Molecular Spectra and Molecular Structure - Lec 0

LECTURE #0: GENERAL INFORMATION Read (before next Lecture) 1/3 2/3 Texts: Bernath Chapters 2.1-2, 3.1-2, 4.1. 6 Problem Sets (with some computer problems) 2 Take home exams Bernath Hougen Handouts HLB-RWF WDC Collaboration OK No Collaboration spectra of Atoms and Molecules NBS 115 very long to be handed out Perturbations in the Spectra of Diatomic Molecules polyatomic vibrations Assumed background: Standard QM Models < e– rigid rotor rotation BJ(J + 1) B ∝ R–2 harmonic oscillator vibration ω(v + 1/2) ω ∝ k1/2 H atom electronic –ℜZ2/n2 Zeff, n* = n–µ MO diagram bond order HOMO, LUMO H+2 LCAO-MO’s Matrix Notation – This is major obstacle to those without 5.73. Schrödinger (ψ) ↔ Heisenberg (matrix) pictures ! Read Bernath 2.1-2, 3.1-2, 4.1 H or Handouts or Merzbacher, “Q.M.” pages 294-324 (2nd edition) nondegenerate Perturbation theory (time independent) [quasidegenerate (Van Vleck)] degenerate Bernath 4.1 ← more handouts General Information Page 2 of 5 pages OUTLINE 1. ATOMS “Electronic Structure” orbitals→configurations→L–S–J terms parametrized in terms of 1e– (εn, ζn) and 2e– (Fk, Gk) parameters periodicity recovered from Heff model in terms of ε, ζ, Fk, Gk 2. DIATOMIC MOLECULES examples of all tricks in molecular structure theory Born-Oppenheimer → potential energy curve V(R) rotation vibration n # R " Re & V (R ) = ! a n % ( $ Re ' Nonrigid, anharmonic, vibrating rotor E vJ = ! Y!m (v +1 / 2 ) [ J ( J +1)] ! m perturbation theory {an} ← {Ym} RKR[WKB] V(R) ← EvJ non-1∑ states (Multiplets, Hund’s cases) 3. Heff MODELS FOR DIATOMICS FIT MODELS ! by Van Vleck Transformation derived from infinite H 1st order effects: “perturbations” (pattern destroyed) 2nd order effects: centrifugal distortion, Λ-doubling ! eff to E by adjusting model parameters. fitting of eigenvalues of H vJ 4. TRANSITION INTENSITIES FOR DIATOMICS Limiting Hund’s cases → Hönl-London linestrength factors !eigenvectors of H eff General cases # "interference effects 5. ELECTRONIC STRUCTURE MODELS: Approximate factorizations of Electronic Schrödinger Equation Quantum Chemists can calculate anything from 1st principles. We want more than a number. We want to know a story, to know why, to know how to transfer insight from one molecule to another or one property to another. + H2 LCAO-MO !Long Range!theory!(atoms-in-molecule) # "Ligand field theory (atomic-ions-in-molecule) Molecular Ion in Molecule Rydberg States Quantum Defect Theory Qualitative MO Theory - Shapes of polyatomic molecules (Walsh) Atom in Molecule General Information 6. Page 3 of 5 pages POLYATOMIC MOLECULES Rotation Symmetric and Asymmetric Tops Vibration Normal Modes Wilson's F & G Matrix Methods Point & CNPI Group Theory Perturbations Anharmonic Local↔Normal Rotational Polyads IVR A, B, C, rotational constants and quantum numbers J K a ,K C (Fermi) (Coriolis) Electronic Transitions Case Studies Vibronic Coupling 7. SPECIAL TOPICS AND TECHNIQUES — SUGGESTIONS? quasi-linear Renner-Teller and Jahn Teller Quantum Beats FM-MRS Degenerate Four Wave Mixing Structure Spectrum balls and springs motion line spectrum transitions between eigenstates eigenstates are stationary, no motion How do we get structure and dynamics from eigenstates? 2 approaches — working toward each other. ! =H !° + H !! Q.M. Models — Simplify exact H * what kind of symmetry - good QN, especially angular momentum patterns are ! ( a, b ) = ĥ°(a) + ĥ°(b) + ĥ!(a ~ b) likely to appear approximate separation of coordinates H in spectrum? Born-Oppenheimer * good guesses of model rotation—vibration—electronic parameters nearly isolated subunits * internal consistency relationships between observables, states, molecules constraints TRICKS General Information Page 4 of 5 pages CRUDE Pattern Recognition in Spectrum Chaos vs. Assignability? at high internal excitation. QM encodes structure and dynamics into spectrum. Our job is to learn how to crack this code! Standard approach " " ! ( q" ,# q" ; Q ,#Q ) ! H ! ( q,Q) Exact H 1 n 1 n n e— N atoms differential replace operator Schrödinger Representation by infinite matrix Heisenberg Representation Factor (block diagonalize) this matrix. !"H !, A ! #$ = 0 * exactly — good QN blocks still infinite ! =H !° + H !! H * approximately ! ° defines complete set of basis functions H standard models are very useful here now we can evaluate all matrix elements. always able to write exact ψ as linear combination of basis functions. ! ° is diagonal (i.e. perfectly block diagonalized into |x| submatrices) H ! ! by some form of perturbation theory nearby perturbers Include effects of infinite H remote perturbers ! eff H How this is done depends on * region of spectrum (i.e. what is remote?) * accuracy sought * whether a phenomenological fit model or an interpretive microscopic model is needed General Information Page 5 of 5 pages ! Algebraic formulas — familiar # or " # $Eigenvalues of computer diagonalized matrices — more general. this course Properties other than E are calculable from computer supplied eigenfunctions ESPECIALLY TRANSITION PROBABILITIES. ! to the exact spectrum. It is a fairly straightforward matter to go from the exact H Of course this is not what a spectroscopist does. We attempt to go from spectrum to a determination of !. all the structural parameters (e.g. force field) that define H This is seldom possible! This is why spectroscopy is still interesting!!! ! invert-ability. Diatomic molecules illustrate nearly complete spectrum → H Polyatomic molecules illustrate the need for an arsenal of old and new tricks. pert. theory diatomic molecules advanced forms of pattern recognition compact generalizations of diatomic tricks We also ask if there is any insight into new kinds of approximate factorizations of many body systems BEYOND THE MOLECULAR CONSTANTS. Why measure something that can be functional groups oxidation states bond E↔bond length barrier systematics computed ab initio? Why measure V(Q) for more than 1 or 2 molecules?