Molecular Spectroscopy 3 Christian Hill Joint ICTP-IAEA School on - PowerPoint PPT Presentation

Molecular Spectroscopy 3 Christian Hill Joint ICTP-IAEA School on Atomic and Molecular Spectroscopy in Plasmas 6 – 10 May 2019 Trieste, Italy

Electronic spectroscopy

The electronic structure of diatomics ๏ A molecular con fi guration is a speci fi cation of the occupied molecular orbitals in a molecule

The electronic structure of diatomics ๏ A molecular con fi guration is a speci fi cation of the occupied molecular orbitals in a molecule

The electronic structure of diatomics ๏ A con fi guration may have one or more states, labelled as molecular term symbols : 2 S +1 | Λ | (+/ − ) ( g / u )

The electronic structure of diatomics ๏ A con fi guration may have one or more states, labelled as molecular term symbols : Total electronic spin angular momentum: S = ∑ s i i 2 S +1 | Λ | (+/ − ) ( g / u )

The electronic structure of diatomics ๏ A con fi guration may have one or more states, labelled as molecular term symbols : 2 S +1 | Λ | (+/ − ) ( g / u ) Total electronic orbital angular momentum about internuclear axis: ∑ | Λ | = = 0,1,2, ⋯ = Σ , Π , Δ , ⋯ λ i i

The electronic structure of diatomics ๏ A con fi guration may have one or more states, labelled as molecular term symbols : Re fl ection symmetry of electronic wavefunction (for Σ states) 2 S +1 | Λ | (+/ − ) ( g / u )

The electronic structure of diatomics ๏ A con fi guration may have one or more states, labelled as molecular term symbols : 2 S +1 | Λ | (+/ − ) ( g / u ) Inversion symmetry of electronic wavefunction (for homonuclear diatomics)

The electronic structure of diatomics ๏ Example 1: a closed-shell con fi guration

The electronic structure of diatomics ๏ Example 1: a closed-shell con fi guration ๏ Easiest case: all electrons paired o ff in their orbitals ๏ No net spin or orbital angular momentum: S = Λ = 0 ๏ Electronic wavefunction is totally symmetric: 1 Σ + g

The electronic structure of diatomics ๏ Example 2: one unpaired σ -electron

The electronic structure of diatomics ๏ Example 2: one unpaired σ -electron ๏ Only contribution is from the partially- fi lled orbital ๏ Λ = 0 and S = ½ , so 2S+1 = 2 (a doublet state): 2 Σ + g

The electronic structure of diatomics ๏ Example 3: one or three unpaired π -electrons ๏ Λ = ±1 and S = ½ , so 2S+1 = 2 (a doublet state): 2 Π u

The electronic structure of diatomics ๏ Example 4: two identical π -electrons

The electronic structure of diatomics ๏ Example 4: two identical π -electrons ๏ Label the valence orbitals π - and π + . Consider some possible spatial wavefunctions: ψ (a 1 ) } spatial = π + (1) π + (2) Λ = 2 ⇒ Δ ψ (a 2 ) spatial = π − (1) π − (2)

The electronic structure of diatomics ๏ Example 4: two identical π -electrons ๏ Label the valence orbitals π - and π + . Consider some possible spatial wavefunctions: ψ (a 1 ) } spatial = π + (1) π + (2) Λ = 2 ⇒ Δ ψ (a 2 ) spatial = π − (1) π − (2) 1 } ψ (b) [ π + (1) π − (2) + π − (1) π + (2) ] Λ = 0 ⇒ spatial = Σ 2

The electronic structure of diatomics ๏ Example 4: two identical π -electrons ๏ Label the valence orbitals π - and π + . Consider some possible spatial wavefunctions: ψ (a 1 ) } spatial = π + (1) π + (2) Λ = 2 ⇒ Δ ψ (a 2 ) spatial = π − (1) π − (2) 1 } ψ (b) [ π + (1) π − (2) + π − (1) π + (2) ] Λ = 0 ⇒ spatial = Σ 2 1 } ψ (c) [ π + (1) π − (2) − π − (1) π + (2) ] spatial = Λ = 0 ⇒ Σ 2

The electronic structure of diatomics ๏ Example 4: two identical π -electrons ๏ Combine with suitable spin wavefunctions: ψ (a 1 ) spatial = π − (1) π − (2) } spatial = π + (1) π + (2) 1 [ α (1) β (2) − β (1) α (2) ] 1 Δ ψ (a 2 ) 2

The electronic structure of diatomics ๏ Example 4: two identical π -electrons ๏ Combine with suitable spin wavefunctions: ψ (a 1 ) } spatial = π + (1) π + (2) 1 [ α (1) β (2) − β (1) α (2) ] 1 Δ ψ (a 2 ) 2 spatial = π − (1) π − (2) 1 1 ψ (b) [ π + (1) π − (2) + π − (1) π + (2) ] [ α (1) β (2) − β (1) α (2) ] 1 Σ spatial = 2 2

The electronic structure of diatomics ๏ Example 4: two identical π -electrons ๏ Combine with suitable spin wavefunctions: ψ (a 1 ) = π + (1) π + (2) } 1 [ α (1) β (2) − β (1) α (2) ] 1 Δ ψ (a 2 ) = π − (1) π − (2) 2 1 1 ψ (b) = [ π + (1) π − (2) + π − (1) π + (2) ] [ α (1) β (2) − β (1) α (2) ] 1 Σ 2 2 1 1 ψ (c) = [ π + (1) π − (2) − π − (1) π + (2) ] [ α (1) β (2) + β (1) α (2) ] 3 Σ 2 2

̂ ̂ ̂ The electronic structure of diatomics ๏ Example 4: two identical π -electrons ๏ ± -re fl ection symmetry (molecular axis system): σ e i Λℏ ϕ = e − i Λℏ ϕ ⇒ σπ ± ( i ) = π ∓ ( i ) σ Λℏ −Λℏ

̂ ̂ ̂ The electronic structure of diatomics ๏ Example 4: two identical π -electrons ๏ ± -re fl ection symmetry (molecular axis system): σ e i Λℏ ϕ = e − i Λℏ ϕ ⇒ σπ ± ( i ) = π ∓ ( i ) σ Λℏ −Λℏ ๏ e.g. 3 Σ −

The electronic structure of diatomics ๏ Example 4: two identical π -electrons X 3 Σ − a 1 Δ g , b 1 Σ + g , g

The electronic structure of diatomics ๏ Example 4: two identical π -electrons X 3 Σ − a 1 Δ g , b 1 Σ + g , g ๏ NB Hund’s rules predict energy ordering ๏ Labelling: ๏ X = ground state ๏ A , B , C , …= excited states with the same spin multiplicity ๏ a , b , c , …= excited states with di ff erent spin multiplicity

The electronic structure of diatomics ๏ Example 4: two identical π -electrons X 3 Σ − a 1 Δ g , b 1 Σ + g , g ๏ Labelling: ๏ X = ground state ๏ A , B , C , …= excited states with the same spin multiplicity ๏ a , b , c , …= excited states with di ff erent spin multiplicity

The electronic structure of diatomics ๏ Example 4: two identical π -electrons X 3 Σ − a 1 Δ g , b 1 Σ + g , g ๏ Labelling: ๏ X = ground state ๏ A , B , C , …= excited states with the same spin multiplicity ๏ a , b , c , …= excited states with di ff erent spin multiplicity ๏ No ± label for states with | Λ | > 0

The electronic structure of diatomics ๏ Hund’s rules predict energy ordering X 3 Σ − g < a 1 Δ g , b 1 Σ + g ๏ State with highest multiplicity is lowest in energy: ๏ “Fermi hole”: 1 1 ψ (c) = [ π + (1) π − (2) − π − (1) π + (2) ] [ α (1) β (2) + β (1) α (2) ] 3 Σ 2 2

The electronic structure of diatomics ๏ Hund’s rules predict energy ordering X 3 Σ − g < a 1 Δ g < b 1 Σ + g ๏ Then, state with highest electronic orbital angular momentum, | Λ |

The electronic structure of diatomics ๏ Example 4: two identical π -electrons

Electronic transitions for diatomics A X

Electronic transitions for diatomics ๏ Transition probability μ | ψ i ⟩ | 2 I fi ∝ | ⟨ ψ f | ̂ μ | χ i , m ϕ i , n ⟩ | 2 = | ⟨ χ f , m ϕ f , n | ̂ ≈ | ⟨ χ f , m | χ i , m ⟩ | 2 | ⟨ ϕ f , n | ̂ μ | ϕ i , n ⟩ | 2

Electronic transitions for diatomics ๏ Transition probability μ | ψ i ⟩ | 2 I fi ∝ | ⟨ ψ f | ̂ μ | χ i , m ϕ i , n ⟩ | 2 = | ⟨ χ f , m ϕ f , n | ̂

Electronic transitions for diatomics ๏ Franck-Condon Principle μ | ψ i ⟩ | 2 I fi ∝ | ⟨ ψ f | ̂ μ | χ i , m ϕ i , n ⟩ | 2 = | ⟨ χ f , m ϕ f , n | ̂ ≈ | ⟨ χ f , m | χ i , m ⟩ | 2 | ⟨ ϕ f , n | ̂ μ | ϕ i , n ⟩ | 2

Electronic transitions for diatomics ๏ Franck-Condon Principle μ | ψ i ⟩ | 2 I fi ∝ | ⟨ ψ f | ̂ μ | χ i , m ϕ i , n ⟩ | 2 = | ⟨ χ f , m ϕ f , n | ̂ ≈ | ⟨ χ f , m | χ i , m ⟩ | 2 | ⟨ ϕ f , n | ̂ μ | ϕ i , n ⟩ | 2 Electronic selection rules ΔΛ = 0, ± 1 g ↔ u Σ + ↔ Σ + , Σ − ↔ Σ −

Electronic transitions for diatomics ๏ Franck-Condon Principle μ | ψ i ⟩ | 2 I fi ∝ | ⟨ ψ f | ̂ μ | χ i , m ϕ i , n ⟩ | 2 = | ⟨ χ f , m ϕ f , n | ̂ ≈ | ⟨ χ f , m | χ i , m ⟩ | 2 | ⟨ ϕ f , n | ̂ μ | ϕ i , n ⟩ | 2 Franck-Condon Electronic Factor selection rules ΔΛ = 0, ± 1 Δ v = unrestricted g ↔ u Σ + ↔ Σ + , Σ − ↔ Σ −

Electronic transitions for diatomics ๏ Franck-Condon Principle R R

Electronic transitions for diatomics ๏ Aurorae

Electronic transitions for diatomics ๏ Aurorae N 2 : B ( 3 Π g ) − A ( 3 Σ + u ) }

Electronic transitions for diatomics ๏ Aurorae

Electronic transitions for diatomics ๏ Aurorae I em ∝ | ⟨ χ f , v ′ � | χ i , v ′ � ′ � ⟩ | 2

The electronic structure of diatomics ๏ Example 5: C 2

The electronic structure of diatomics ๏ Example 5: C 2 ๏ Nonetheless: Swan bands d ( 3 Π g ) − a ( 3 Π u ) ๏ ab initio calcualtions of hot line lists (e.g. exomol.com)

Nuclear spin statistics ๏ There are two kinds of H 2 molecule.

Nuclear spin statistics ๏ There are two kinds of H 2 molecule. ๏ 1 H has a nuclear spin; quantum number I = 1 2

Nuclear spin statistics ๏ There are two kinds of H 2 molecule. I = 1 ๏ 1 H has a nuclear spin; quantum number 2 ๏ Just as for identical electrons, the nuclear angular momentum couples: I = 1 I = 0