Molecular Spectra and Molecular Structure - Lec 2

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 (1, 2 ) = hˆ(1) + hˆ(2) + H′(1, 2) H° ψ°(1, 2) = φ(1)φ(2) * E° = ε1 + ε2 Matrix elements of H° diagonal, H′ 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. CO2) ( l2 Q , P l2 same for H 2 l1 = 1 P l 12 µ + 1 kQ 2 H 1 2 2 l 1φ ( Q ) = =ω ( v + 1 / 2 ) H v1 1 1 ) ω1 = (k/µ)1/2 1 reduced mass l 2 φ (Q ) = =ω ( v + 1/ 2 ) same for H v2 2 2 2 ( ) ( v1 = 0, 1, … ω2 = ω1 ≡ ω ) l =H l1 Q , P l1 + H l2 Q , P l 2 + aQ Q H 2  1   1 2 H′ H° o define basis set ψv1v2 (Q1,Q 2 ) = φ v1 (Q1 )φ v2 (Q 2 ) complete, orthonormal, and convenient l °ψ o = =ω [ v + 1/ 2 + v + 1/ 2] = E o = =ω ( v + v + 1) H v1v2 v1v2 1 2 1 2 Page 2 Matrix of H° is diagonal H v1 v1′ ;v2 v2′ = hω (v1 + v2 +1)δ v1v1′ δ v2 v2′ We also know all matrix elements of Qi (Q = x in H–O Handout) in H–O basis set 1/ 2 ⎡ ⎤ Qv ,v ±1 = ∫ ψ Qψ v ±1dQ = ⎢ ⎥ ⎣ 4 µω ⎦ * v ( ) 2v + 1 ± 1 2 v> 1/ 2 All of the non-zero matrix elements of Q follow the “selection rule” ∆v = ±1 ′ So it is an easy matter to write down all matrix elements of H = aQ1Q2 in ψov1 v2 basis set. H′v1 v2 ; v1′ v2′ = hω2b [v1> v2 > ] δ v1′ v1 ±1δ v2′ v2 ±1 1/ 2 a b ≡ 4k a and k have same units, so b is unitless [aQ1Q2 + kQ21/2 are both energies]. So we have all formulas needed to write H but we need to think about how to organize the matrix according to the FOUR indices v1 v2 v′1 v′2 . Arrange matrix so that E ov1 v2 along diagonal increases monotonically * we usually look at E levels from bottom up H′ij * perturbation theory - near degeneracies require special treatment E oi − E oj E° = =ω(v1 + v2 + 1) v1v2 E° = =ω(1) 00 (2) 10, 01 (3) (4) … (n) block index ↑ So we are done with H°. dimension of block 1×1 20, 11, 02 30, 21, 12, 03 n0, ……0n 2×2 3×3 4×4 nxn n ← block index ⇑ What do we know abou t H′? ² v1 = ±1, ²v 2 = ±1, ² (v1 + v2) = 0, ±2 * fill in blocks along diagonal ∆n = 0 for H′ * off--diagonal ∆n = 2 elements between blocks (see top of handout, page 1) Page 3 In H′, blocks are connected only by ∆n = ±2. ∴ rigorous factorization of H into even n and odd n blocks (consequence of operator form of H′) There is also another symmetry. Oscillators 1 and 2 are identical. Construct new basis functions that are eigenfunctions (consequence of symmetry) of pertmutation operator, P(1,2) Since ⎡⎣ H, P (1, 2 ) ⎤⎦ = 0 permutation symmetry (+1 or –1) is a rigorous (GOOD) QN v1v2 , ± = 2 −1/2 ⎡⎣ v1 = v v2 = v′ ± v′ v ⎤⎦ also P(1,2) |v〉|v〉 = +|v〉|v〉 0,0 1,1 2,2 3,3 etc. all even + and – symmetry blocks all H+,– = 0 even n, odd n all Heven n,odd n = 0 doubly factored H — see bottom of handout on page 1. FOUR RIGOROUSLY SEPARATE BLOCKS even n, + even n, – odd n, + odd n, – Each of these four blocks is partly block-diagonalized. * off-diagonal elements within sub-blocks * off-diagonal elements between adjacent (∆n = 2) sub-blocks. Look at (odd n, +) block in more detail. v1 v2 n 0 2–1/2(20 1 –1/2 2 (40 2–1/2(31 2 + + + 0 02) 1 1 3 3 04) 13) 2 5 5 5 ω 1 0 2b 0 3 4b 2b 4b 3 0 0 0 0 12b 0 0 0 0 0 12b 0 0 0 4b 5 4b 0 4b 5 4 3b degenerate blocks 0 0 4b 0 4 3b 5 note: even-n blocks are only quasi-degenerate Now we have simplified as much as is possible rigorously. Each of the four blocks is still infinite and can’t be solved exactly. Perturbation Theory is needed to get rid of high-n part of matrix. Perturbation Theory!