Lecture 7: Electronic Spectra of Diatomics Fortrat parabola, 1 ∆←1∆ (Symmetric Top) 1. Term symbols for diatomic molecules 2. Common molecular models for diatomics 3. Improved treatments 4. Quantitative absorption 1. Term symbols for diatomic molecules Term symbols characterize key features of electron spin and orbital angular momentum For an atom: 2S 1 LJ Important terms 2S 1 For a diatomic: projection of orbital angular momentum onto the internuclear axis. Magnitude: | | L012 Symbols: Λ 012 Atoms Symbol ΣΠ∆ Symbol S P D total electronic spin angular momentum (the sum of electron spin in unfilled shells) S Magnitude:| S | S , S will have 1/2-integer values projection of S onto the internuclear axis (only defined when Λ≠0). Magnitude: | | Allowed values: S, S 1,...,S 2S 1 values sum of projections along the internuclear axis of electron spin and orbital angular momentum S, S 1,..., S 2S 1 values for S 2 1. Term symbols for diatomic molecules Examples 2S 1 NO The ground state for NO is X2Π For a diatomic: S = 1/2, Λ = 1, Ω = 3/2, 1/2 2 2 There are two spin-split sub-states: Π1/2, Π3/2 Separation: 121cm-1 CO The ground state for CO is X1Σ+ S = 0 and Λ = 0, therefore Ω is unnecessary. This is a rigid rotor molecule. Easiest case! 3 – O2 The ground state for O2 is X Σg S = 1, Λ = 0 – The and g are notations about symmetry properties of wave functions. This is an example of a molecule that is modeled by Hund’s case b 3 2. Common molecular models for diatomics Four common molecular models Rigid Rotor Λ = 0, S = 0 2S+1 = 1 “singlets” no influence of electron spin Symmetric Top Λ≠ 0, S = 0 on spectra Hund’s a Λ≠ 0, S ≠ 0 Spin important through Hund’s b Λ = 0, S ≠ 0 interaction of Λ and Σ . This lecture: Rigid Rotor Symmetric Top Followed by: Hund’s a Hund’s b 4 2. Common molecular models for diatomics 1 Rigid rotor ( Σ) Axis of B IA ≈ 0 rotation C IB = IC J m1 m2 A-axis Center of mass . Λ = 0, S = 0 1Σ type, Ω is not defined . Λ = 0 means the projection of the orbital angular momentum onto the A-axis is zero, and rotation must thus be around the B-axis 5 2. Common molecular models for diatomics 1 Rigid rotor ( Σ) 2 2 Rotational Energy FJ Bv J J 1 Dv J J 1 Total Energy E Te G F ETe , v, J Te G v F J Selection Rules Rotational spectra: J J 'J" 1 Rovibrational spectra: v v'v" 1 J 1 Rovibronic spectra: v determined by Frank- Condon factors J 1 Note: an alternate form is sometimes used final initial J or v 6 2. Common molecular models for diatomics 1 Rigid rotor ( Σ) Intensity Distribution Within each band (v', v"), the intensity distribution follows the Boltzmann distribution for J modified by a J-dependent branching ratio (i.e., for the P and R branch), known as the Hönl-London factor. The relative intensities among all the vibrational bands originating from a single initial level vinitial to all possible final levels vfinal are given by Franck-Condon factors. The relative total emission or absorption from vinitial depends directly on the Boltzmann fraction in that level, i.e., nv initial/n Examples Most stable diatomics: CO, Cl2, Br2, N2, H2 are rigid rotors 2 3 Exceptions: NO (X Π), O2(X Σ) Note: no X∆ states for diatomics – all X states are Σ or Π! ~X1 Σ ~X1 Σ g Some linear polyatomics: CO2( ), HCN and N2O ( ) are rigid rotors with 1Σ ground states. Nuclear spin will have an impact on the statistics of homonuclear diatomic molecules 7 2. Common molecular models for diatomics Symmetric top N J IA ≠ 0 IA ≠ IB = IC . Λ≠ 0, S = 0 (non-zero projection of orbital angular momentum on the internuclear axis and zero spin) ground states 1Π, 1∆ . Important components N angular momentum of nuclei A-axis projection of electron orbital angular momentum J total angular momentum; J N Only the axial component of orbital angular momentum is used, because only is a “good” quantum number, i.e., a constant of the motion 8 2. Common molecular models for diatomics Symmetric top (Λ≠0, S = 0) Rotational Energy FJ BJ J 1 A B 2 , J , 1,... h A, B Same spacing as the rigid 2 rotor, but with a constant 8 cI A,B offset Since IA < IB, A > B, lines with J < Λ are missing, as J = Λ, Λ+1, … Selection Rules 0,J 1,0 J 0 is weak 1,J 1,0 As a result of having a Q branch (i.e., ∆J = 0), the bands for a symmetric top will be double- headed, in contrast to the single-headed character of rigid rotor bands 9 2. Common molecular models for diatomics Symmetric top (Λ≠0, S = 0) Spectra for ∆Λ = 0 (1Π←1Π or 1∆←1∆) T ' B' J ' J '1 A'B' 2 G v' T ' e = 0 for ground state 2 T" B"J"J"1 A"B" G v" Te" upperfor J ' 0 lower for J" 0 constant 2 PJ" B'B"J B'B"J 2 QJ" B'B" J B'B" J 2 RJ" B'B" J 1 B'B" J 1 mP J 2 P and R branches: am bm m J Q Q branch: bm bm2 m J 1 R where a B'B",b B'B" 10 2. Common molecular models for diatomics Symmetric top (Λ≠0, S = 0) Spectra for ∆Λ = 0 Fortrat parabola, 1∆←1∆ 2 P and R branches: am bm 2 Q branch: bm bm where a B'B",b B'B" Notes: . Band heads in the Q and R branches for the typical case of B'<B". mP = -J, mQ = +J, mR = J+1 1 1 . Jmin = 2 for ∆← ∆ mmin=3 for R branch mmin=2 for Q branch |mmin|=3 for P branch missing lines near the origin Intensity Distribution Relative intensities depend on nJ/n, and P,Q,R Hönl-London factors (S J ) - “relative intensity factors / line strengths” breakdown of the principle of equal probability 11 2. Common molecular models for diatomics Example – 1: Hönl-London Factors for Symmetric Top (see Herzberg) J 1 J 1 For ∆Λ = 0 S R J 1 J J J 1 2 2 Q 2J 1 2 SJ 0 for large J SJ 2J 1 J J 1 J P J J SJ J for large J Notes: J 1. ΣSJ = 2J+1, the total degeneracy! 2. The R-branch line for a specific J, is ~ J+1/J times as strong as the P-branch line 3. For ∆Λ=±1, J>>Λ 2J 1 S R Q branch lines are twice as J 4 strong as P and R lines! Q 2J 1 SJ SJ 2J 1 ∆Λ value is important in 2 determining the relative line P 2J 1 and branch strengths of SJ 4 rovibronic spectra. 12 2. Common molecular models for diatomics Example – 2: Symmetric Top Ground State If X = 1Π, possible transitions (Recall ∆Λ = 0, 1) 1Π←1Π 1∆←1Π 1Σ←1Π ∆Λ = 0 ∆Λ = 1 ∆Λ = –1 1. Three separate “systems” of bands possible from X1Π 2. Hönl-London factors for ∆Λ = 1 differ from for ∆Λ = 0 (see previous page) 13 3. Electronic Spectra of Diatomic Molecules: Improved Treatments (add Spin) Hund’s case a 1. Review of angular momentum 2. Interaction of Λ and Σ 3. Hund’s case a (Λ≠0, S ≠ 0) 4. Hund’s case b (Λ = 0, S ≠ 0) Hund’s case b 5. Λ-doubling 14 3.1. Review of angular momentum Review – then add spin Term symbol “Multiplicity” of state 2S 1 Term Symbol Sum of projections on A axis Λ 012 when Λ≠0 Projection of electron Symbol ΣΠ∆ orbital angular momentum on A axis S, S 1,..., S 4 models 1 Rigid Rotor Λ = S = 0 e.g., N2, H2: X Σ 1 Symmetric Top Λ≠ 0; S = 0 e.g., Π 2 Hund’s a Λ≠ 0; S ≠ 0 e.g., OH, NO (both X Π) Add spin 3 Hund’s b Λ = 0; S ≠ 0 e.g., O2: X Σ 15 3.1. Review of angular momentum Electronic angular momentum for molecules Orbital angular momentum of electrons 1. Separate from spin and nuclear rotation 2. Strong electrostatic field exists between nuclei. So L precesses about field direction (internuclear axis) with “allowed” components along axis ml = L, L-1, …, -L ≡ Λ L E 3. If we reverse direction of electron orbit in E field, we get the same energy but Λ→ –Λ (Λ doubling) 16 3.1. Review of angular momentum Electronic angular momentum for molecules Spin of electrons 1. To determine L and S for molecule, we usually sum l & s for all electrons. e.g., S si i So even number of electrons integral spin odd number of electrons half-integral spin 2. For Λ≠ 0, precession of L about internuclear axis magnetic field along axis. So ms is defined. ms ≡ Σ = S, S-1, … -S. Note for change of orbital direction, energy of electron spinning in magnetic field changes no degeneracy 2S+1 possibilities (multiplets) 3. For Λ = 0, no magnetic field exists and the projection of S on the nuclear axis is not conserved (Σ not defined) 17 3.1. Review of angular momentum Electronic angular momentum for molecules Total electronic angular momentum 1.