ( ± , ² ) “centrifugal barrier” ² nucleus-electron H central force H reduces to universal angular part Y m ( ¹, ´ ) and atom-specific radial part R n (r) (still universal for 1 e atoms)
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5.80 Lecture #4 Fall, 2008 Page 2 of 9 pages Z is charge on nucleus μ = m N m e is “reduced mass” μ m e because m N m e m N + m e m N = 1±amu μ = 5.4828 ± 10 ² 4 ~ 1 part in 10 3 ² seems small but electronic spectra are m N = 200±amu μ = 5.4858 ± 10 ² 4 typically measured to 1 part in 10 6 Basis sets: sets of mutually commuting operators that also commute with H ° * uncoupled | n m sm ± ± ± s j ± + s * coupled | n sjm j complete basis only if we include continuum | ± sjm j ± > 0 · s = z s z + 1/2( + s + s + ) + | m = [ ( + 1) – m (m + 1)] 1/2 | m + 1 H SO not diagonal in uncoupled basis because of 1 ( ± + s ² + ± ² s + ) [ ² = x ² i y ] 2 j 2 = ( + s ) 2 ± ·
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04_580ln_fa08 - MIT OpenCourseWare http:/ocw.mit.edu 5.80...

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