Molecular Spectra and Molecular Structure - Lec 3

Page 1 Lecture #3: Coupled Harmonic Oscillators (Part II) and Atoms H° = ωnδv1v1′ δv2v2′ n = v1 + v2 + 1 H′ = aQˆ1Qˆ 2 H′ = ω2b [ v1> v2> ] 1/ 2 δv1 ±1v1′ δv2 ±1v2′ b= a 4k ² n = 0,±2 ⎫ ⎬ factorization of H into 4 blocks P(1,2) ⎭ symmetrization odd n odd n even n even n + – + – 2 –1/2 ⎡⎣ ij ± ji ⎤⎦ “Wang Transformation” ⎛ −1 ⎜ ⎜0 −1/2 ⎜ =2 0 ⎜ ⎜0 ⎜ ⎝1 ⎛ ⎜ T =⎜ ⎜ ⎜ ⎝ T†HT ⎞ ⎟ ⎟ ⎟ etc. ⎟ ⎠ 0 0 1⎞ ⎛−1 0 0 1 ⎞ ⎟ 0⎟ ⎜ ⎟ −1 1 0 ⎟ –1/2 ⎜ 0 1/2 ⎟ 0 or 2 0 2 ⎜ 0 1 1 0⎟ ⎟ 1 0 0⎟ ⎜ ⎟ ⎝ 1 0 0 1⎠ ⎟ 0 0 0 1⎠ odd-n even-n 0 0 −1 0 0 1 0 1 Now we have maximally simplified H, but each block is still ∞. Often each block of H is a very large and complicated matrix. Even super computers will gag on these without symmetry and tricks. Perturbation Theory to cut each block down to size. Nondegenerate 2 ⎡ H′ij o ⎢E i = E j + i H′ i + ∑ o o ⎢ j≠i E i − E j ⎢ H′ ⎢ i = i °+ ∑ j ° E o −ijE o ⎢ i j ⎣ j≠i etc. This allows you to get rid of remote states by correcting for effect of |j〉 on Hii [corrections to diagonal elements]. [Also allows calculation of effect of neglected terms on expectation value of other operators than H .] Page 2 Degenerate Perturbation Theory Solve secular equation (diagonalize) for near-degenerate blocks after correcting for effects of remote states. e.g. d/2−E V =0 V −d / 2 − E –d2/4 + E2 – V2 = 0 E± = ±[V2 + d2/4]1/2 Quasi-degenerate perturbation theory plus Van Vleck transformation (see Van Vleck handout for derivation). H′′block n, block n ′ <1 E oblock n − E oblock n ′ * Van Vleck correction to elements in energy region to be diagonalized — generalization of 2nd order Perturbation Theory. * diagonalize entire region of H where H V-V nn ′ = H nn ′ + ∑ m≠n, n ′ out of block states H′nm H′mn ′ o o E n + E n′ − E om 2 corrections both on and off diagonal On second page of handout I worked through the energy levels through n = 4. Here I will deal only with the odd-n, + symmetry group (but must include effects of n = 5 on n = 3). * Correct each quasi-degenerate block for the effect of the two adjacent ones. * diagonalize each corrected sub-block. n=1 n=3 ⎡ (2b)2 ⎤ = hω [1 − 2b 2 ] . ω 1 becomes hω ⎢1+ ⎣ 1 − 3 ⎥⎦ ⎛ 12b 2 ⎞ 4b 3− ⎜ ⎟ 3 4b 2 ⎜ ⎟ ω becomes hω 2 2 4b 3 ⎜ 16b 4b ⎟ 3− + ⎜ 4b ⎟ ⎝ 2 2 ⎠ ⎛ 3 − 6b 2 4b ⎞ E + = 3 + 4b − 6b 2 ω⎜ ⎟⇒ 3 − 6b 2 ⎠ E − = 3 − 4b − 6b 2 ⎝ 4b Page 3 You should work out the corrections to the n = 5 block and try to diagonalize the corrected 3 × 3 secular 1/2 determinant. You should also think about a different H′ = aQ12Q22 or a ⎡⎣Q12 + Q 22 ⎤⎦ . v1 v2 0 0 0 0 1 0 0 0 2b 0 1 0 0 0 0 0 0 0 1 0 2 2b 2b 2 0 0 0 0 0 2 0 0 0 0 3 8b 0 0 0 0 0 1 1 0 2 2b 0 0 0 0 0 8b 3 8b 0 8b 3 0 0 0 0 0 0 0 0 3 0 0 2 1 1 2 0 3 0 0 0 0 0 + 0 0 20 + 02 1 11 – + 20 − 02 n n 10 + 01 4 12b 0 0 8b 0 0 0 0 0 0 0 0 0 0 12b 4 4b 0 0 4b 4 12b 0 0 12b 4 8b 0 0 2b 0 0 3 0 0 0 0 0 0 0 0 3 2b 3 4b 4b 3 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 4 0 0 0 0 0 4 0 0 0 0 8b 2 0 0 0 30 − 03 4 0 0 0 0 21 −12 4 0 0 0 0 2 + 2b 30 + 03 10 – 01 0 0 1 ω 2 21+12 8b 0 0 0 0 0 0 n 0 0 0 v1 v2 8b 0 0 0 0 0 0 8b 4 12b 8b 4 + 4b 0 0 0 0 0 0 0 0 2 − 2b 0 0 0 0 4 0 0 0 8b 12b 4 − 4b 0 8b – 12b Page 4 Level Diagram: Coupled (H′ = 4kbQ1Q2) Identical Harmonic Oscillators n ≡ v1 + v2 +1 E°/ ω MAXIMAL FACTORIZATION 4 + 6b 4 4 Symmetry Labels ± odd/even +o 4 + 2b +o 4 – 2b –o 4 – 6b –o SECOND-ORDER PERTURBATION CORRECTION 4 + 6b – 8b2 4 + 2b – 8b2 4 – 2b – 8b2 4 – 6b – 8b2 3 2 1 3 2 1 3 + 4b 3 3 – 4b +e –e +e 2 + 2b +o 2 – 2b –o 1 +e 3 + 4b – 6b2 3 – 6b2 3 – 4b – 6b2 2 + 2b – 4b2 2 – 2b – 4b2 1 – 2b2 0 General Formula: ⎛ 2n − 3 ⎞ ⎜ ⎟ 2n − 6 ⎟ (n ) + ⎜ b − 2 (n ) b 2 ⎜ M ⎟ ⎜ ⎟ ⎝ −2n + 2 ⎠ Page 5 Atoms — ideas to represent “electric structure” beyond Bernath Tinkham Weissbluth Group Theory and Quantum Mechanics Atoms and Molecules pages 154-188 pages 413-454 My goal is a survey of the key ideas — whet your appetite. 3-Lecture Outline 1 1e– atoms |n m sms〉 |n sjm〉 all properties of all states → f(n, , j; Z, µ) SCALING 2. alkali 1e– outside closed shell Z → Zeff(r) → Z eff nl n → n* = n – δ ← quantum defect δ π is a phase shift 1 : 1 corresponding n orbital ↔ electronic state modified scaling 3. many - e– atoms configurations → L,S terms (several) εn , Fk(n , n′ ′), Gk(n , n′ ′), ζn limiting coupling cases [characteristic patterns of levels] Heff models scaling in ε, F, G, ζ Many of these ideas will be used (and developed) for molecules.