Molecular Spectra and Molecular Structure - 1985 PS3

Problem Set #3 1 Hund’s Coupling Cases. (a) Write the case (a) e and f –symmetry 3 × 3 effective Hamiltonian matrices for the 2 Π, 2 Σ+ problem we have considered in Lecture. Include only the zeroth and first order matrix elements of HROT and HSO . Show that the effective rotational constants for 2 Π3/2 and 2 Π1/2 are B ± B2 /A near the case (a) limit. (b) Consider the case (b) limit where A � BJ. Form the approximate case (b) eigenfunctions for 2 Π as �� � � �� ψ± = 2−1/2 ��2 Π1/2 ± ��2 Π3/2 and re-express the full 3 × 3 matrix in this basis. (i) You will find that both of the 2 Π eigenstates follow a BN(N + 1) rotational energy level expression. Which group of states E+ , ψ+ or E− , ψ− corresponds to N = J + 1/2 and which to N = J − 1/2? (ii) What is the ΔN selection rule for spin-orbit 2 Π ∼2 Σ+ perturbations? (iii) What is the ΔN selection rule for BJ·L 2 Π ∼2 Σ+ perturbations? (c) Consider the case (c) limit for a “p–complex”. This means that the 2 Π and 2 Σ+ states correspond to � � the λ = 1 and λ = 0 projections of an isolated � = 1 atomic orbital. In this case 2 Π| BL+ |2 Σ+ = � � B[1 · 2 − 0 · 1]1/2 = 21/2 B. BΠ = BΣ = B, 2 Π|AL+ |2 Σ+ = 21/2 A, AΠ = A. Write the case (c) matrix and find the eigenvalues for EΠ = EΣ = E. What is the pattern forming rotational quantum number when A � BJ? For each J–value you should find two near degenerate pairs of e, f levels above one e, f pair. What is the splitting of these two groups of molecular levels? How does this compare to the level pattern (degeneracies and splitting) for a 2 P atomic state? (d) Consider the case (d) limit for a “p–complex”. Use the same definitions of EΠ , EΣ , BΠ , BΣ , AΠ , α, β as for case (c) but set A = 0. Your transformed case (b) matrix will be helpful here. Show that R is the pattern forming quantum number by finding the relation between R and J for each of the six same-J, e/ f eigenvalues. page 2 2. Effective Hamiltonian Matrices. (a) Set up the Hamiltonian, H = HROT + HSPIN−ORBIT for the 9 basis functions: 3Π 3 Σ+ Λ 1 −1 1 −1 1 −1 0 0 0 S 1 1 1 1 1 1 1 1 1 Σ 1 −1 0 0 −1 1 1 0 −1 Ω 2 −2 1 −1 +0 −0 1 0 −1 � �� �3 Π2 |vΠ � � �� �3 Π−2 |vΠ � � �� �3 Π1 |vΠ � � �� �3 Π−1 |vΠ � � �� �3 Π0 |vΠ � � �� �3 Π−0 |vΠ � � �� �3 Σ+1 |vΣ � � �� �3 Σ+0 |vΣ � � �� �3 Σ+−1 |vΣ � . Let α ≡ �Λ = 1�AL+ �Λ = 0� β ≡ �Λ = 1�L+ �Λ = 0� and use �1|H|2� = (−1)2J+S 1 +S 2 +σ1 +σ2 �−1|H| − 2� where σ = 1 for Σ− states and 0 for all other states, to ensure phase consistency for �Λ = 1|H|Λ = 0� and �Λ = −1|H|Λ = 0� matrix elements. (b) Construct the e/ f parity basis using the following phase definitions σv |v, nΛσ S Σ, ΩJM� = (−1) J−2Σ+S +σ |v, n − Λσ S − Σ, −ΩJM� e-levels σv ψ = +(−1) J ψ f-levels σv ψ = −(−1) J ψ. (c) Factor the 9 × 9 Hamiltonian into a 5 × 5 and a 4 × 4 matrix using the e/ f basis functions. � � � � � � (d) Obtain the centrifugal distortion correction terms for the 3 Π0 |H|3 Π0 , 3 Π1 |H|3 Π1 , and 3 Π1 |H|3 Π0 e matrix elements. DΠ ≡ − � � � 2 � vΠ |B|vΠ vΠ� 0 − E0 EΠ Π� v v . e e page 3 (e) Obtain the correction terms for the effect of remote 3 Σ+ levels on 3 Π for the � � 3 Π |H|3 Π matrix elements. 1 1 � � 3 Π |H|3 Π 0 0 e and f and e and f � o≡ � v� Σ � �� 1 � 2 α v | v Π 2 Σ 0 0 EΠv − EΣv � � p≡4 � 1 2 αβ � �� �� vΠ |v�Σ vΠ |B|v�Σ 0 − E0 EΠ v Σv� � � ��2 � β vΠ |B|v�Σ v� Σ q≡2 v� Σ 0 − E0 EΠ v Σv� Express diagonal contributions to the Λ–doubling of the Ω = 0 and 1 3 Π substates in terms of o, p, and q.